JP2014123272A - Simulation method, program, and information processing method - Google Patents

Abstract

PROBLEM TO BE SOLVED: To perform a simulation by efficiently using a plurality of processors.SOLUTION: A control part 11 selects an index value of reference, selects temporary index values v1 to v3 on the basis of the index value of reference, and distributes tasks 13-1 to 13-3 corresponding to the temporary index values v1 to v3 to processors 12-1 to 12-3. Each processor generates probability data corresponding to the temporary index values as a task corresponding to any allocated temporary index value by using a model for generating a sample of an index value from random numbers and a distribution of biased random numbers. The control part 11 compares probability data respectively corresponding to the temporary index values v1 to v3 with a predetermined probability condition to select one temporary index value, determines the one temporary index value as an index value satisfying the predetermined probability condition, or uses the one temporary index value as a new index value of reference to perform selection of a plurality of temporary index values and distribution of a plurality of tasks again.

Description

  The present invention relates to a simulation method, a program, and an information processing system.

  As one of the simulation methods, a Monte Carlo simulation which is approximately obtained using random numbers instead of analytically obtaining a solution of a problem is often used. In the Monte Carlo simulation, a large number of samples are generated by generating a large number of random numbers according to a distribution of random numbers and substituting these large numbers of random numbers into a model that generates a sample of index values from the random numbers. Thereby, the probability distribution of the index value can be estimated. Various statistical values such as expected value and variance of the index value can be obtained from the probability distribution of the index value. In the Monte Carlo simulation, the calculation accuracy is improved by increasing the number of samples to be generated.

  Monte Carlo simulation is one of the effective numerical analysis techniques for finding solutions to problems that are difficult to solve analytically or complex problems. For example, when it is desired to analyze the yield of a product at a certain manufacturing site, it is conceivable to calculate the occurrence probability of a defective product using Monte Carlo simulation. Monte Carlo simulations can also be used when calculating risk indicators by predicting profits and losses arising from a portfolio of financial assets, or when calculating the present value of complex financial derivatives.

  However, in the Monte Carlo simulation, the calculation accuracy gradually improves as the number of samples increases. Therefore, if the calculation accuracy is simply increased by increasing the number of samples, the calculation amount becomes enormous and the simulation time becomes longer. Therefore, methods for suppressing the amount of calculation of Monte Carlo simulation include, for example, an importance sampling (Importance Sampling) method and a stratified extraction (Stratified Sampling) method.

  In the importance sampling method, instead of generating a sample group with a large variance, a sample group with a small variance is generated around a specific index value of interest by biasing the distribution of random numbers. When the importance sampling method is used, a calculation result with higher accuracy can be obtained in the vicinity of the target index value even if the number of samples is reduced as compared with the normal Monte Carlo simulation. In the stratification extraction method, the index value range is divided into multiple sections so that the probabilities (integration of probability density) of each section are almost equal, and the same number of samples are adopted from each section. Estimate the probability distribution of values. When the stratified extraction method is used, if the range can be appropriately divided, a highly accurate calculation result can be obtained even if the number of samples is reduced as compared with a normal Monte Carlo simulation.

  A risk quantification system has been proposed in which risk factors are divided according to the magnitude of the impact on risk, and the amount of loss is calculated by a different method for each divided risk factor. This risk quantification system calculates a loss amount by a Monte Carlo simulation for a risk factor having a large impact on the risk, and calculates a loss amount by a method having a small calculation amount for a risk factor having a small impact on the risk. Further, the risk quantification system generates a loss distribution based on the calculated plurality of losses.

JP 2002-092311 A

  As described above, the calculation amount of the simulation can be suppressed by using the importance sampling method or the stratified extraction method. On the other hand, it is conceivable that the simulation time is shortened by distributing the tasks to a plurality of processors to parallelize the simulation. Depending on the problem to be solved, the simulation based on the above-mentioned importance sampling method or stratified extraction method may be parallelized to reduce the simulation time. However, the problem is how to distribute tasks to multiple processors to make parallel processing more efficient.

  For example, in the importance sampling method, it is easy to calculate a probability corresponding to a specific index value, but conversely, it is not easy to search for an index value that satisfies a predetermined probability condition. When solving the latter type of problem, the question is how parallel processing can reduce the simulation time. In addition, for example, in the stratified extraction method, the index value range is divided into a plurality of sections. Therefore, as a method of parallel processing, a method of assigning processors for each section is conceivable. However, if a task that is independent of other sections is set for each section, each processor discards the sample when a sample that does not belong to the section that it is responsible for is generated, and a large number of samples are discarded. Has become inefficient.

  In one aspect, an object of the present invention is to provide a simulation method, a program, and an information processing system capable of performing simulation using a plurality of processors efficiently.

  In one aspect, there is provided a simulation method for estimating an index value probability distribution and determining an index value satisfying a predetermined probability condition, which is executed by an information processing system including a plurality of processors. A reference index value is selected, and a plurality of temporary index values are selected based on the reference index value. A plurality of tasks corresponding to a plurality of temporary index values are allocated to a plurality of processors. As a task corresponding to each temporary index value, a model that generates index value samples from random numbers and a distribution of random numbers biased so that the index value samples are generated around the temporary index value are used. Then, processing for generating probability data corresponding to the temporary index value is executed in the processor to which the task is assigned. One temporary index value is selected by comparing the probability data corresponding to each of the plurality of temporary index values with a predetermined probability condition. The selected temporary index value is determined as an index value that satisfies a predetermined probability condition, or a plurality of temporary index values are selected using the selected temporary index value as a new reference index value. And assigning multiple tasks again.

  In one aspect, a simulation method executed by an information processing system including a plurality of processors is provided. The index value range is divided into a plurality of sections, and a lower limit number s (s is an integer of 2 or more) of index value samples used in each of the plurality of sections is determined. P tasks (p is an integer of 2 or more) are allocated to a plurality of processors. As each task, a process of generating index value samples from random numbers is executed in a processor to which the task is assigned until at least s / p index value samples belonging to each of a plurality of sections are reached. A sample of s / p index values is collected from each task for each of a plurality of sections, and a probability distribution of the index values is estimated using the collected sample of index values for each section.

  In one aspect, a program to be executed by a computer is provided. In one aspect, an information processing system having a plurality of processors and a control unit is provided.

  In one aspect, a solution calculated from the distribution of calculated values can be calculated at high speed.

It is the figure which showed the example of the information processing system which concerns on 1st Embodiment. It is the figure which showed the example of the information processing system which concerns on 2nd Embodiment. It is the 1st figure showing the example of the hardware of the information processing system concerning a 3rd embodiment. It is the 2nd figure showing the example of the hardware of the information processing system concerning a 3rd embodiment. It is the 3rd figure showing the example of the hardware of the information processing system concerning a 3rd embodiment. It is the figure which showed the example of the functional block of the information processing system which concerns on 3rd Embodiment. It is a figure for demonstrating the calculation method of VaR. It is a figure for demonstrating the importance sampling method. It is a 1st figure for demonstrating parallelization of the importance sampling method. It is a 2nd figure for demonstrating parallelization of the importance sampling method. It is the figure which showed the flow of the process by the information processing system which concerns on 3rd Embodiment. It is a figure which shows the example of a calculation which compared the generation frequency of the calculated value obtained by performing the simulation calculation of the same precision. It is the figure which showed the example of the functional block of the information processing system which concerns on 4th Embodiment. It is a figure for demonstrating the stratification extraction method. It is a figure for demonstrating the parallelization of the stratification extraction method. It is the figure which showed the flow of distribution calculation by the stratification extraction method.

Hereinafter, embodiments will be described with reference to the drawings.
[First Embodiment]
A first embodiment will be described.

FIG. 1 is a diagram illustrating an example of an information processing system according to the first embodiment.
The information processing system 10 according to the first embodiment estimates a probability distribution of index values and determines an index value that satisfies a predetermined probability condition. The simulation executed by the information processing system 10 is, for example, a Monte Carlo simulation using an importance sampling method.

  The information processing system 10 includes a control unit 11 and a plurality of processors including processors 12-1 to 12-3. Examples of the “plurality of processors” include the following. (1) A plurality of processor cores included in one Central Processing Unit (CPU). (2) A plurality of GPUs for a plurality of CPUs and general-purpose computing on graphics processing units (GPGPU) included in one information processing apparatus. (3) A set of GPUs for CPU and GPGPU provided in a plurality of information processing devices connected to the network. However, the “processor” may be a specific purpose electronic circuit such as a digital signal processor (DSP), an application specific integrated circuit (ASIC), or a field programmable gate array (FPGA). Each processor executes a task according to a program stored in a memory such as a random access memory (RAM), for example.

  The control unit 11 performs a simulation using the processors 12-1 to 12-3. The control unit 11 includes a processor that executes a program stored in a memory such as a RAM, for example. The control unit 11 and the processors 12-1 to 12-3 may belong to the same information processing apparatus or may belong to different information processing apparatuses. When the control unit 11 includes a processor, at least one of the tasks described below may be assigned to the control unit 11.

  The control unit 11 selects a reference index value from the range of index values. The reference index value is selected as follows, for example. First, the control unit 11 estimates a probability distribution of index values using a model that generates a sample of index values from random numbers and a random number distribution that is not biased. This probability distribution may be low in accuracy, and the number of samples to be generated may be sufficiently small (for example, about 1/100 to 1/1000) compared to a normal Monte Carlo simulation that does not use the importance sampling method. . Then, the control unit 11 selects an index value that satisfies a predetermined probability condition from the estimated probability distribution as a reference index value. The predetermined probability condition is, for example, a condition relating to a probability such that cumulative probability = 1%. However, the reference index value may be selected by referring to the past simulation result or the like without estimating the probability distribution of the index value.

  Next, the control unit 11 selects a plurality of temporary index values based on the reference index value. For example, the control unit 11 selects a reference index value and two or more index values that deviate from the reference index value by a predetermined width back and forth as temporary index values. In the example of FIG. 1, the index value v2 is selected as the reference index value, and the index values v1 to v3 are selected as temporary index values. The control unit 11 allocates a plurality of tasks corresponding to a plurality of temporary index values to a plurality of processors. In the example of FIG. 1, the task 13-1 corresponding to the index value v1 is assigned to the processor 12-1, the task 13-2 corresponding to the index value v2 is assigned to the processor 12-2, and corresponds to the index value v3. Task 13-3 is assigned to processor 12-3.

  A processor to which any task is assigned performs the following processing as the task. Here, the task 13-1 will be described as a representative. The processor 12-1 biases the distribution of random numbers so that a sample of index values is generated around the index value v1, which is a temporary index value. Then, the processor 12-1 generates probability data corresponding to the index value v1, which is a temporary index value, using the model for generating index value samples from random numbers and the biased distribution of random numbers. . The probability data indicates, for example, a cumulative probability (probability that the index value is v1 or less) in the index value v1. The number of samples generated here may be a sufficiently small number compared to a normal Monte Carlo simulation that does not use the importance sampling method.

  For other temporary index values, probability data is generated in the same manner as the index value v1. The plurality of tasks can be executed independently of each other, and tasks assigned to different processors can be executed in parallel. For example, tasks 13-1 to 13-3 are executed in parallel.

  When the plurality of tasks allocated to the plurality of processors are completed, the control unit 11 compares the probability data corresponding to each temporary index value with a predetermined probability condition, thereby selecting one of the plurality of temporary index values. Select one. For example, when the probability data and the predetermined probability condition indicate the cumulative probability, the temporary index value that provides the cumulative probability closest to the predetermined cumulative probability is selected. At this time, even if the reference index value is included in the plurality of temporary index values, there is a possibility that a temporary index value that is not the reference index value is selected. This is because probability data can be calculated with higher accuracy than when a reference index value is selected by using a biased distribution of random numbers.

  Then, the control unit 11 determines the selected one temporary index value as an index value that satisfies a predetermined probability condition, or uses the one temporary index value as a new reference index value. The index value is selected and the task is executed again. By repeating the selection of the temporary index value and the execution of the task, it can be expected that the probability data approaches the predetermined probability condition. The repetition may be automatically terminated when a predetermined stop condition is satisfied. The stop condition may be determined based on the total number of generated samples, the simulation time, the number of repetitions, and the like.

  According to the information processing system 10 according to the first embodiment, a plurality of temporary index values are selected, and a task for generating a sample of index values around each temporary index value is set. With such a task, the probability data corresponding to the temporary index value can be accurately calculated with a small number of samples. A plurality of tasks can be executed independently of each other, and the simulation can be efficiently processed in parallel by distributing the plurality of tasks to a plurality of processors. In addition, simulation accuracy can be improved by selecting one index value from a plurality of provisional index values based on probability data calculated for each of a plurality of tasks. In particular, if the above process is repeated, the accuracy can be increased step by step. Note that the total number of samples generated by the simulation method according to the first embodiment can be suppressed to a sufficiently small number compared to a normal Monte Carlo simulation that does not use the importance sampling method.

[Second Embodiment]
Next, a second embodiment will be described. Differences from the above-described first embodiment will be mainly described, and description of matters similar to those in the first embodiment will be omitted as appropriate.

FIG. 2 is a diagram illustrating an example of an information processing system according to the second embodiment.
The information processing system 20 according to the second embodiment divides the index value range into a plurality of sections and estimates the probability distribution of the index value. The simulation executed by the information processing system 20 is, for example, a Monte Carlo simulation using a stratified extraction method.

  The information processing system 20 includes a control unit 21 and a plurality of processors including processors 22-1 to 22-3. As described in the first embodiment, various “processors” such as a plurality of processor cores included in one CPU can be considered, and “processors” include various processors such as DSPs and ASICs. Things can be considered. Each processor executes a task according to a program stored in a memory such as a RAM, for example.

  The control unit 21 performs simulation using the processors 22-1 to 22-3. For example, the control unit 21 includes a processor that executes a program stored in a memory such as a RAM. The control unit 21 and the processors 22-1 to 22-3 may belong to the same information processing apparatus or may belong to different information processing apparatuses. When the control unit 21 includes a processor, at least one of the tasks described below may be assigned to the control unit 21.

  The control unit 21 divides the index value range into a plurality of sections. For example, the range is divided as follows. First, the control unit 21 estimates a probability distribution of index values using a model that generates a sample of index values from random numbers and a random number distribution. This probability distribution may be low in accuracy, and the number of samples to be generated is sufficiently small (for example, about 1/100 to 1/1000) compared to a normal Monte Carlo simulation that does not use a stratified extraction method. Good. And the control part 21 divides | segments the value range of an index value into a some area so that the probability (integration of the probability density of the said area) of each area becomes the same according to the estimated probability distribution. However, the range of index values may be divided by referring to past simulation results or the like without estimating the probability distribution of index values.

  The number of sections can be about 10 to 20, for example. When the number of sections is 10, ideally, the probability of all sections is equal to 0.1 (10%). In the example of FIG. 2, the index value range is divided into three in order to simplify the description. That is, as shown in FIG. 2, the index value range is divided into sections T1, T2, and T3. In this case, ideally, the probabilities of the sections T1, T2, and T3 are equally 0.33 (33%).

  Next, the control unit 21 determines the lower limit number s of index value samples used in each section. The lower limit number s is an integer of 2 or more determined in consideration of the accuracy of simulation, and may be specified by the user. If the lower limit total number m of samples used in all sections is designated, it may be calculated as s = m / number of sections. However, the lower limit total number m may be a sufficiently small number compared with a normal Monte Carlo simulation in which the stratified extraction method is not performed. In addition, the control unit 21 allocates p tasks to a plurality of processors. The number of tasks p is an integer equal to or greater than 2, and for example, is matched with the number of processors to which tasks are assigned. In the example of FIG. 2, the task 23-1 is assigned to the processor 22-1, the task 23-2 is assigned to the processor 22-2, and the task 23-3 is assigned to the processor 22-3.

  A processor to which any task is assigned performs the following processing as the task. Here, the task 23-1 will be described as a representative. The processor 22-1 uses a model for generating index value samples from random numbers and the distribution of random numbers, and generates index value samples until a predetermined number condition is satisfied. The number condition is that the number of index value samples belonging to each section reaches s / p. It can also be said that the number of samples in the section with the fewest samples has reached s / p. When the above number condition is satisfied, more than s / p samples may be generated in one or more intervals.

  Other tasks also generate at least s / p samples for each interval. The plurality of tasks can be executed independently of each other, and tasks assigned to different processors can be executed in parallel. For example, tasks 23-1 to 23-3 are executed in parallel.

  When the plurality of tasks allocated to the plurality of processors are completed, the control unit 21 collects s / p index value samples from the processor that executed each task for each of the plurality of sections. Thereby, s samples in each section are collected. Then, the control unit 21 estimates the probability distribution of the index value using the collected samples. If the number of tasks is p = 3 in the example of FIG. 2, s / 3 samples are collected from the processors 22-1 to 22-3 for the section T1, and a total of s samples are collected. Also in the sections T2 and T3, s / 3 samples are collected from the processors 22-1 to 22-3, and a total of s samples are collected.

  In this way, each task is not in charge of any one index value section, but is in charge of all index value sections. In the example of FIG. 2, each of the tasks 23-1 to 23-3 is not in charge of any one of the sections T1, T2, and T3, but is in charge of all the sections. However, the number of samples to be generated in each task can be reduced by increasing the number of tasks.

  In the simulation method of dividing the index value range into a plurality of sections, the number of samples in each section may be the same, and the number of samples in each section may be larger than the lower limit value s. Rather, the accuracy of estimation improves as the number of samples in each section increases. Therefore, the control unit 21 collects extra samples exceeding s / p generated by each task, and part or all of the extra samples are subjected to probability distribution so that the number of samples in each section is the same. It may be used for estimation.

  For example, it is assumed that the processor 22-1 generates s / 3 samples in the section T1, s / 3 + α samples in the section T2, and s / 3 + α samples in the section T3. Further, it is assumed that the processor 22-2 generates s / 3 + α samples in the section T1, s / 3 samples in the section T2, and s / 3 + α samples in the section T3. Further, it is assumed that the processor 22-3 generates s / 3 + α samples in the section T1, s / 3 samples in the section T2, and s / 3 samples in the section T3. Then, 2α extra samples are collected in the interval T1, α extra samples are collected in the interval T2, and 2α extra samples are collected in the interval T3. In this case, since α in the interval T2 is the minimum value of the number of extra samples, a maximum of each interval s + α can be used for estimation of the probability distribution. At this time, the control part 21 should just discard the excess sample which was not used.

  According to the information processing system 20 according to the second embodiment, the range of index values is divided into a plurality of sections, and a plurality of tasks are set so that each task is responsible for all sections. These multiple tasks can be executed independently of each other, and the simulation time can be shortened by distributing the multiple tasks to multiple processors. Further, compared to the case where a task is set for each divided section, it is possible to suppress the generation of useless samples that are not used for estimation of the probability distribution, and the simulation can be efficiently processed in parallel.

  Also, if extra samples exceeding the number of samples assigned to each task are collected and used for estimating the probability distribution, the accuracy of the simulation can be further improved. Note that the total number of samples generated by the simulation method according to the second embodiment can be suppressed to a sufficiently small number compared to a normal Monte Carlo simulation that does not use the stratified extraction method.

[Third Embodiment]
Next, a third embodiment will be described.
FIG. 3 is a first diagram illustrating an example of hardware of the information processing system according to the third embodiment. The functions of the information processing system 100 according to the third embodiment can be realized using, for example, computer hardware as shown in FIG.

  The computer shown in FIG. 3 includes a CPU 901, a RAM 902, an HDD 903, an image signal processing unit 904, an input signal processing unit 905, a disk drive 906, and a communication interface 907. Further, the CPU 901 has a plurality of cores C1, ..., Cn.

  The CPU 901 is an example of the control unit 11 and the processors 12-1, 12-2, and 12-3 according to the first embodiment. The CPU 901 is an example of the control unit 21 and the processors 22-1, 22-2, and 22-3 according to the second embodiment.

  The CPU 901 is a processor including an arithmetic unit that executes instructions described in a program. The CPU 901 loads at least a part of the program and data stored in the HDD 903 into the RAM 902 and executes instructions described in the program. Note that the information processing system 100 may include a plurality of CPUs 901.

  The RAM 902 is a volatile memory for temporarily storing programs executed by the CPU 901 and data used for processing. Note that the information processing system 100 may have a different type of memory from the RAM 902. Further, the information processing system 100 may include a plurality of memories.

  The HDD 903 is an example of a nonvolatile storage device that stores programs such as an operating system (OS), firmware, or application software, data used for processing, and the like. Note that the information processing system 100 may include a different type of storage device from the HDD 903, such as a flash memory or a solid state drive (SSD). The information processing system 100 may have a plurality of storage devices.

  The image signal processing unit 904 is controlled by the CPU 901 and outputs an image to the display device 91 connected to the computer. The display device 91 is a display device such as a Cathode Ray Tube (CRT) display, a Liquid Crystal Display (LCD), a Plasma Display Panel (PDP), or an Organic Electro-Luminescence Display (OELD).

  The input signal processing unit 905 acquires an input signal from the input device 92 connected to the computer and notifies the CPU 901 of the input signal. As the input device 92, for example, a mouse, a keyboard, a touch panel, a touch pad, a trackball, a remote controller, a button switch, or the like can be used.

  The disk drive 906 is a reading device that reads programs and data recorded on the recording medium 93. As the recording medium 93, for example, a magnetic disk such as a Flexible Disk (FD) or HDD, an optical disk such as a Compact Disc (CD) or a Digital Versatile Disc (DVD), or a magneto-optical disk such as a Magneto-Optical disk (MO) is used. be able to. For example, the disk drive 906 is controlled by the CPU 901 and stores a program and data read from the recording medium 93 in the RAM 902 or the HDD 903.

  The communication interface 907 is an interface for communicating with other computers via the network 94. The communication interface 907 may be a wired interface or a wireless interface.

  Further, the functions of the information processing system 100 according to the third embodiment can be realized by using, for example, computer hardware as shown in FIG. FIG. 4 is a second diagram illustrating an example of hardware of the information processing system according to the third embodiment.

  4 includes a plurality of GPUs 911, 912, and 913 in addition to a CPU 901, a RAM 902, an HDD 903, an image signal processing unit 904, an input signal processing unit 905, a disk drive 906, and a communication interface 907. The GPUs 911, 912, and 913 are, for example, GPUs for general-purpose computing on graphics processing units (GPGPU).

  The CPU 901 is an example of the control unit 11 according to the first embodiment. The GPUs 911, 912, and 913 are examples of the processors 12-1, 12-2, and 12-3 according to the first embodiment. The CPU 901 is an example of the control unit 21 according to the second embodiment. The GPUs 911, 912, and 913 are examples of the processors 22-1, 22-2, and 22-3 according to the second embodiment.

  Further, the functions of the information processing system 100 according to the third embodiment can be realized by using a plurality of computer hardwares as shown in FIG. 5, for example. FIG. 5 is a third diagram illustrating an example of hardware of the information processing system according to the third embodiment.

  As shown in FIG. 5, the plurality of computers 90-1, 90-2 and 90-3 are connected via a network 94. In addition, the computer 90-1 includes a CPU 921, a RAM 922, an HDD 923, and a communication interface 924. Further, the computer 90-2 includes a CPU 931, a RAM 932, and a communication interface 933. The computer 90-3 includes a CPU 941, a RAM 942, and a communication interface 943.

  The CPUs 921 and 931 are substantially the same as the CPU 901 described above. However, the CPU 931 may be a GPGPU, for example. The RAMs 922 and 932 are substantially the same as the RAM 902 described above. The HDD 923 is substantially the same as the HDD 903 described above. The communication interfaces 924 and 933 are substantially the same as the communication interface 907 described above. The computers 90-1, 90-2, 90-3 may include the image signal processing unit 904, the input signal processing unit 905, and the disk drive 906 described above.

  The CPU 921 is an example of the control unit 11 according to the first embodiment. The CPU 931 is an example of the processors 12-1, 12-2, and 12-3 according to the first embodiment. The CPU 921 is an example of the control unit 21 according to the second embodiment. The CPU 931 is an example of the processors 22-1, 22-2, and 22-3 according to the second embodiment.

FIG. 6 is a diagram illustrating an example of functional blocks of the information processing system according to the third embodiment.
As illustrated in FIG. 6, the information processing system 100 includes a first processor 110, a plurality of second processors 120-1, 120-2,..., 120 -N, and a storage device 130. The plurality of second processors 120-1, 120-2,..., 120-N have the same function. Therefore, the second processor 120-1 is taken as a representative example, and in the following, this will be described as the second processor 120.

  Note that the functions of the first processor 110 can be realized by the CPUs 901 and 921. The functions of the second processors 120-1,..., 120-N can be realized by the CPU 901, the GPUs 911, 912, 913, or the CPU 931. In addition, some or all of the functions of the first processor 110 and the second processor 120 can be realized as a module of a program executed by the CPUs 901 and 921. Also, some or all of the functions of the first processor 110 and the second processor 120 can be realized as an electronic circuit instead of software. The storage device 130 is a storage area secured in the RAMs 902, 922, 932 and HDDs 903, 923 or an external storage device such as a file server connected to the network 94.

  The first processor 110 approximately calculates a probability distribution taken by a value of a calculation formula including a random variable, and calculates an index value obtained from the probability distribution. The first processor 110 has functions of a random number generation unit 111, a distribution calculation unit 112, an index calculation unit 113, an index selection unit 114, and an index output unit 115.

  The random number generation unit 111 generates a random number group based on the density function p followed by the random variable included in the above calculation formula. Information on the density function p is stored in the storage device 130. The random number group generated by the random number generation unit 111 is input to the distribution calculation unit 112.

  The distribution calculation unit 112 approximately calculates the probability distribution that the value of the above calculation formula takes, using the random number group generated by the random number generation unit 111. The distribution calculation unit 112 substitutes each random number included in the random number group as the value of the random variable included in the above calculation formula, and calculates the value of the calculation formula (hereinafter, the first calculation value). In addition, the distribution calculation unit 112 generates a density distribution of the first calculation value. The density distribution generated by the distribution calculation unit 112 is input to the index calculation unit 113.

The index calculation unit 113 calculates an index value that satisfies a certain condition based on the density distribution generated by the distribution calculation unit 112. As the index value, for example, various statistical values such as the lower 1 percentile point, the lower 5 percentile point of the density distribution, the expected value, the variance value, and the half value width can be considered. In the following, a calculation using the P ₀ percentile point as an index value will be considered as an example.

  The index calculation unit 113 sets a plurality of temporary index values based on the calculated index value. The temporary index value is set so as to satisfy a preset condition such as a value 5% or 10% larger than the index value or a value 5% or 10% smaller than the index value. A plurality of temporary index values set by the index calculation unit 113 are stored in the storage device 130.

The second processor 120 reads one of the temporary index values from the storage device 130, uses the temporary index value to approximately calculate the probability distribution represented by the value of the above formula, and is obtained from the probability distribution. A difference value between the cumulative density and the probability P ₀ as the index value calculation reference is calculated. The second processor 120 has functions of a random number generation unit 121, a distribution calculation unit 122, and a cumulative density calculation unit 123.

  The random number generation unit 121 sets a density function q that generates a random number group in which calculated values are concentrated in the vicinity of the temporary index value. Further, the random number generation unit 121 generates a random number group based on the density function q. The random number group generated by the random number generation unit 121 is input to the distribution calculation unit 122. Information about the density function q is input to the distribution calculation unit 122.

The distribution calculation unit 122 approximately calculates the probability distribution that the value of the above calculation formula takes, using the random number group generated by the random number generation unit 121. The distribution calculation unit 122 acquires information on the density function p from the storage device 130, and sets a conversion formula for calculating the value of the above formula using the information on the density functions p and q. For example, when the random variable X follows the density function p, the expected value E ^p (F (X)) of the calculation formula F (X) is expressed as the following formula (1). However, E ^p (·) represents an expected value related to the random variable X according to the density function p.

Furthermore, the above equation (1) can be transformed into an expected value E ^q (•) related to the random variable Y according to the density function q, as shown in the following equation (2). This means that the expected value for the random variable X according to the density function p can be replaced with the expected value for the random variable Y according to the density function q by using the conversion formula p (Y) / q (Y). To do. Incidentally, the solution I _MC of the integral I by the Monte Carlo simulation is obtained by the following equation (3), where y ₁ ,..., Y _N are independent random numbers according to the density function q. In addition, although the expected value was mentioned here as an example, the random variable can be replaced by the conversion formula in the same manner for other statistical values.

The distribution calculation unit 122 substitutes each random number included in the random number group as the value of the random variable included in the product of the above calculation formula and conversion formula, and calculates the value of the product (hereinafter, the second calculation value). In addition, the distribution calculation unit 122 generates a density distribution of the second calculation value. The density distribution generated by the distribution calculation unit 122 is input to the cumulative density calculation unit 123. The cumulative density calculation unit 123 calculates the cumulative density up to the temporary index value for the density distribution generated by the distribution calculation unit 122. Further, the cumulative density calculation unit 123 calculates a difference value between the calculated cumulative density and the probability P ₀ as the index value calculation reference. The difference value calculated by the cumulative density calculation unit 123 is stored in the storage device 130.

  The storage device 130 stores a plurality of difference values calculated by the plurality of second processors 120-1,..., 120-N. The index selection unit 114 extracts the minimum difference value from the plurality of difference values stored in the storage device 130, and selects a temporary index value corresponding to the extracted difference value. In addition, the index selection unit 114 sets a plurality of new temporary index values based on the selected temporary index value. The temporary index value is set so as to satisfy a preset condition such as a value 4% or 8% larger than the index value or a value 4% or 8% smaller than the index value. A plurality of temporary index values newly set by the index selection unit 114 are stored in the storage device 130.

  In addition, the plurality of second processors 120-1,..., 120-N calculate the above difference values for the plurality of newly set temporary index values, and store them in the storage device 130. The index selection unit 114 extracts the minimum difference value from the plurality of difference values stored in the storage device 130, and selects a temporary index value corresponding to the extracted difference value. The calculation of the difference value by the plurality of second processors 120-1,..., 120-N and the selection of the temporary index value by the index selection unit 114 are repeatedly performed until the end condition is satisfied. As the termination condition, for example, a stop operation by the user, a preset number of repetitions, a threshold determination of the minimum difference value, or a convergence determination of the difference value can be considered.

  When the end condition is satisfied, the index selection unit 114 inputs the selected temporary index value to the index output unit 115 as a calculation result. The index output unit 115 outputs the temporary index value input by the index selection unit 114 as a calculation result of the index value. For example, the index output unit 115 causes the display device 91 to display calculation result information. Further, the index output unit 115 may provide information on the calculation result to other devices via the network 94. Further, the index output unit 115 may sequentially output the temporary index values selected by the index selection unit 114 even when the end condition is not satisfied.

  As described above, according to the information processing system 100 according to the third embodiment, the first processor 110 calculates the index value to be calculated. Further, by setting a plurality of temporary index values around the index value and generating a random number group in which the calculated values are concentrated in the vicinity of each temporary index value, a more accurate density distribution is calculated. Further, a temporary index value close to the desired index value is selected from a plurality of temporary index values, and the temporary index value is output as a calculation result.

  Thus, since the calculation with higher accuracy is executed using the calculation result of the first processor 110, the calculation by the first processor 110 may be executed using a small random number. Further, since the calculation by the second processor 120 is executed using a random number group in which the calculated values are concentrated in the vicinity of the temporary index value, the number of random numbers used for this calculation may be small. Furthermore, since the calculations by the plurality of second processors 120-1,..., 120-N are independent of each other, they are executed in parallel.

  Therefore, according to the information processing system 100 according to the third embodiment, the number of random numbers used for calculating the index value can be reduced as compared with the case where the index value is calculated by a normal Monte Carlo simulation. The number of calculations can be reduced. In addition, since a part of the simulation calculation is executed in parallel, it is possible to perform a high-speed calculation by efficiently using the calculation resources.

Hereinafter, the processing by the information processing system 100 according to the third embodiment will be described by taking the calculation method of Value at Risk (VaR) as an example.
FIG. 7 is a diagram for explaining a method of calculating VaR.

VaR is used as a risk index for market risk management. For example, as shown in FIG. 7, a point giving a lower P ₀ percentile in the density distribution of future profits and losses from a portfolio in which a plurality of financial assets are combined is used as VaR. The probability P ₀ is often set to 1% or 5%. VaR can be calculated using Monte Carlo simulation.

As an example, a method for calculating VaR of a portfolio composed of one stock and its European call option (hereinafter referred to as option) by Monte Carlo simulation is introduced. It should be noted, is referred to as the maturity of the option T, K the exercise price of the option, the stock price S _t at the time t, the interest rate r _d, the volatility of the stock return on investment σ. Also consider the risk-neutral stock price stochastic process. In this case, if a random term following the Wiener process is expressed as dW _t , the probability process of the stock price is expressed as the following equation (4). The cash flow C generated by the option is expressed as in the following equation (5).

In order to apply the Monte Carlo simulation, the stochastic differential equation shown in the above equation (4) is approximated by a difference equation. The period up to the maturity T is divided into N equal parts, and the time increment divided into N parts is expressed as Δt. Also, a random term dW _t following the Wiener process is expressed using a random number ε according to a standard normal distribution with an average of 0 and a variance of 1. By such approximation, the stochastic differential equation shown in the above equation (4) can be expressed by a differential equation as shown in the following equation (6).

If the current stock price S ₀ given, SΔ _t according to the above equation _{(6), S 2 Δ t} , ..., path stock is obtained by calculating the S _N delta _t in order. In addition, cash flow C (S _T ) generated by the option is obtained based on the above equation (5). Also, the value of the portfolio consisting of stocks and options at maturity T is obtained by the following equation (7). In addition, the distribution of portfolio values can be calculated by generating a sufficient number of stock price paths.

Once the portfolio value distribution is obtained, the portfolio value corresponding to the lower P ₀ percentile point of the distribution can be calculated. In the case where the probability P ₀ is 5%, the value of the portfolio corresponding to 5% of the total value counted in descending order from the value of the portfolio calculated for the generated many paths corresponds to VaR. For example, when 10,000 stock price paths are generated, the value of the portfolio corresponding to the 500th counted from the smallest value is VaR in this portfolio.

  With the above method, VaR can be calculated using Monte Carlo simulation. However, in the case of the simple Monte Carlo simulation as described above, sufficient calculation accuracy cannot be obtained unless a large number of paths are generated.

Therefore, a method for allowing VaR to be calculated accurately with a small number of paths will be considered. From the above equation (5), the path to maturity stock S _T is less than the exercise price K is, it can be seen that does not contribute to the value of the options. It can also be seen that a path whose portfolio value greatly exceeds VaR contributes little to the calculation of VaR. In consideration of these situations, it is effective to apply an importance sampling method and generate a path in which calculation results are concentrated in the vicinity of VaR.

FIG. 8 is a diagram for explaining the importance sampling method.
The importance sampling method is a method for efficiently calculating a desired solution by transforming a density function followed by random numbers and concentrating the generation of random numbers in a specific range. For example, if an appropriate density function q is selected, as shown in FIG. 8, it is possible to generate a random number such that calculation results are concentrated in the vicinity of VaR, and to efficiently calculate VaR that is a desired solution. it can.

Here, how to select the density function q will be considered. For example, when calculating the integral value I shown in the above equation (2), if the density function q can be selected so as to reduce the error variance of the solution I _MC in the Monte Carlo simulation, the weighted sampling method is realized. Density function q that minimizes the error variance of the solution I _MC is given by approximately the following equation (8). When calculating VaR, the density function q that minimizes the error variance can be obtained in the same manner by setting the calculation formula F to the above formula (7) and the solution I as the value of VaR.

Here, a method for improving the efficiency of parallel execution of the above simulation calculation will be considered. Here, the explanation will be made taking VaR calculation as an example.
FIG. 9 is a first diagram for explaining parallelization of the importance sampling method.

As shown in FIG. 9, the parallel simulation calculation according to the third embodiment is executed in stages.
In the first stage (# 1), a normal Monte Carlo simulation (MC calculation) is executed, and VaR ₁ indicating the lower P ₀ percentile point is calculated. However, in the first stage (# 1), VaR ₁ with a small number of random numbers (passes) is calculated because it is not necessary to calculate VaR with high accuracy. For example, in the case where 100,000 paths are required to calculate VaR only by normal Monte Carlo simulation, VaR ₁ is calculated in about 100 to 1000 paths in the first stage (# 1) calculation. VaR ₁ calculated in the first stage (# 1) is used for setting temporary index values (VaR ₁₁ , VaR ₁₂ ,..., VaR _1M ) used in the second stage (# 2).

The simulation calculation executed in the first stage (# 1) is executed by the first processor 110. Further, the number M of temporary index values can be set arbitrarily.
The temporary index value used in the second stage (# 2) is set to a value shifted by several percent in the vertical direction with respect to VaR ₁ . For example, VaR ₁₁ is set to 0.9 * VaR ₁ ,..., And VaR _1M is set to 1.1 * VaR ₁ . The setting method of the temporary index value is not limited to this example, and may be arbitrarily determined. When the temporary index value is set, the calculation of the second stage (# 2) is executed.

In the second stage (# 2), a Monte Carlo simulation based on the importance sampling method (IS) is executed using a plurality of temporary index values VaR ₁₁ ,..., VaR _1M . Since the simulation calculations executed for each temporary index value in the second stage (# 2) are independent from each other, a plurality of simulation calculations can be executed in parallel. For example, it is possible to realize parallel processing in which simulation calculation using VaR ₁₁ is executed by the second processor 120-1 and simulation calculation using VaR ₁₂ is executed by the second processor 120-2.

The plurality of simulation calculations executed in the second stage (# 2) have substantially the same calculation contents except that the temporary index values to be used are different. Therefore, description will be made by taking an example of simulation calculation using VaR ₁₁ .

When VaR ₁₁ is set, a density function q for realizing the importance sampling method is set based on VaR ₁₁ . The density function q is given by the above equation (8) using VaR ₁₁ as an approximate solution I _MC by Monte Carlo simulation.

Further, based on the set density function q, a Monte Carlo simulation by the importance sampling method is executed, and the density distribution of the portfolio value is calculated. In addition, the area (cumulative density P ₁₁ ) up to the lower VaR ₁₁ of the calculated density distribution is calculated. Similarly, density distributions are calculated for the other temporary index values VaR ₁₂ ,..., VaR _1M , and cumulative densities P _{12 ,.} The simulation calculation executed in the second stage (# 2) is performed using the same sampling method as the simulation calculation in the first stage (# 1) because the importance sampling method is applied. Good.

The plurality of cumulative densities P ₁₁ ,..., P _1M calculated in the second stage (# 2) are used for the VaR selection process in the third stage (# 3). In the third stage (# 3), a difference value between each of the plurality of cumulative densities P ₁₁ ,..., P _1M and P ₀ is calculated, and a temporary index value corresponding to the cumulative density that minimizes the difference value is selected. The For example, when the cumulative density that minimizes the difference value is P _1k , the temporary index value VaR _1k is selected in the third stage (# 3). The selected temporary index value VaR _1k becomes the index value VaR ₂ used for setting the temporary index values (VaR ₂₁ , VaR ₂₂ ,..., VaR _2M ) used in the fourth stage (# 4).

Note that the number of temporary index values used in the fourth stage (# 4) may not be M. In addition, as the temporary index value used in the fourth stage (# 4), for example, a value shifted by several percent in the vertical direction with respect to VaR ₂ is set. For example, VaR ₂₁ is set to 0.95 * VaR ₂ ,..., VaR _2M is set to 1.05 * VaR _2, etc. However, the setting method of the temporary index value is not limited to this example and may be arbitrarily determined. Further, the deviation amount used for setting the temporary index value used in the fourth stage (# 4) is smaller than the deviation amount used for setting the temporary index value used in the second stage (# 2) (for example, half). May be set).

When the temporary index value is set, the calculation in the fourth stage (# 4) is executed. The simulation calculation executed in the fourth stage (# 4) has substantially the same contents as the simulation calculation executed in the second stage (# 2) except that the temporary index values are different. In the fourth stage (# 4), density distributions are calculated for the temporary index values VaR ₂₁ ,..., VaR _2M , and cumulative densities P _{21 ,.} Note that the simulation calculation executed in the fourth stage (# 4) may be executed in the same or fewer paths as the simulation calculation in the first stage (# 1) or the second stage (# 2). .

After that, as in the third stage (# 3), a temporary index value is selected based on the accumulated density P ₂₁ ,..., P _2M, and a plurality of temporary index values used in the next stage are set. Then, as in the fourth stage (# 4), the cumulative density is calculated based on a plurality of temporary index values. In this way, the same processing as in the third stage (# 3) and the fourth stage (# 4) is repeatedly executed. This iterative process is executed until the end condition is satisfied. When the end condition is satisfied, the temporary index value selected by the same process as in the third stage (# 3) is output as the calculation result.

  In addition, as said completion | finish conditions, the stop operation by a user, expiration of the preset repeat count etc. can be considered, for example. Another possible method is to end the iterative process when the number of paths used for the simulation calculation reaches about one-tenth of the number of paths required when using only a normal Monte Carlo simulation. Further, a method of ending the iterative process when the difference between a plurality of difference values falls below a threshold value may be considered.

In the example of FIG. 9, a normal Monte Carlo simulation is executed in the first stage (# 1) to calculate VaR _1. For example, a method of calculating VaR ₁ using the delta gamma method or the like. There is also. The delta-gamma method is a method of calculating a risk index such as VaR using a model in which profit / loss generated from a portfolio is regarded as a function of the underlying asset price and the function is approximated up to the second-order term of the Taylor expansion. Although the accuracy of the delta-gamma method is lower than that of the Monte Carlo simulation, it can be used as a reference for determining a temporary index value for performing the priority sampling method, as in the method illustrated in FIG.

FIG. 10 is a second diagram for explaining parallelization of the importance sampling method.
As shown in FIG. 10A, a density distribution is obtained by a normal Monte Carlo simulation. From this density distribution, VaR ₁ which is the lower P ₀ percentile point is obtained. Further, when the VaR ₁ is obtained, a plurality of temporary index values VaR ₁₁ based on VaR _1, ..., it is possible to set the VaR _1M. Further, as shown in FIG. 10B, for example, a density distribution is obtained by executing a simulation calculation based on the priority sampling method based on the temporary index value VaR _1k , and the area up to the lower VaR _1k in the density distribution The cumulative density P _1k can be calculated. Further, a difference value between the cumulative densities P _1k and P ₀ can be calculated.

Similarly, a difference value is calculated for a plurality of temporary index values VaR ₁₁ ,..., VaR _1M , and a temporary index value that minimizes the difference value is selected. A plurality of new temporary index values are set based on the selected temporary index value, and a difference value is calculated for each of the new plurality of temporary index values in the same manner as described above. Also, a temporary index value that minimizes the difference value is selected. As described above, the setting of the temporary index value and the calculation of the difference value are repeatedly performed, and when the end condition is satisfied, the selected temporary index value is output as the calculation result. Note that each time a temporary index value is selected, the selected temporary index value may be output as the progress of the calculation result.

FIG. 11 is a diagram showing a flow of processing by the information processing system according to the third embodiment. The flow of processing will be described by taking the calculation method of VaR of portfolio value as an example.
(S101) The random number generation unit 111 generates a random number group according to a certain density function.

(S102) The distribution calculation unit 112 calculates the density distribution of the portfolio value using the random number group generated by the random number generation unit 111. The index calculation unit 113 calculates VaR that is the lower P ₀ percentile point of the density distribution calculated by the distribution calculation unit 112.

  (S103) The index calculation unit 113 sets a plurality of temporary VaRs based on the VaR calculated in the process of S102. For example, the index calculation unit 113 sets a value obtained by shifting the VaR calculated in the process of S102 by 5% up and down as the temporary VaR.

  (S104) The random number generation unit 121 sets a density function based on the temporary VaR. For example, the random number generation unit 121 sets a density function that generates a random number group in which calculated values are concentrated in the vicinity of the temporary VaR.

(S105) The random number generation unit 121 generates a random number group according to the density function set in S104.
(S106) The distribution calculation unit 122 uses the random number group generated in S105 to calculate a portfolio value density distribution by Monte Carlo simulation based on the priority sampling method. The cumulative density calculation unit 123 calculates an area (cumulative density) up to the provisional VaR in the density distribution calculated by the distribution calculation unit 122.

  In addition, the process of S104-S106 is performed by several 2nd processor 120-1, ..., 120-N about each of several temporary VaR. The cumulative density value corresponding to each provisional VaR is stored in the storage device 130.

(S107) the index selection unit 114 calculates a difference value between the value and P ₀ of the cumulative density corresponding to each formal VaR.
(S108) The index selection unit 114 selects the temporary VaR corresponding to the minimum difference value among the difference values calculated in S107.

  (S109) If the end condition is reached, the process proceeds to S110. On the other hand, if the end condition has not been reached, the process proceeds to S103. As the termination condition, for example, a stop operation by the user or expiration of a preset number of repetitions can be considered.

(S110) The index output unit 115 outputs the temporary VaR selected by the index selection unit 114 as a calculation result.
When the processing of S110 ends, a series of processing is completed. In the processing flow shown in FIG. 11, the calculation result is output when the end condition is satisfied in S109. However, the temporary VaR selected in the processing of S108 is output every time it is selected. May be.

FIG. 12 is a diagram showing a calculation example in which the occurrence frequency of the calculated value obtained by executing the simulation calculation with the same accuracy is compared.
As shown in FIG. 12, when a normal Monte Carlo simulation (normal MC calculation) is executed, the distribution of calculated values spreads to a position where the amount of profit / loss is far from VaR. On the other hand, when the simulation calculation based on the importance sampling method (MC calculation based on IS) is executed, the distribution of the calculated values is concentrated in the vicinity of VaR. In addition, the frequency of calculation values used to obtain calculation results with the same accuracy is much lower in simulation calculations based on the priority sampling method than in normal Monte Carlo simulations. That is, by applying the importance sampling method, the number of simulation calculations can be reduced. According to the information processing system 100 described above, the same effect as shown in FIG. 15 is obtained, and the speeding up and efficiency of the simulation calculation are realized.

The third embodiment has been described above.
[Fourth Embodiment]
Next, a fourth embodiment will be described.

  FIG. 13 is a diagram illustrating an example of functional blocks of the information processing system according to the fourth embodiment. Note that the information processing system 200 according to the fourth embodiment can be realized by using the same hardware as the information processing system 100 according to the third embodiment.

  As illustrated in FIG. 13, the information processing system 200 includes a first processor 210, a plurality of second processors 220-1, 220-2, ..., 220-N, and a storage device 230. The plurality of second processors 220-1, 220-2,..., 220-N have the same function. Therefore, the second processor 220-1 is taken as a representative example, and hereinafter, this will be described as the second processor 220.

  Note that the functions of the first processor 210 can be realized by the CPUs 901 and 921. The functions of the second processors 220-1,..., 220-N can be realized by the CPU 901, the GPUs 911, 912, 913, or the CPU 931. Further, part or all of the functions of the first processor 210 and the second processor 220 can be realized as a module of a program executed by the CPUs 901 and 921. Also, some or all of the functions of the first processor 210 and the second processor 220 can be realized as an electronic circuit instead of software. The storage device 230 is a storage area secured in the RAMs 902, 922, 932 and HDDs 903, 923, or an external storage device such as a file server connected to the network 94.

  The first processor 210 approximately calculates a probability distribution represented by a value of a calculation formula including a random variable, and divides the probability distribution into a plurality of sections. The first processor 210 has functions of a random number generation unit 211, a distribution calculation unit 212, an interval division unit 213, a distribution synthesis unit 214, and a result output unit 215. The functions of the random number generation unit 211 and the distribution calculation unit 212 are the same as the functions of the random number generation unit 111 and the distribution calculation unit 112 according to the third embodiment, respectively. Therefore, the description regarding the functions of the random number generation unit 211 and the distribution calculation unit 212 is omitted.

  The section dividing unit 213 divides the density distribution generated by the distribution calculating unit 212 into a plurality of sections. For example, the section dividing unit 213 divides the density distribution into a plurality of sections so that the areas of the density distribution regions corresponding to the sections have the same area. Information indicating a plurality of sections divided by the section dividing unit 213 is stored in the storage device 230.

In addition, the section dividing unit 213 sets the number of calculation values to be calculated by the plurality of second processors 220-1, ..., 220-N. First, the section dividing unit 213 sets a lower limit value M ₁ of the total number of calculation values used for calculating the density distribution. Note that this lower limit value may be set in advance. Further, the section dividing unit 213 sets a lower limit value M ₂ of the number of calculated values of each section calculated by each of the plurality of second processors 220-1,..., 220-N. For example, when the density distribution is divided into L sections, the lower limit value M ₂ is set to M ₁ / N / L.

The second processor 220 reads the information indicating the interval from the storage unit 230, executes the computation of the generator and the calculated value of the random number to the number of calculated values belonging to each section reaches the lower limit value M _2. The second processor 220 has functions of a random number generation unit 221, a distribution calculation unit 222, and a result holding unit 223.

The random number generation unit 221 generates a random number group based on a predetermined density function. The random number group generated by the random number generation unit 211 is input to the distribution calculation unit 222. The distribution calculation unit 222 calculates a calculated value using the input random number group. In addition, the distribution calculation unit 222 reads information indicating a section from the storage device 230 and determines a section to which the calculated value belongs. Distribution calculation unit 222 repeats the calculation of the calculated value to the number of calculated values belonging to each section reaches the lower limit value M _2.

Incidentally, the distribution calculator 222, a result the calculated value as an extra calculation value when the number of calculated values belonging to a certain interval after reaching the lower limit value M _2, calculated values are further calculated that belong to the section The holding unit 223 holds it. On the other hand, the distribution calculation unit 222 causes the result holding unit 223 to hold the calculation values calculated until the number of calculation values belonging to a certain section reaches the lower limit M ₂ . When the number of calculated values reaches the lower limit M ₂ for all the sections, the distribution calculation unit 222 stores the calculated values held in the result holding unit 223 in the storage device 230.

  Similarly, the calculated values are calculated by the plurality of second processors 220-2,..., 220 -N, and the calculated values are stored in the storage device 230. The distribution synthesis unit 214 reads the calculated values stored in the storage device 230, and generates a density distribution for the set of calculated values calculated by the plurality of second processors 220-1, ..., 220-N.

  First, the distribution synthesis unit 214 extracts calculated values from the set of extra calculated values so that the number of calculated values belonging to each section is equal, and discards the remaining calculated values. For example, when the minimum number of extra calculated values belonging to each section is b, the distribution synthesis unit 214 extracts b extra calculated values belonging to each section. Further, the distribution synthesis unit 214 calculates a set of calculated values extracted from the set of extra calculated values and a set of calculated values obtained as a calculation result by the plurality of second processors 220-1,. To create a density distribution. Information on the density distribution generated by the distribution synthesis unit 214 is input to the result output unit 215. The result output unit 215 outputs density distribution information. For example, the result output unit 215 causes the display device 91 to display calculation result information. Further, the result output unit 215 may provide calculation result information to other devices via the network 94.

  As described above, according to the information processing system 200 according to the fourth embodiment, the first processor 210 calculates the density distribution used for segment division. Therefore, even when the density distribution is not given in advance, it is possible to extract the calculated values used for calculating the density distribution so that the number of calculated values belonging to each section is equal. Further, by performing calculation value calculation so that the number of calculation values belonging to each section becomes a predetermined number by the plurality of second processors 220-1,. Realized.

Further, in the plurality of second processors 220-1,..., 220-N, when the calculation value is repeatedly calculated until the calculation value of each section reaches a predetermined number, an extra calculation value calculated exceeding the predetermined number Is also stored and used by the distribution synthesis unit 214 for calculating the density distribution. As a result, it becomes possible to use as many calculation values as possible for the extra calculation values that are effectively used for calculating the density distribution, and it is possible to improve the accuracy. In addition, it is expected that an extra calculated value of about 10% or less of the whole will be generated. For example, if M ₁ / L is 50, about 5 or less extra calculation values are calculated for each of the plurality of second processors 220-1,..., 220-N.

  If the plurality of second processors 220-1,..., 220-N calculate the calculated values by the stratified extraction method and discard the extra calculated values, many calculated values are discarded. become. However, according to the information processing system 200, it is possible to reduce the number of calculated values to be discarded and use more calculated values by effectively using a part of the extra calculated values without discarding them. As a result, calculation efficiency and high accuracy are realized.

  Since the calculation result by the first processor 210 is used for dividing the section, the calculation by the first processor 210 may be executed using a small random number. In addition, since the accuracy can be improved by the stratified extraction method, the number of random numbers used for the calculation by the second processor 220 may be smaller than that in the case of executing a normal Monte Carlo simulation.

Hereinafter, processing by the information processing system 200 according to the fourth embodiment will be described.
FIG. 14 is a diagram for explaining the stratified extraction method.
The stratified extraction method is a method in which the density distribution is divided into a plurality of regions having an equal area, and the generated random numbers are adjusted to be equally distributed to each region. For example, as shown in FIG. 14, it is assumed that a density distribution is given and the area is divided into _six areas (A ₁ ,..., A ₆ ) having the same area. Moreover, as a result of generating 100 random numbers and calculating the calculated value for each random number, 18 for A ₁ , 16 for A ₂ , 12 for A ₃ , 12 for A ₄ , 20 for A ₅ , Assume that A ₆ includes 22 calculated values. In this case, the stratified extraction method extracts the calculated values of 12 from A ₁ , 12 from A ₂ , 12 from A ₃ , 12 from A ₄ , 12 from A _5, and 12 from A _6. The extracted calculated values are used for later calculations, and other calculated values are discarded. Then, a solution to be obtained is calculated using the selected calculated value.

  According to the stratified extraction method, since a set of calculated values close to the density distribution is used for calculating a desired solution, sufficient calculation accuracy can be obtained with fewer random numbers. However, knowledge of density distribution is a prerequisite for applying the stratified extraction method. In addition, there are many random numbers that are wasted. Therefore, the following method is proposed to solve these problems.

FIG. 15 is a diagram for explaining parallelization of the stratified extraction method.
As shown in FIG. 15, the stratified extraction method (SS) proposed here includes three stages. In the first stage (# 1), a temporary density distribution D ₀ is calculated by a normal Monte Carlo simulation. Note that the simulation calculation executed in the first stage (# 1) is executed using a smaller number of random numbers than in the case where the final density distribution is calculated using only the normal Monte Carlo simulation. .

In the second step (# 2), dividing the density distribution D ₀ provisional into a plurality of regions such that the area becomes equal to sample the calculated value so that the number of computed values belonging to each area are equal. Note that the processing of the second stage (# 2) is executed in parallel as shown in FIG. In addition, the lower limit value M ₁ of the total number of calculation values used for calculation of the division number L and the density distribution, the lower limit value M ₂ of the number of calculation values for each area calculated in each process executed in parallel, and the like are set in advance. It is assumed that

For example, in the first process, the second processor 220-1, the number of calculated values belonging to each area obtained by dividing repeatedly executes the calculation of the calculated value to a lower limit value M _2. At this time, after the number of calculated values belonging to a certain section reaches the lower limit M ₂ , the second processor 220-1 uses the calculated value as the remainder E ₁ when the calculated value belonging to that section is further calculated. Hold as. In addition, the second processor 220-1 holds the calculation value calculated until the number of calculation values belonging to a certain section reaches the lower limit M ₂ as the calculation result D ₁ . When the number of calculated values reaches the lower limit M ₂ for all the intervals, the second processor 220-1 outputs the result D ₁ and the remainder E ₁ .

Similarly, the result D ₂ and the remainder E ₂ are output in the second process, and the result D ₃ and the remainder E ₃ are output in the _third process. In this way, the second step (# 2), the result D _2, ..., D _L and remainder E _1, ..., E _N is obtained. Note that the first to Nth processes are independent of each other. Therefore, the first to Nth processes are executed independently by the second processors 220-1,..., 220-N.

In the third step (# 3), the result D _1, ..., a first density distribution is calculated from D _N. Further, from the set of remainders E ₁ ,..., E _N , a part of the calculation results is sampled so that the calculation results are equally included in each region, and the second density distribution is calculated. For example, when the minimum number of extra calculated values belonging to each region is b, b extra calculated values belonging to each region are extracted, and the second density distribution is calculated. Then, the first and second density distributions are combined and a final density distribution is calculated. That is, the result D _1, ..., D _N, and the remainder E _1, ..., a final density distribution is created using a set of sampled calculated values from E _N.

As described above, since a plurality of divided regions can be calculated by obtaining the provisional density distribution D ₀ , it becomes possible to realize a stratified extraction method. In addition, the simulation calculation based on the stratified extraction method is executed in parallel to speed up the simulation calculation. Further, by reusing the remainder in the third stage (# 3), the calculated value is effectively used, and the efficiency and accuracy of the simulation calculation can be improved. For example, it is possible to improve calculation efficiency and accuracy compared to the case of discarding an extra calculation value in each process executed in parallel.

  Note that the stratified extraction method shown in FIG. 15 can be combined with a simulation calculation based on the priority sampling method according to the third embodiment. For example, the method shown in FIG. 15 can be applied to each simulation calculation executed in the second stage (# 2) of FIG. In this case, since the density distribution is obtained in the first stage (# 1) of FIG. 9, the first stage (# 1) of FIG. 15 can be omitted. In addition, the above stratified extraction method may be applied to the simulation calculation in the fourth stage (# 4) in which the calculation is substantially the same as the second stage (# 2) in FIG. 9 and subsequent stages. it can.

FIG. 16 is a diagram showing the flow of distribution calculation by the stratified extraction method.
(S201) The first processor 210 calculates a density distribution by executing a normal Monte Carlo simulation to which the stratified extraction method is not applied, using a small number of random numbers.

(S202) Based on the division number L, the first processor 210 divides the density distribution calculated in the process of S201 into a plurality of regions having the same area.
(S203) the first processor 210, based on the lower limit value M ₁ of the total number of calculated values utilized in the calculation of the density distribution, the plurality of second processors 220-1, ..., the calculation of each area 220-N are respectively calculated Set the lower limit M ₂ of the number of values. For example, the lower limit value M ₂ is set to M ₁ / N / L.

(S204) a plurality of second processors 220-1, ..., 220-N calculates the calculated value until the number of calculated values belonging to each area, respectively reaches the lower limit value M _2. For example, the second processor 220-1 executes repeatedly the calculation of the calculated value to the number of calculated values belonging to each region is the lower limit value M _2. At this time, after the number of calculated values belonging to a certain section reaches the lower limit M ₂ , the second processor 220-1 holds the calculated value as a remainder when the calculated value belonging to that section is further calculated. To do. In addition, the second processor 220-1 holds the calculated value calculated until the number of calculated values belonging to a certain section reaches the lower limit M ₂ as a result. The same applies to the second processors 220-2, ..., 220-N.

  (S205) The first processor 210 sets the calculated values so that the number of calculated values belonging to each region is equal from the set of calculated values held by the plurality of second processors 220-1,. Sampling. For example, when the minimum number of the remaining calculated values belonging to each area is b, the remaining calculated values belonging to each area are sampled b times.

  (S206) The first processor 210 generates a density distribution using the set of calculated values held as a result by the plurality of second processors 220-1, ..., 220-N and the calculated values sampled in the processing of S205. When the process of S206 is completed, the distribution calculation by the stratified extraction method is completed.

  As described above, according to the fourth embodiment, it is possible to speed up the calculation by parallelizing a part of the simulation calculation. In addition, the simulation calculation can be made more efficient by applying the stratified extraction method. Further, by reducing the number of calculation values to be discarded, more calculation values can be used and calculation accuracy can be improved.

  The fourth embodiment has been described above.

10, 20, 100, 200 Information processing system 11, 21 Control unit 12-1 to 12-3, 222-1 to 22-3 Processor 13-1 to 13-3, 233-1 to 23-3 Task v1 to v3 Index values T1 to T3 Intervals 110, 210 First processors 111, 211, 121 Random number generation units 112, 122, 212, 222 Distribution calculation units 113 Index calculation units 114 Index selection units 115 Index output units 130, 230 Storage devices 120, 120 -1, 120-2, 120-N, 220, 220-1, 220-2, 220-N Second processor 123 Cumulative density calculation unit 213 Section division unit 214 Distribution composition unit 215 Result output unit

Claims (9)

A simulation method for estimating an index value probability distribution and determining an index value satisfying a predetermined probability condition, executed by an information processing system including a plurality of processors,
Select a reference index value, select a plurality of temporary index values based on the reference index value,
Assigning a plurality of tasks corresponding to the plurality of provisional index values to the plurality of processors;
As a task corresponding to each temporary index value, a model for generating index value samples from random numbers and a distribution of random numbers biased so that the index value samples are generated around the temporary index value Processing to generate probability data corresponding to the temporary index value using the processor to which the task is assigned,
Comparing the probability data corresponding to each of the plurality of provisional index values with the predetermined probability condition to select one provisional index value;
The selected temporary index value is determined as an index value that satisfies the predetermined probability condition, or the selected temporary index value is used as a new reference index value, A simulation method for selecting an index value and allocating the plurality of tasks again. The index value probability distribution is estimated using the model and a random number distribution without bias, and the first index value of the reference is selected based on the probability distribution and the predetermined probability condition. The simulation method described. The probability data corresponding to each temporary index value indicates a cumulative probability in the temporary index value,
The temporary index value closest to the predetermined cumulative probability indicated by the predetermined probability condition among the plurality of temporary index values is selected as the temporary index value. 3. The simulation method according to 1 or 2. A simulation method executed by an information processing system including a plurality of processors,
The index value range is divided into a plurality of sections, and a lower limit number s (s is an integer of 2 or more) of index value samples used in each of the plurality of sections is determined.
P tasks (p is an integer of 2 or more) are allocated to the plurality of processors,
As each task, a process of generating index value samples from random numbers until at least s / p index value samples belonging to each of the plurality of sections is executed in the processor to which the task is allocated,
A simulation method of collecting s / p index value samples from each task for each of the plurality of sections and estimating an index value probability distribution using the collected index value samples of each section. In each task, for each of the plurality of sections, s / p index value samples are distinguished from other index value samples generated in excess of s / p,
Collect other index value samples from a plurality of tasks, and part or all of the collected other index value samples so that the number of other index value samples belonging to each of the plurality of sections is the same. Select all other index value samples,
The simulation according to claim 4, wherein a probability distribution of the index value is estimated using a sample of the index values of each section s in addition to the sample of some or all of the selected other index values. Method. A computer used in an information processing system for estimating an index value probability distribution and determining an index value satisfying a predetermined probability condition,
Select a reference index value, select a plurality of temporary index values based on the reference index value,
Allocating a plurality of tasks corresponding to the plurality of provisional index values to a plurality of processors;
As a task corresponding to each temporary index value, a model for generating index value samples from random numbers and a distribution of random numbers biased so that the index value samples are generated around the temporary index value Processing to generate probability data corresponding to the temporary index value using the processor that allocated the task,
Comparing the probability data corresponding to each of the plurality of provisional index values with the predetermined probability condition to select one provisional index value;
The selected temporary index value is determined as an index value that satisfies the predetermined probability condition, or the selected temporary index value is used as a new reference index value, A program for executing a process of selecting an index value and allocating the plurality of tasks again. On the computer,
The index value range is divided into a plurality of sections, and a lower limit number s (s is an integer of 2 or more) of index value samples used in each of the plurality of sections is determined.
Allocate p tasks (p is an integer of 2 or more) to multiple processors,
As each task, the processor to which the task is allocated executes processing for generating index value samples from random numbers until at least s / p index value samples belonging to each of the plurality of sections are reached.
A program for collecting a sample of s / p index values from each task for each of the plurality of sections and estimating a probability distribution of the index values using the collected sample of index values for each section s . An information processing system that estimates a probability distribution of index values and determines an index value that satisfies a predetermined probability condition,
Multiple processors,
A control unit that selects a reference index value, selects a plurality of temporary index values based on the reference index value, and allocates a plurality of tasks corresponding to the plurality of temporary index values to the plurality of processors; Have
Each of the plurality of processors, as a task corresponding to any assigned temporary index value, generates a model of an index value from a random number and the index value sample is generated around the temporary index value. Using the distribution of random numbers biased so as to generate probability data corresponding to the temporary index value,
The control unit compares the probability data corresponding to each of the plurality of temporary index values with the predetermined probability condition to select one temporary index value, and selects the selected temporary index value. The index value is determined as an index value that satisfies the predetermined probability condition, or the selection of the plurality of temporary index values and the plurality of tasks are performed using the selected temporary index value as a new reference index value. Information processing system that performs allocation again. Multiple processors,
The index value range is divided into a plurality of sections, a lower limit number s (s is an integer of 2 or more) of index value samples used in each of the plurality of sections is determined, and p tasks ( p is an integer greater than or equal to 2), and
Each of the plurality of processors performs, as an assigned task, a process of generating index value samples from random numbers until at least s / p index value samples belonging to each of the plurality of sections are reached,
The control unit collects s / p index value samples from each task for each of the plurality of sections, and estimates a probability distribution of the index values using the collected index value samples of each section. Information processing system.

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