CN113988370A - Solving condition nonlinear optimal disturbance method based on analytical four-dimensional set variation - Google Patents

Solving condition nonlinear optimal disturbance method based on analytical four-dimensional set variation Download PDF

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CN113988370A
CN113988370A CN202111128897.2A CN202111128897A CN113988370A CN 113988370 A CN113988370 A CN 113988370A CN 202111128897 A CN202111128897 A CN 202111128897A CN 113988370 A CN113988370 A CN 113988370A
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刘思远
李威
邵祺
梁康壮
龚延天
刘涵宇
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Abstract

A solving condition nonlinear optimal disturbance method based on analytical four-dimensional set variation is characterized in that a condition nonlinear optimal disturbance target function is constructed according to an ocean forecast model, and a target function gradient formula is obtained; solving an increment matrix at the disturbance value matrix forecasting time; rewriting an accompanying operator in the gradient formula, and partially rewriting the accompanying operator in the gradient formula of the conditional nonlinear optimal disturbance objective function into a form related to the generalized ocean background state error covariance matrix; and circularly and iteratively solving the condition nonlinear optimal disturbance of the ocean forecast model. The method avoids the writing of an accompanying mode, has good portability, uses the continuously updated error covariance matrix, ensures more accurate solution, ensures the equivalence with the traditional algorithm when the prediction mode has longer integration time and stronger nonlinearity, is even superior to the traditional algorithm, greatly improves the applicability of the CNOP method, and has great significance for developing atmospheric-ocean predictability research.

Description

Solving condition nonlinear optimal disturbance method based on analytical four-dimensional set variation
Technical Field
The invention relates to a method for solving nonlinear optimal disturbance under conditions. In particular to a solving condition nonlinear optimal disturbance method based on analytical four-dimensional set variational for atmospheric ocean forecasting.
Background
Solving a class of initial disturbances that grow fastest is one of the key problems in studying atmospheric ocean predictability, and Conditional Nonlinear disturbance (CNOP) describes such a class of initial disturbances: under certain dynamic-physical constraint conditions, the nonlinear system is enabled to develop maximally at the forecast moment. The CNOP method considers the influence of a nonlinear physical process on the atmospheric marine motion, and provides a new idea for improving the atmospheric-marine forecasting skill.
The CNOP method is widely applied to the research of atmospheric ocean predictability, how to solve the CNOP is a key problem for applying the method, the traditional method for solving the CNOP needs to calculate the gradient information of an objective function through an accompanying mode, the development and maintenance of the accompanying mode needs to invest a large amount of manpower and material resources, the transportability is poor, and the defects limit the business application of the CNOP method.
Aiming at the limitations of the traditional algorithm, the patent provides a solution CNOP algorithm which is based on an analytic four-dimensional set variation method, does not need to accompany and is equivalent to the traditional algorithm.
Disclosure of Invention
The invention aims to overcome the defects of the prior art and provide a solving condition nonlinear optimal disturbance method which can ensure the equivalence with the traditional method when the prediction mode integration time is longer and has stronger nonlinearity, and is even superior to the traditional method based on the analysis of four-dimensional set variation.
The technical scheme adopted by the invention is as follows: a solving condition nonlinear optimal disturbance method based on analytical four-dimensional set variation is characterized by comprising the following steps:
1) constructing a nonlinear optimal disturbance target function according to an ocean forecast model and acquiring a target function gradient formula;
2) solving an increment matrix of the disturbance value matrix forecasting time, namely adding an initial guess value of the conditional nonlinear optimal disturbance and an initial ocean background state to obtain a new ocean background state, superposing the new ocean background state on a disturbance value matrix which obeys normal distribution to form a set sample, and substituting the new ocean background state and the set sample into a numerical mode for operation to obtain the increment matrix of the disturbance value matrix forecasting time;
3) the companion operator in the gradient formula is rewritten, a generalized ocean background state error covariance matrix is solved through a disturbance value matrix and an increment matrix of the disturbance value matrix at the forecasting time, and the companion operator part in the gradient formula of the conditional nonlinear optimal disturbance objective function is rewritten into a form related to the generalized ocean background state error covariance matrix;
4) and circularly and iteratively solving the condition nonlinear optimal disturbance of the ocean forecast model.
The solving condition nonlinear optimal disturbance method based on the analytical four-dimensional set variation introduces dynamic background field error covariance matrix information in the analytical four-dimensional set variation to calculate an adjoint operator, and then solves CNOP. Compared with the traditional method, the method avoids the writing of the accompanying mode, has good portability, uses the continuously updated error covariance matrix, ensures more accurate solution, ensures the equivalence with the traditional algorithm when the prediction mode has longer integration time and stronger nonlinearity, even is superior to the traditional algorithm, greatly improves the applicability of the CNOP method, and has great significance for developing atmospheric-ocean predictability research.
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FIG. 1 is a flow chart of a solution condition nonlinear optimal perturbation method based on analytical four-dimensional set variation.
Detailed Description
The following describes in detail the solving condition nonlinear optimal perturbation method based on the analytic four-dimensional set variation according to the present invention with reference to the embodiments and the accompanying drawings.
As shown in fig. 1, the method for solving the condition nonlinear optimal disturbance based on the analytical four-dimensional set variation includes the following steps:
1) constructing a nonlinear optimal disturbance (CNOP) objective function according to the ocean forecast model and acquiring an objective function gradient formula; the method comprises the following steps:
(1) giving an ocean forecast model formula:
Figure BDA0003279791110000021
wherein X is the ocean background state, including the ocean surface temperature, salinity and flow rate; t is a time variable, and F is a nonlinear partial differential operator; x0The state is the ocean background state at the initial moment;
setting M0→TFor the evolution operator of the background state from time 0 to time T, the state of equation (1) at time T is:
X|t＝T＝M0→T(X0) (2)
setting x0Is a CNOP initial guess value, X, superimposed on a sea background state XTIs x0Development at time T, i.e.:
xT＝M0→T(X0+x0)-M0→T(X0) (3)
and giving a conditional nonlinear optimal disturbance objective function for solving the ocean forecast model according to the definition of the conditional nonlinear optimal disturbance:
I(x0)＝||M0→T(X0+x0)-M0→T(X0)||2 (4)
wherein, X0For initial moments of non-linear optimal perturbation of the condition, the ocean background state, x0As initial guess value of conditional nonlinear optimum disturbance, M0→TAn evolution operator of the ocean background state from 0 time to T time;
(2) in order to find the maximum value of the target function, the target function is converted into the minimum value of the reciprocal of the target function, and the rewritten target function is obtained:
J(x0)＝1/||M0→T(X0+x0)-M0→T(X0)||2 (5)
(3) and solving the gradient of the rewritten target function to obtain a gradient formula of the nonlinear optimal disturbance target function under the following conditions:
Figure BDA0003279791110000022
wherein the content of the first and second substances,
Figure BDA0003279791110000023
is a companion operator.
2) Solving an increment matrix of the disturbance value matrix forecasting time, namely adding an initial guess value of the conditional nonlinear optimal disturbance and an initial ocean background state to obtain a new ocean background state, superposing the new ocean background state on a disturbance value matrix which obeys normal distribution to form a set sample, and substituting the new ocean background state and the set sample into a numerical mode for operation to obtain the increment matrix of the disturbance value matrix forecasting time; the method comprises the following steps:
(1) ocean background state X at initial moment of nonlinear optimal disturbance to condition0Sum conditional nonlinear optimal disturbance initial guess value x0Summing to obtain new ocean background state X with condition nonlinear optimal disturbance0+x0；
(2) Selecting an initial disturbance value matrix with the set membership n and obeying normal distribution
Figure BDA0003279791110000031
(3) Normal distribution-compliant initial disturbance value matrix
Figure BDA0003279791110000032
New ocean background state X perturbed with conditional non-linear optimality0+x0Adding to form a set sample:
Figure BDA0003279791110000033
(4) the samples are assembled:
Figure BDA0003279791110000034
with new sea background state X0+x0Substituting into the following numerical pattern formula,
Figure BDA0003279791110000035
obtaining an increment matrix corresponding to the aggregate sample at the forecast time T
Figure BDA0003279791110000036
Wherein M is0→TEvolution operators of state variables from 0 to T;
disturbance value matrix in the pair formula (4)
Figure BDA0003279791110000037
And each collection member
Figure BDA0003279791110000038
Taylor expansion is carried out on the terms at the time T, and the high-order terms are ignored to obtain an increment matrix at the forecasting time of the following initial disturbance value matrix:
Figure BDA0003279791110000039
3) the companion operator in the gradient formula is rewritten, a generalized ocean background state error covariance matrix is solved through a disturbance value matrix and an increment matrix of the disturbance value matrix at the forecasting time, and the companion operator part in the gradient formula of the conditional nonlinear optimal disturbance objective function is rewritten into a form related to the generalized ocean background state error covariance matrix; the method comprises the following steps:
introducing an initial field concept in analytical four-dimensional set variation, wherein a generalized ocean background state error covariance matrix of the initial field is
Figure BDA00032797911100000310
Constructing a generalized ocean background state error covariance matrix B between the T-th moment and the 0-th moment ocean state variables according to the increment matrix of the initial disturbance value matrix forecasting moment of the formula (8) and the initial disturbance value matrixT0And a generalized sea-background state error covariance matrix B between the state variables at time 0 and time T0T；
Figure BDA00032797911100000311
The two generalized ocean background state error covariance matrixes actually respectively comprise a tangential evolution operator matrix
Figure BDA00032797911100000312
And corresponding companion operator matrix
Figure BDA00032797911100000313
The gradient formula of the conditional nonlinear optimal perturbation objective function of equation (6) becomes:
Figure BDA00032797911100000314
4) circularly and iteratively solving the conditional nonlinear optimal disturbance of the ocean forecasting model; the method comprises the following steps:
substituting the gradient formula of the conditional nonlinear optimal disturbance objective function of the formula (10) into a sequence quadratic programming method (SQP) or a spectral projection gradient method (SPG2) or an L-BFGS method for optimization to obtain a new initial guess value x of the conditional nonlinear optimal disturbance0 *Judging whether the value is converged, if yes, x0 *For conditional non-linear optimal perturbation of the ocean forecast model to be solved, and vice versa x0 *And (3) returning to the step 2) as a new initial guess value until the value meets the convergence condition, and obtaining a conditional nonlinear optimal disturbance value of the ocean forecast model.

Claims (5)

1. A solving condition nonlinear optimal disturbance method based on analytical four-dimensional set variation is characterized by comprising the following steps:
1) constructing a nonlinear optimal disturbance target function according to an ocean forecast model and acquiring a target function gradient formula;
2) solving an increment matrix of the disturbance value matrix forecasting time, namely adding an initial guess value of the conditional nonlinear optimal disturbance and an initial ocean background state to obtain a new ocean background state, superposing the new ocean background state on a disturbance value matrix which obeys normal distribution to form a set sample, and substituting the new ocean background state and the set sample into a numerical mode for operation to obtain the increment matrix of the disturbance value matrix forecasting time;
3) the companion operator in the gradient formula is rewritten, a generalized ocean background state error covariance matrix is solved through a disturbance value matrix and an increment matrix of the disturbance value matrix at the forecasting time, and the companion operator part in the gradient formula of the conditional nonlinear optimal disturbance objective function is rewritten into a form related to the generalized ocean background state error covariance matrix;
4) and circularly and iteratively solving the condition nonlinear optimal disturbance of the ocean forecast model.
2. The method for solving the nonlinear optimal perturbation based on the analytical four-dimensional set variation as recited in claim 1, wherein the step 1) comprises:
(1) giving an ocean forecast model formula:
Figure FDA0003279791100000011
wherein X is the ocean background state, including the ocean surface temperature, salinity and flow rate; t is a time variable, and F is a nonlinear partial differential operator; x0The state is the ocean background state at the initial moment;
setting M0→TFor the evolution operator of the background state from time 0 to time T, the state of equation (1) at time T is:
X|t＝T＝M0→T(X0) (2)
setting x0Is a CNOP initial guess value, X, superimposed on a sea background state XTIs x0Development at time T, i.e.:
xT＝M0→T(X0+x0)-M0→T(X0) (3)
and giving a conditional nonlinear optimal disturbance objective function for solving the ocean forecast model according to the definition of the conditional nonlinear optimal disturbance:
I(x0)＝||M0→T(X0+x0)-M0→T(X0)||2 (4)
wherein, X0For initial moments of non-linear optimal perturbation of the condition, the ocean background state, x0As initial guess value of conditional nonlinear optimum disturbance, M0→TAn evolution operator of the ocean background state from 0 time to T time;
(2) in order to find the maximum value of the target function, the target function is converted into the minimum value of the reciprocal of the target function, and the rewritten target function is obtained:
J(x0)＝1/||M0→T(X0+x0)-M0→T(X0)||2 (5)
(3) and solving the gradient of the rewritten target function to obtain a gradient formula of the nonlinear optimal disturbance target function under the following conditions:
Figure FDA0003279791100000012
wherein the content of the first and second substances,
Figure FDA0003279791100000021
is a companion operator.
3. The method for solving the nonlinear optimal perturbation based on the analytical four-dimensional set variation as recited in claim 1, wherein the step 2) comprises:
(1) ocean background state X at initial moment of nonlinear optimal disturbance to condition0Sum conditional nonlinear optimal disturbance initial guess value x0Summing to obtain new ocean background state X with condition nonlinear optimal disturbance0+x0；
(2) Selecting an initial disturbance value matrix with the set membership n and obeying normal distribution
Figure FDA0003279791100000022
(3) Normal distribution-compliant initial disturbance value matrix
Figure FDA0003279791100000023
New ocean background state X perturbed with conditional non-linear optimality0+x0Adding to form a set sample:
Figure FDA0003279791100000024
(4) the samples are assembled:
Figure FDA0003279791100000025
with new sea background state X0+x0Substituting into the following numerical pattern formula,
Figure FDA0003279791100000026
obtaining an increment matrix corresponding to the aggregate sample at the forecast time T
Figure FDA0003279791100000027
Wherein M is0→TEvolution operators of state variables from 0 to T;
disturbance value matrix in the pair formula (4)
Figure FDA0003279791100000028
Each of the set members
Figure FDA0003279791100000029
Taylor expansion is carried out on the terms at the time T, and the high-order terms are ignored to obtain an increment matrix at the forecasting time of the following initial disturbance value matrix:
Figure FDA00032797911000000210
4. the method for solving the nonlinear optimal perturbation based on the analytical four-dimensional set variation as recited in claim 1, wherein the step 3) comprises:
introducing an initial field concept in analytical four-dimensional set variation, wherein a generalized ocean background state error covariance matrix of the initial field is
Figure FDA00032797911000000211
Constructing a generalized ocean background state error covariance matrix B between the T-th moment and the 0-th moment ocean state variables according to the increment matrix of the initial disturbance value matrix forecasting moment of the formula (8) and the initial disturbance value matrixT0And a generalized sea-background state error covariance matrix B between the state variables at time 0 and time T0T；
Figure FDA00032797911000000212
The two generalized ocean background state error covariance matrixes actually respectively comprise a tangential evolution operator matrix
Figure FDA00032797911000000213
And corresponding companion operator matrix
Figure FDA00032797911000000214
Information of (2), the conditional nonlinear optimal perturbation target of equation (6)The gradient formula of the function becomes:
Figure FDA00032797911000000215
5. the method for solving the nonlinear optimal perturbation based on the analytical four-dimensional set variation as recited in claim 1, wherein the step 4) comprises:
substituting the gradient formula of the conditional nonlinear optimal disturbance objective function of the formula (10) into a sequence quadratic programming method (SQP) or a spectral projection gradient method (SPG2) or an L-BFGS method for optimization to obtain a new initial guess value x of the conditional nonlinear optimal disturbance0 *Judging whether the value is converged, if yes, x0 *For conditional non-linear optimal perturbation of the ocean forecast model to be solved, and vice versa x0 *And (3) returning to the step 2) as a new initial guess value until the value meets the convergence condition, and obtaining a conditional nonlinear optimal disturbance value of the ocean forecast model.
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CN110298375A (en) * 2019-05-16 2019-10-01 同济大学 The parallel gradient of solving condition nonlinear optimal perturbation defines data processing method
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