An Inference Method for Inverse Problems of Partial Differential Equations with Laplace Noise

Abstract

The invention discloses an inference method for an inverse problem of a partial differential equation containing Laplace noise, comprising: assuming that the observation noise obeys the Laplace distribution, expressing the probability density function of the noise as a mixture of Gaussian scales; obtaining a likelihood function; using a Markov random field to represent the prior distribution of unknown variables and multi-layer modeling to model the scale parameter as a random variable; remodeling the posterior probability density function of the inverse problem into a multi-layer Bayesian model through the Gaussian scale mixture representation of the Laplace distribution; based on the Kullback-Leibler divergence, using a variational Bayesian inference method to convert the calculation problem of the posterior probability density function into a variational problem of probability distribution; applying a mean field hypothesis, finding an approximate estimate of the true posterior probability distribution in a probability distribution family, and inverting the approximate distribution of the unknown variable, the noise variance parameter and the scale parameter; selecting an appropriate approximate probability distribution family to obtain a non-degenerate method and a degenerate method.

Description

Technical Field

[0001] The invention relates to the technical field of inverse problem inference, and in particular to an inference method for an inverse problem of a partial differential equation containing Laplace noise. Background Art

[0002] With the advancement of science and technology and the continuous deepening of mathematical research, inverse problems have been widely used in many scientific and technological fields such as geology and environmental science, life science and medicine, geophysical science, engineering control, signal processing, etc. In the field of mathematical physics research, inverse problems are used to describe physical changes in the space-time domain in a counterclockwise direction, and the internal laws of things can be explored through direct observation information. For example, in the field of geophysical exploration, seismic wave signals are received, and the propagation speed of seismic waves at different depths is obtained through travel time imaging. The acoustic wave equation or one-way wave equation migration imaging method is further used to obtain the position and shape of the reflection interface and other stratum geomorphology. This type of problem is a mathematical physics inverse problem.

[0003] The solution of inverse problems is relatively difficult. The essence is that inverse problems are usually ill-posed, that is, one or more of the existence, uniqueness and stability of the solution is not satisfied. For ill-posed problems, researchers have proposed two major effective solutions: regularization methods and Bayesian methods. The starting point of traditional regularization methods for solving inverse problems is that the error characteristics of the measurement data are known or the error is very small and negligible. However, when dealing with practical problems, it is difficult to accurately obtain the error characteristics of the measurement data, or it is difficult to obtain the specific error size of the measurement data. In addition, from a statistical point of view, the regularization method solves a certain value, which is equivalent to the "point estimate" of the Bayesian statistical method. It gives less information and ignores the randomness of the error and the uncertainty information of the model. Therefore, the regularization method has certain limitations. In order to overcome the above shortcomings, researchers use random methods to quantify the uncertainty of the model and solve ill-posed problems. The most commonly used method is the Bayesian statistical inference method, referred to as the Bayesian method. The Bayesian method transforms the inverse problem into a statistical inference problem, uses the prior probability to characterize the known information of the function to be estimated, and effectively combines the known information with the information obtained from the measured data with the help of the Bayesian formula, thereby obtaining a posterior probability distribution that can fully characterize the uncertainty of the function to be estimated. Compared with the traditional regularization method, the Bayesian method provides uncertainty quantitative analysis.

[0004] An important issue in the Bayesian inversion method is the acquisition of posterior information. Sampling algorithms represented by Markov chain Monte Carlo (MCMC) sampling have received widespread attention in statistical inversion, but their computational cost is too high when massive data or complex models are involved. At this time, the variational inference method, which is widely used in machine learning, is a good alternative method. It can provide uncertainty analysis of the function to be estimated with a computational cost comparable to that of point estimation. Therefore, this method has wide applicability for inverse problems of partial differential equations.

[0005] Variational Bayesian inference provides a flexible framework for solving various inverse problems. First, it can generate a set of solutions that are consistent with the observed data, thereby quantifying the uncertainty of the solution. Second, it solves the problem of hyperparameter selection through hierarchical modeling, thus providing a flexible regularization method. In existing studies on Bayesian inference of inverse problems, the noise prior distribution is mostly assumed to be a Gaussian distribution. Although the Gaussian prior distribution makes calculations convenient and easy to explore the posterior state space, the limitation of the Gaussian distribution is that it lacks robustness to outliers in the observed data. One abnormal data point can significantly affect the estimated values ​​of all parameters in the model, thus seriously damaging the accuracy of the estimate. In addition, not all data noise can be fully described by the Gaussian model, and sometimes Laplace noise may be generated when acquiring the signal. Summary of the invention

[0006] In order to solve at least one of the problems existing in the above-mentioned background technology, the present invention provides an inference method for the inverse problem of partial differential equations containing Laplace noise.

[0007] An inference method for an inverse problem of a partial differential equation with Laplace noise provided by an embodiment of the present invention includes the following eight steps:

[0008] Step 1: Assume that the observation noise follows a Laplace distribution and express the probability density function of the noise as a mixture of Gaussian scales;

[0009] Step 2: Obtain likelihood function;

[0010] Step 3: Use Markov random fields to represent the prior distribution of unknown variables;

[0011] Step 4: Use multi-level modeling to model the scale parameter as a random variable;

[0012] Step 5: Remodel the posterior probability density function of the inverse problem into a multi-layer Bayesian model through the Gaussian scale mixture representation of the Laplace distribution;

[0013] Step 6: Based on the Kullback-Leibler divergence, the variational Bayesian inference method is used to transform the calculation problem of the posterior probability density function into a variational problem about probability distribution;

[0014] Step 7: Apply the mean field hypothesis, find an approximate estimate of the true posterior probability distribution in the probability distribution family, and invert the approximate distribution of the unknown variables, noise variance parameters, and scale parameters;

[0015] Step 8: Select an appropriate family of approximate probability distributions to obtain a non-degenerate method that can simultaneously characterize the uncertainty of unknown variables, noise variance parameters and scale parameters, as well as a degenerate method that only characterizes the uncertainty of noise variance parameters and scale parameters.

[0016] Furthermore, in the above-mentioned inference method for the inverse problem of partial differential equations containing Laplace noise, in step 1, it is assumed that the observation noise obeys the Laplace distribution, and the probability density function of the noise is expressed as a mixture of Gaussian scales, which includes the following two steps:

[0017] Step (1) establishes a variational Bayesian model for the inverse problem with Laplace noise. For a finite-dimensional linear inverse problem: Hm = d, where Represent the system matrix, unknown variables, and observed data respectively; let d = d + +ζ, where d + represents noise-free data, ζ represents noise, ζ=[ζ 1 ,ζ 2 ,···ζ n ] T , each component ζ i are independent and identically distributed with an expectation of 0 and a variance of The Laplace distribution of Its probability density function is expressed as:

[0018] Step (2) expresses the probability density function of the noise ζ as a mixture of Gaussian scales, specifically in the form of:

[0019]

[0020] Among them, i The expected value is 0 and the variance is w i The Gaussian distribution of w i is a hidden variable, w i Subordinate to exponential distribution w i ~Exponential(τ), its probability density function is w=[w 1 ,w 2 ,···wn ] T , the probability density function of w

[0021] Furthermore, in the above-mentioned inference method for the inverse problem of partial differential equations containing Laplace noise, obtaining the likelihood function in step 2 includes:

[0022] When considering the additive noise model, the likelihood function is a translation of the noise ζ probability density function and can be expressed as: That is, the probability density function of d when m and w are known, where m is the unknown variable, d is the observed data, and w is a random variable that obeys an exponential distribution; d obeys the expectation Hm and the covariance matrix diag(w 1 ,w 2 ,···w n )’s n-dimensional Gaussian distribution, w i is a hidden variable, w i Subordinate to exponential distribution w i ~Exponential(τ).

[0023] Furthermore, in the above-mentioned inference method for the inverse problem of partial differential equations containing Laplace noise, the prior distribution of unknown variables is represented by using a Markov random field in step 3, including:

[0024] The Markov random field, a priori modeling tool, is used to represent the prior distribution p(m|λ) of the unknown variable m, which is in the form of: Among them, λ is the scale parameter describing the strength of interaction, the matrix L represents the structure of interaction between adjacent points, and s is the rank of the matrix L;

[0025] In step 4, multi-layer modeling is used to model the scale parameter as a random variable, including: selecting the gamma distribution G(λ; α, β) as the prior distribution of the scale parameter λ: Here, Γ(·) is the standard gamma function, and α and β are non-negative constants.

[0026] Furthermore, in the above-mentioned inference method for the inverse problem of partial differential equations containing Laplace noise, in step 5, the posterior probability density function of the inverse problem is remodeled as a multi-layer Bayesian model through the Gaussian scale mixture representation of the Laplace distribution, including: using multi-layer Bayesian modeling to express the posterior probability density function as:

[0027]

[0028] Where w is the noise variance parameter, W is a diagonal matrix with w as diagonal elements, and the norm Defined as (α, β) is the parameter pair of the gamma distribution, m is the unknown variable, d is the observed data, λ is the scale parameter describing the strength of the interaction, the matrix L represents the structure of the interaction between adjacent points, s is the rank of the matrix L, α and β are non-negative constants, d obeys the expectation Hm, and the covariance matrix is ​​diag(w 1 ,w 2 ,···w n )’s n-dimensional Gaussian distribution, w i is a hidden variable, w i Subordinate to exponential distribution w i ~Exponential(τ);

[0029] In step 6, based on the Kullback-Leibler divergence, the variational Bayesian inference method is used to transform the calculation problem of the posterior probability density function into a variational problem about the probability distribution, including:

[0030] Convert the posterior probability density function into an optimization problem Find an approximate distribution q(m,w,λ) in a family of easily computable probability distributions to approximate the true posterior probability density function p(m,w,λ|d).

[0031] Furthermore, in the above-mentioned inference method for the inverse problem of partial differential equations containing Laplace noise, in step eight, an appropriate approximation probability distribution family is selected to obtain a non-degenerate method that can simultaneously characterize the uncertainty of unknown variables, noise variance parameters and scale parameters, including: applying a mean field assumption to the unknown variable m, the noise variance parameter w and the scale parameter λ, the approximate distribution q(m,w,λ) can be separated into q(m)q(w)q(λ), the mean field assumption simultaneously characterizes the uncertainty of the unknown variable m, the noise variance parameter w and the scale parameter λ, and the approximation under this condition is called a non-degenerate approximation;

[0032] In step eight, an appropriate family of approximate probability distributions is selected to obtain a degenerate method that can only characterize the uncertainty of the noise variance parameter w and the scale parameter λ, including: adding additional assumptions to reduce computational complexity, and when q(m) is degraded, that is, when q(m) = 1, only the uncertainty of the noise variance parameter w and the scale parameter λ can be characterized. The approximation under this condition is called degenerate approximation.

[0033] Furthermore, in the above-mentioned inference method for the inverse problem of partial differential equations containing Laplace noise, the non-degenerate approximation comprises the following steps:

[0034] (1) The solution for calculating the approximate distribution is q(m,w,λ)=q * (m)q * (w)q * (λ), where

[0035] (2) Calculate the logarithmic function of the posterior probability density function ln p(m,w,λ|d):

[0036]

[0037] (3) Inversion of unknown variable m:

[0038] Substituting the formula in (2) into the first formula in (1), we get:

[0039]

[0040] Among them, X(w,λ) is a term independent of m;

[0041] make Then the first two terms of (3) are simplified to:

[0042] Then let m * =(H T H+η * L T L) -1 H T d, then the above formula is:

[0043]

[0044] The last term contributes to the normalization constant, m follows the mean m * , the variance is (H T W * H+λ * L T L) -1 Gaussian distribution,

[0045] Right now:

[0046] (4) Inversion noise variance parameter w:

[0047] Substituting the formula in (2) into the second formula in (1), we get:

[0048]

[0049] Among them, T(m,λ) is the term in ln p(m,w,λ) that is independent of w;

[0050] Make a transformation z follows a mean of μ z , with variance λ z The inverse Gaussian distribution of is: in,

[0051] (5) Inversion scale parameter λ: Substituting the formula in (2) into the third formula in (1), we get:

[0052]

[0053] Among them, F(m,w) is a term independent of λ, and λ obeys the mean The covariance is The gamma distribution is:

[0054] (6) Assume the parameter τ, which is related to the latent variable w i Directly related, carefully adjusted in specific applications to obtain reasonable results;

[0055] (7) Through the parameter inversion of steps (3) to (5), the non-explicit analytical solution of the marginal distribution of each variable of the approximate distribution q(m, w, λ) is obtained; when solving q(m), q(w) and q(λ) respectively, the alternating iteration method is adopted, that is, firstly initializing the parameters involved in q(m), q(w), q(λ) and q(m), q(w), q(λ), and then starting the alternating iterative calculation from one of q(m), q(w), q(λ) and the parameters involved until the algorithm converges; the alternating iterative algorithm estimates q(m), q(w) and q(λ);

[0056] Among them, w is a random variable that follows an exponential distribution; d follows an expectation of Hm and a covariance matrix of diag(w 1 ,w 2 ,···w n )’s n-dimensional Gaussian distribution, w i is a hidden variable, w i Subordinate to exponential distribution w i ~Exponential(τ), W is a diagonal matrix with w as diagonal element, norm Defined as (α, β) is the parameter pair of the gamma distribution, m is the unknown variable, d is the observed data, λ is the scale parameter describing the strength of the interaction, the matrix L represents the structure of the interaction between adjacent points, s is the rank of the matrix L, and α and β are non-negative constants.

[0057] Furthermore, in the above-mentioned inference method for the inverse problem of partial differential equation containing Laplace noise, the degenerate approximation comprises the following steps:

[0058] (1) To further reduce the computational complexity, we add the assumption that q(m) is degenerate based on the conditional independence assumptions for m, w, and λ, i.e., q(m) = 1. For a fixed m, minimizing the KL divergence can be transformed into the formula: Introducing the Lagrangian function of KL divergence The definition is as follows:

[0059]

[0060] Through calculation, we can get the solution:

[0061] m * =(H T H+λ * L T L) -1 H T d

[0062] Similar to the calculation in the non-degenerate approximation, we get:

[0063]

[0064] (2) Specifically estimate the noise variance parameter w,

[0065] make We get z to obey the mean μ z , with variance λ z The inverse Gaussian distribution of

[0066]

[0067] (3) Specifically estimate the scale parameter λ,

[0068] λ follows the mean The covariance is The gamma distribution of

[0069] Right now Where m * =(H T H+λ * L T L) -1 H T d;

[0070] (4) Using an alternating iterative algorithm to perform a degenerate approximate estimate of q(w) and q(λ);

[0071] Among them, w is a random variable that follows an exponential distribution; d follows an expectation of Hm and a covariance matrix of diag(w 1 ,w 2 ,···w n ), n-dimensional Gaussian distribution, w i is a hidden variable, w i Subordinate to exponential distribution w i ~Exponential(τ), W is a diagonal matrix with w as diagonal element, norm Defined as (α, β) is the parameter pair of the gamma distribution, m is the unknown variable, d is the observed data, λ is the scale parameter describing the strength of the interaction, the matrix L represents the structure of the interaction between adjacent points, s is the rank of the matrix L, and α and β are non-negative constants.

[0072] The beneficial effects of the present invention are as follows: the present invention assumes that the noise obeys the Laplace distribution, uses the Gaussian scale mixture representation of the Laplace distribution to re-express the inverse problem as a multi-layer Bayesian model, and simultaneously inverts the solution and the approximate distribution of the noise variance parameter and the scale parameter; by selecting different approximate probability distribution families, a non-degenerate method that can simultaneously characterize the uncertainty of the solution and hyperparameters and a degenerate method that only characterizes the uncertainty of the hyperparameters are given. Iterative loop solution achieves faster estimation speed and stronger robustness under the premise of higher estimation precision and accuracy, fully overcomes the problem of sensitivity to abnormal data and poor robustness under the Gaussian noise assumption, and is more suitable for engineering applications with real-time requirements. BRIEF DESCRIPTION OF THE DRAWINGS

[0073] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the conventional technology, the drawings required for use in the embodiments or the conventional technology descriptions are briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying any creative work.

[0074] Figure 1 A flow chart of an inference method for an inverse problem of a partial differential equation containing Laplace noise provided by an embodiment of the present invention;

[0075] Figure 2 is the calculation area Ω;

[0076] Figure 3 When θ(x)=sin(πx)e π +x+1, when the noise variance parameter is 0.5, the implementation effect of the non-degenerate and degenerate model algorithms under the Gaussian model and Laplace model;

[0077] Figure 4 When θ(x)=sin(πx)e π +x+1, Laplace noise with a noise variance parameter of 5, the implementation effects of the non-degenerate and degraded model algorithms under the Gaussian model and Laplace model;

[0078] Figure 5 When When the noise variance parameter is 0.5, the implementation effect of the non-degenerate and degenerate model algorithms under the Gaussian model and Laplace model;

[0079] Figure 6 When When the noise variance parameter is 5, the implementation effect of the non-degenerate and degraded model algorithms under the Gaussian model and Laplace model. DETAILED DESCRIPTION

[0080] In order to make the above-mentioned objects, features and advantages of the present invention more obvious and easy to understand, the specific embodiments of the present invention are described in detail below in conjunction with the accompanying drawings. In the following description, many specific details are set forth to facilitate a full understanding of the present invention. However, the present invention can be implemented in many other ways different from those described herein, and those skilled in the art can make similar improvements without violating the connotation of the present invention, so the present invention is not limited by the specific implementation disclosed below.

[0081] Unless otherwise defined, all technical and scientific terms used herein have the same meaning as those commonly understood by those skilled in the art of the present invention. The terms used herein in the specification of the present invention are only for the purpose of describing specific embodiments and are not intended to limit the present invention. The term "and / or" used herein includes any and all combinations of one or more related listed items.

[0082] Figure 1 A flow chart of an inference method for an inverse problem of a partial differential equation containing Laplace noise provided by an embodiment of the present invention.

[0083] An inference method for an inverse problem of a partial differential equation of Laplace noise provided by an embodiment of the present invention is combined with Figure 1 , including eight steps from S101 to S108:

[0084] S101: Assuming that the observation noise follows a Laplace distribution, the probability density function of the noise is represented as a mixture of Gaussian scales.

[0085] S102: Obtain likelihood function;

[0086] S103: Using Markov random fields to represent the prior distribution of unknown variables;

[0087] S104: Modeling the scale parameter as a random variable using multi-layer modeling;

[0088] S105: Remodel the posterior probability density function of the inverse problem into a multi-layer Bayesian model through a Gaussian scaled mixture representation of the Laplace distribution;

[0089] S106: Based on Kullback-Leibler divergence, the variational Bayesian inference method is used to transform the calculation problem of the posterior probability density function into a variational problem about probability distribution;

[0090] S107: Apply the mean field hypothesis to find an approximate estimate of the true posterior probability distribution in the probability distribution family, and invert the approximate distribution of the unknown variables, noise variance parameters, and scale parameters;

[0091] S108: Select an appropriate family of approximate probability distributions to obtain a non-degenerate method that can simultaneously characterize the uncertainty of unknown variables, noise variance parameters and scale parameters, and a degenerate method that only characterizes the uncertainty of noise variance parameters and scale parameters.

[0092] Furthermore, in the above-mentioned inference method for the inverse problem of partial differential equations containing Laplace noise, it is assumed in step S101 that the observation noise obeys the Laplace distribution, and the probability density function of the noise is expressed as a mixture of Gaussian scales, which includes the following two steps:

[0093] Step (1) establishes a variational Bayesian model for the inverse problem with Laplace noise. For a finite-dimensional linear inverse problem: Hm = d, where Represent the system matrix, unknown variables, and observed data respectively; let d = d + +ζ, where d + represents noise-free data, ζ represents noise, ζ=[ζ 1 ,ζ 2 ,···ζ n ] T , each component ζ i are independent and identically distributed with an expectation of 0 and a variance of The Laplace distribution of Its probability density function is expressed as:

[0094] Step (2) expresses the probability density function of the noise ζ as a mixture of Gaussian scales, specifically in the form of:

[0095]

[0096] Among them, i The expected value is 0 and the variance is w i The Gaussian distribution of w i is a hidden variable, w i Subordinate to exponential distribution w i ~Exponential(τ), its probability density function is w=[w 1 ,w 2 ,···w n ] T , the probability density function of w

[0097] Furthermore, in the above-mentioned inference method for the inverse problem of partial differential equation containing Laplace noise, obtaining the likelihood function in step S102 includes:

[0098] When considering the additive noise model, the likelihood function is a translation of the noise ζ probability density function and can be expressed as: That is, the probability density function of d when m and w are known, where m is the unknown variable, d is the observed data, and w is a random variable that obeys an exponential distribution; d obeys the expectation Hm and the covariance matrix diag(w 1 ,w 2 ,···w n ), n-dimensional Gaussian distribution, w i is a hidden variable, w i Subordinate to exponential distribution w i ~Exponential(τ).

[0099] Furthermore, in the above-mentioned inference method for the inverse problem of partial differential equations containing Laplace noise, the prior distribution of unknown variables is represented by using a Markov random field in step S103, including:

[0100] The Markov random field, a priori modeling tool, is used to represent the prior distribution p(m|λ) of the unknown variable m, which is in the form of: Among them, λ is the scale parameter describing the strength of interaction, the matrix L represents the structure of interaction between adjacent points, and s is the rank of the matrix L;

[0101] In step S104, the scale parameter is modeled as a random variable using multi-layer modeling, including: selecting a gamma distribution G(λ; α, β) as the prior distribution of the scale parameter λ: Here, Γ(·) is the standard gamma function, and α and β are non-negative constants.

[0102] Furthermore, in the above-mentioned inference method for the inverse problem of partial differential equations containing Laplace noise, in step S105, the posterior probability density function of the inverse problem is remodeled as a multi-layer Bayesian model by using the Gaussian scale mixture representation of the Laplace distribution, including: using the multi-layer Bayesian modeling to express the posterior probability density function as:

[0103]

[0104] Where w is the noise variance parameter, W is a diagonal matrix with w as diagonal elements, and the norm Defined as is the parameter pair of the gamma distribution, m is the unknown variable, d is the observed data, λ is the scale parameter describing the strength of the interaction, the matrix L represents the structure of the interaction between adjacent points, s is the rank of the matrix L, α and β are non-negative constants, d obeys the expectation Hm, and the covariance matrix is ​​diag(w1 ,w 2 ,···w n )’s n-dimensional Gaussian distribution, w i is a hidden variable, w i Subordinate to exponential distribution w i ~Exponential(τ);

[0105] In step S106, based on the Kullback-Leibler divergence, a variational Bayesian inference method is used to transform the calculation problem of the posterior probability density function into a variational problem about probability distribution, including:

[0106] Convert the posterior probability density function into an optimization problem Find an approximate distribution q(m,w,λ) in a family of easily computable probability distributions to approximate the true posterior probability density function p(m,w,λ|d).

[0107] Further, in the above-mentioned inference method for the inverse problem of partial differential equations containing Laplace noise, in step S108, an appropriate approximation probability distribution family is selected to obtain a non-degenerate method that can simultaneously characterize the uncertainty of unknown variables, noise variance parameters and scale parameters, including: applying a mean field assumption to the unknown variable m, the noise variance parameter w and the scale parameter λ, the approximate distribution q(m,w,λ) can be separated into q(m)q(w)q(λ), the mean field assumption simultaneously characterizes the uncertainty of the unknown variable m, the noise variance parameter w and the scale parameter λ, and the approximation under this condition is called a non-degenerate approximation;

[0108] In step S108, an appropriate approximate probability distribution family is selected to obtain a degradation method that can only characterize the uncertainty of the noise variance parameter w and the scale parameter λ, including: adding additional assumptions to reduce computational complexity, and when q(m) is degraded, that is, when q(m) = 1, only the uncertainty of the noise variance parameter w and the scale parameter λ can be characterized. The approximation under this condition is called a degenerate approximation.

[0109] Furthermore, in the above-mentioned inference method for the inverse problem of partial differential equations containing Laplace noise, the non-degenerate approximation comprises the following steps:

[0110] (1) The solution for calculating the approximate distribution is q(m,w,λ)=q * (m)q * (w)q * (λ), where

[0111]

[0112] (2) Calculate the logarithmic function of the posterior probability density function lnp(m,w,λd):

[0113]

[0114] (3) Inversion of unknown variable m:

[0115] Substituting the formula in (2) into the first formula in (1), we get:

[0116]

[0117]

[0118] Among them, X(w,λ) is a term independent of m;

[0119] make Then the first two terms of (3) are simplified to:

[0120] Then let m * =(H T H+η * L T L) -1 H T d, then the above formula is:

[0121]

[0122] The last term contributes to the normalization constant, m follows the mean m * , the variance is (H T W * H+λ * L T L) -1 Gaussian distribution,

[0123] Right now:

[0124] (4) Inversion noise variance parameter w:

[0125] Substituting the formula in (2) into the second formula in (1), we get:

[0126]

[0127] Among them, T(m,λ) is the term in lnp(m,w,λ) that is independent of w;

[0128] Make a transformation z follows a mean of μ z , with variance λ z The inverse Gaussian distribution of is: in,

[0129] (5) Inversion scale parameter λ: Substituting the formula in (2) into the third formula in (1), we get:

[0130]

[0131] Among them, F(m,w) is a term independent of λ, and λ obeys the mean The covariance is The gamma distribution is:

[0132] (6) Assume the parameter τ, which is related to the latent variable w i Directly related, carefully adjusted in specific applications to obtain reasonable results;

[0133] (7) Through the parameter inversion of steps (3) to (5), the non-explicit analytical solution of the marginal distribution of each variable of the approximate distribution q(m, w, λ) is obtained; when solving q(m), q(w) and q(λ) respectively, the alternating iteration method is adopted, that is, firstly initializing the parameters involved in q(m), q(w), q(λ) and q(m), q(w), q(λ), and then starting the alternating iterative calculation from one of q(m), q(w), q(λ) and the parameters involved until the algorithm converges; the alternating iterative algorithm estimates q(m), q(w) and q(λ);

[0134] Specifically, in the embodiment of the present invention, the steps of estimating q(m), q(w) and q(λ) by the alternating iterative algorithm are as follows:

[0135] (7a) Initialize the parameters involved in q(m), q(w), q(λ) and q(m), q(w), q(λ), and input the initial value q 0 (w) and q 0 (λ), let k = 0;

[0136] (7b) Let k = k + 1;

[0137] (7c) Calculate q k (m)

[0138]

[0139]

[0140] Where m k =(H T W k H+λ k L T L) -1 H T W k d,

[0141] (7d) Calculate q k (w)

[0142]

[0143]

[0144] in

[0145] (7e) Calculate q k (λ)

[0146]

[0147]

[0148] (7f) Setting stopping criteria;

[0149] (7g) Output q k (m)q k (w)q k (λ);

[0150] Among them, w is a random variable that follows an exponential distribution; d follows an expectation of Hm and a covariance matrix of diag(w 1 ,w 2 ,···w n ), n-dimensional Gaussian distribution, w i is a hidden variable, w i Subordinate to exponential distribution w i ~Exponential(τ), W is a diagonal matrix with w as diagonal element, norm Defined as (α, β) is the parameter pair of the gamma distribution, m is the unknown variable, d is the observed data, λ is the scale parameter describing the strength of the interaction, the matrix L represents the structure of the interaction between adjacent points, s is the rank of the matrix L, and α and β are non-negative constants.

[0151] Furthermore, in the above-mentioned inference method for the inverse problem of partial differential equation containing Laplace noise, the degenerate approximation comprises the following steps:

[0152] (1) To further reduce the computational complexity, we add the assumption that q(m) is degenerate based on the conditional independence assumptions for m, w, and λ, i.e., q(m) = 1. For a fixed m, minimizing the KL divergence can be transformed into the formula: Introducing the Lagrangian function of KL divergence The definition is as follows:

[0153]

[0154] Through calculation, we can get the solution:

[0155] m *=(H T H+λ * L T L) -1 H T d

[0156] Similar to the calculation in the non-degenerate approximation, we get:

[0157]

[0158] (2) Specifically estimate the noise variance parameter w,

[0159] make We get z to obey the mean μ z , with variance λ z The inverse Gaussian distribution of

[0160] m * =(H T H+λ * L T L) -1 H T d;

[0161] (3) Specifically estimate the scale parameter λ,

[0162] λ follows the mean The covariance is The gamma distribution of

[0163] Right now Where m * =(H T H+λ * L T L) -1 H T d;

[0164] (4) Using an alternating iterative algorithm to perform a degenerate approximate estimate of q(w) and q(λ);

[0165] Specifically, in the embodiment of the present invention, the steps of using the alternating iterative algorithm to perform degenerate approximate estimation on q(w) and q(λ) are as follows:

[0166] (4a) Initialize the parameters involved in q(m), q(w), q(λ) and q(m), q(w), q(λ), and input the initial value q 0 (w) and q 0 (λ), let k = 0;

[0167] (4b) Let k = k + 1;

[0168] (4c) Calculate m k ;

[0169] m k =(H T H+λ * L T L) -1 H T d

[0170] (4d) Calculate q k (w)

[0171]

[0172]

[0173] in

[0174] (4e) Calculate q k (λ)

[0175]

[0176]

[0177] (4f) Setting stopping criteria;

[0178] (4g) Output q k (w)q k (λ);

[0179] Among them, w is a random variable that follows an exponential distribution; d follows an expectation of Hm and a covariance matrix of diag(w 1 ,w 2 ,···w n )’s n-dimensional Gaussian distribution, w i is a hidden variable, w i Subordinate to exponential distribution w i ~Exponential(τ), W is a diagonal matrix with w as diagonal element, norm Defined as (α, β) is the parameter pair of the gamma distribution, m is the unknown variable, d is the observed data, λ is the scale parameter describing the strength of the interaction, the matrix L represents the structure of the interaction between adjacent points, s is the rank of the matrix L, and α and β are non-negative constants.

[0180] The following is a specific example of a Cauchy-type problem to illustrate the performance of the variational Bayesian inference method with Laplace noise inverse problem. In the heat transfer process, Cauchy-type problems often arise in which the unknown boundary conditions on the boundary part are determined by the overdetermined boundary data of the remaining part. For example, in the study of reentry space shuttles, it is necessary to infer the temperature of the outer surface of the space shuttle based on the measurement of the temperature of the inner surface. In the quenching process, it is necessary to estimate the convective heat transfer coefficient through the boundary by measuring the temperature and flow rate of the accessible part of the boundary.

[0181] The steady-state heat conduction problem can be expressed as follows:

[0182]

[0183] where α(x) is the conductivity and f(x) is the source term.

[0184] The restrictions of the above formula are as follows:

[0185]

[0186] Where n is the direction perpendicular to Γ c The inverse problem is to find the boundary Γ i The unknown temperature u(x) on the surface.

[0187] Consider the Laplace equation, that is, let α=1, f=0 in the above formula. Figure 2 To calculate the area Ω, Ω is defined as a square unit open domain (0,1)×(0,1), and the inaccessible boundary part Γ is experimentally measured i and the accessible boundary portion Γ c , respectively defined as: Γ i =[0,1]×{1},Γ c =Γ\Γ i , boundary Γ 0 Defined as Γ 0 ={0,1}×(0,1). The problem is discretized using 3200 triangular finite elements. 0 Take up to 80 measurements.

[0188] Consider the case of reconstructing a smooth solution, let θ(x) = sin(πx)e π +x+1. When the noise is Gaussian noise, by comparing the reconstruction of the true solution by non-degenerate and degenerate approximation assuming Gaussian noise and Laplace noise under different noise variance parameters, it is found that the difference is not obvious, and both can achieve good reconstruction results. The subtle difference is that the degenerate approximation under the two noise assumptions is more accurate and more stable than the non-degenerate approximation algorithm. When the noise variance parameter is 0.5, the implementation effect under the Gaussian model and Laplace model is as follows Figure 3As shown, the comparison between the consumed time and the reconstruction error is shown in Table 1. Figure 3 In the figure, the solid black line 1 is the true solution, the solid black line 2 is the non-degenerate approximation assuming that the noise is Gaussian noise, the dashed black line 1 is the degenerate approximation assuming that the noise is Gaussian noise, the dashed black line 2 is the non-degenerate approximation assuming that the noise is Laplace noise, and the dashed black line 3 is the degenerate approximation assuming that the noise is Laplace noise. In the figure below, unless otherwise specified, the lines in the figure have the same meaning. Figure 5 From Table 1, we can conclude that all four approximations can quickly reconstruct the true solution with relatively small errors. In comparison, the degenerate approximation is more accurate and more stable.

[0189]

[0190] Table 1

[0191] When the noise is Laplace noise, by comparing the reconstruction of the true solution by assuming Gaussian noise and Laplace noise under different noise variance parameters, it is found that the model assuming Gaussian noise can no longer reconstruct the true solution well, while the Laplace model proposed in the present invention can reconstruct the true solution with high accuracy. When the noise variance parameter is 5, the implementation effects under the Gaussian model and the Laplace model are as follows Figure 4 As shown, the comparison of the consumed time and reconstruction error is shown in Table 2. Figure 4 It can be concluded from Table 2 that when the noise is Laplace noise, the degenerate approximation algorithm of the Gaussian noise model is completely unable to reconstruct the true solution, and the reconstruction error of the non-degenerate approximation algorithm is large. However, the two Laplace model approximation algorithms proposed in the present invention both achieve the reconstruction of the true solution with very high accuracy and speed.

[0192]

[0193] Table 2

[0194] Consider the case of reconstructing a non-smooth solution, let Since the equation cannot be solved analytically, in order to avoid inversion errors, the numerical solution of the forward problem on the fine grid is used as the precise data and inverted on the coarse grid. When the noise is Gaussian noise, by comparing the reconstruction of the true solution by assuming non-degenerate and degenerate approximations of Gaussian noise and Laplace noise under different noise variance parameters, it is found that the difference is not obvious and both can achieve good reconstruction effects. When the noise variance parameter is 0.5, the implementation effects under the Gaussian model and Laplace model are as follows Figure 5 As shown, the comparison of the consumed time and reconstruction error is shown in Table 3. Figure 5 From Table 3, we can conclude that all four approximations can quickly reconstruct the true solution with relatively small errors.

[0195]

[0196] Table 3

[0197] When the noise is Laplace noise, by comparing the reconstruction of the true solution by assuming Gaussian noise and Laplace noise under different noise variance parameters, it is found that the model assuming Gaussian noise can no longer reconstruct the true solution, while the Laplace model proposed in the present invention can reconstruct the true solution with high accuracy. When the noise variance parameter is 5, the implementation effects under the Gaussian model and the Laplace model are as follows Figure 6 As shown, the comparison of the consumed time and reconstruction error is shown in Table 4. Figure 6 It can be concluded from Table 4 that when the noise is Laplace noise, the two approximate algorithms of the Gaussian noise model are completely unable to reconstruct the true solution, while the two Laplace model approximate algorithms proposed in the present invention both achieve the reconstruction of the true solution with very high accuracy and speed.

[0198]

[0199] Table 4

[0200] Through practice, the application of the present invention can fully overcome the problems of the model being sensitive to abnormal data and having poor robustness under the Gaussian noise assumption, and effectively improve the accuracy and robustness of the calculation results.

[0201] Those skilled in the art will appreciate that although some embodiments described herein include certain features included in other embodiments but not other features, the combination of features from different embodiments is meant to be within the scope of the present invention and to form different embodiments.

[0202] Those skilled in the art will appreciate that the description of each embodiment has its own emphasis, and for parts not described in detail in a certain embodiment, reference can be made to the relevant descriptions of other embodiments.

[0203] The above are only specific embodiments of the present invention, but the protection scope of the present invention is not limited thereto. Any technician familiar with the technical field can easily think of various equivalent modifications or replacements within the technical scope disclosed by the present invention, and these modifications or replacements should be included in the protection scope of the present invention. Therefore, the protection scope of the present invention shall be based on the protection scope of the claims.

Claims

1. An inference method for an inverse problem of a partial differential equation containing Laplace noise, characterized in that: In the study of reentry space shuttles, it is necessary to infer the temperature of the outer surface of the space shuttle based on the measurement of the temperature of the inner surface. During the quenching process, it is necessary to estimate the convective heat transfer coefficient through the boundary by measuring the temperature and flow rate of the accessible part of the boundary. The inverse problem is to find the unknown temperature on the boundary, which includes the following eight steps: Step 1: Assume that the temperature observation noise follows the Laplace distribution and express the probability density function of the noise as a mixture of Gaussian scales; Step 2: Obtain likelihood function; Step 3: Use Markov random fields to represent the prior distribution of unknown variables; Step 4: Use multi-level modeling to model the scale parameter as a random variable; Step 5: Remodel the posterior probability density function of the inverse problem into a multi-layer Bayesian model through the Gaussian scale mixture representation of the Laplace distribution; Step 6: Based on the Kullback-Leibler divergence, the variational Bayesian inference method is used to transform the calculation problem of the posterior probability density function into a variational problem about probability distribution; Step 7: Apply the mean field hypothesis, find an approximate estimate of the true posterior probability distribution in the probability distribution family, and invert the approximate distribution of the unknown variables, noise variance parameters, and scale parameters; Step 8: Select an appropriate family of approximate probability distributions to obtain a non-degenerate method that can simultaneously characterize the uncertainty of unknown variables, noise variance parameters, and scale parameters, as well as a degenerate method that only characterizes the uncertainty of noise variance parameters and scale parameters; In step 1, it is assumed that the temperature observation noise obeys the Laplace distribution, and the probability density function of the noise is expressed as a mixture of Gaussian scales, which includes the following two steps: Step (1) establishes a variational Bayesian model for the inverse problem with Laplace noise. For a finite-dimensional linear inverse problem: Hm = d, where Represent the system matrix, unknown variables, and temperature observation data respectively; Let d = d + +ζ, where d + represents noise-free data, ζ represents noise, ζ=[ζ1,ζ2,···ζ n ] T , each component ζ i are independent and identically distributed with an expectation of 0 and a variance of The Laplace distribution of Its probability density function is expressed as: Step (2) expresses the probability density function of the noise ζ as a mixture of Gaussian scales, specifically in the form of: Among them, i The expected value is 0 and the variance is w i The Gaussian distribution of w i is a hidden variable, w i Subordinate to exponential distribution w i ~Exponential(τ), its probability density function is w=[w1,w2,···w n ] T , the probability density function of w Obtaining the likelihood function in step 2 includes: When considering the additive noise model, the likelihood function is a translation of the noise ζ probability density function and can be expressed as: That is, the probability density function of d when m and w are known, where w is a random variable that obeys an exponential distribution; d obeys expectation Hm and covariance matrix diag(w1,w2,···w n )’s n-dimensional Gaussian distribution; In the step 3, the Markov random field is used to represent the prior distribution of the unknown variable, including: The Markov random field, a priori modeling tool, is used to represent the prior distribution p(m|λ) of the unknown variable m, which is in the form of: Among them, λ is the scale parameter describing the strength of interaction, the matrix L represents the structure of interaction between adjacent points, and s is the rank of the matrix L; In the step 4, the scale parameter is modeled as a random variable using multi-layer modeling, including: selecting a gamma distribution G(λ; α, β) as the prior distribution of the scale parameter λ: Where Γ(·) is the standard gamma function, α and β are non-negative constants; In the step 5, the posterior probability density function of the inverse problem is remodeled into a multi-layer Bayesian model by using the Gaussian scale mixture representation of the Laplace distribution, including: using the multi-layer Bayesian modeling to express the posterior probability density function as: Where w is the noise variance parameter, W is a diagonal matrix with w as diagonal elements, and the norm Defined as (α, β) is the parameter pair of the gamma distribution; In step 6, based on the Kullback-Leibler divergence, a variational Bayesian inference method is used to transform the calculation problem of the posterior probability density function into a variational problem about probability distribution, including: Convert the posterior probability density function into an optimization problem Find an approximate distribution q(m,w,λ) in a family of easily computable probability distributions to approximate the true posterior probability density function p(m,w,λ|d); In the step eight, an appropriate approximate probability distribution family is selected to obtain a non-degenerate method that can simultaneously characterize the uncertainty of the unknown variable, the noise variance parameter and the scale parameter, including: applying a mean field assumption to the unknown variable m, the noise variance parameter w and the scale parameter λ, the approximate distribution q(m,w,λ) can be separated into q(m)q(w)q(λ), and the mean field assumption simultaneously characterizes the unknown variable m, the noise variance parameter w and the scale parameter λ. w and the uncertainty of the scale parameter λ. The approximation under this condition is called a non-degenerate approximation. In step 8, an appropriate approximate probability distribution family is selected to obtain a degradation method that can only characterize the uncertainty of the noise variance parameter w and the scale parameter λ, including: adding additional assumptions to reduce the computational complexity, when q(m) is degraded, that is, when q(m) = 1, only the noise variance parameter can be characterized w and scale parameter λ are uncertain. The approximation under this condition is called degenerate approximation.

2. The inference method for an inverse problem of a partial differential equation containing Laplace noise according to claim 1, characterized in that: The non-degenerate approximation includes the following steps: (1) The solution for calculating the approximate distribution is q(m,w,λ)=q * (m)q * (w)q * (λ), where (2) Calculate the logarithmic function of the posterior probability density function lnp(m,w,λd): (3) Inversion of unknown variable m: Substituting the formula in (2) into the first formula in (1), we get: Among them, X(w,λ) is a term independent of m; make Then the first two terms of (3) are simplified to: Then let m * =(H T H+η * L T L) -1 H T d, then the above formula is: The last term contributes to the normalization constant, m follows the mean m * , the variance is (H T W * H+λ * L T L) -1 Gaussian distribution, Right now: (4) Inversion noise variance parameter w: Substituting the formula in (2) into the second formula in (1), we get: Among them, T(m,λ) is the term in lnp(m,w,λ) that is independent of w; Make a transformation z follows a mean of μ z , with variance λ z The inverse Gaussian distribution of is: in, (5) Inversion scale parameter λ: Substituting the formula in (2) into the third formula in (1), we get: Among them, F(m,w) is a term independent of λ, and λ obeys the mean The covariance is The gamma distribution is: (6) Assume the parameter τ, which is related to the latent variable w i Directly related, carefully adjusted in specific applications to obtain reasonable results; (7) Through the parameter inversion of steps (3) to (5), the non-explicit analytical solution of the marginal distribution of each variable of the approximate distribution q(m,w,λ) is obtained; when solving q(m), q(w) and q(λ) respectively, the alternating iteration method is adopted, that is, firstly initialize the parameters involved in q(m), q(w), q(λ) and q(m), q(w), q(λ), and then start the alternating iterative calculation from one of q(m), q(w), q(λ) and the parameters involved until the algorithm converges; the alternating iterative algorithm estimates q(m), q(w) and q(λ).

3. The inference method for an inverse problem of a partial differential equation containing Laplace noise according to claim 1, characterized in that: The degenerate approximation includes the following steps: (1) To further reduce the computational complexity, we add the assumption that q(m) is degenerate on the basis of the conditional independence assumptions for m, w, and λ, that is, q(m) = 1. For a fixed m, we consider minimizing the KL divergence and convert it into the formula: Introducing the Lagrangian function of KL divergence The definition is as follows: Through calculation, we can get the solution: m * =(H T H+λ * L T L) -1 H T d Similar to the calculation in the non-degenerate approximation, we get: (2) Specifically estimate the noise variance parameter w, make We get z to obey the mean μ z , with variance λ z The inverse Gaussian distribution of (3) Specifically estimate the scale parameter λ, λ follows the mean The covariance is The gamma distribution of (4) An alternating iterative algorithm is used to perform degenerate approximate estimation of q(w) and q(λ).

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