A large-scale space random variation feature generation method based on mesh pseudo point approximation

Abstract

The application discloses a large-scale space random variation characteristic generation method based on grid pseudo-point approximation, and belongs to the field of non-homogeneous material space variation characteristic generation. The method utilizes pseudo-points to reduce the size of a covariance matrix, simultaneously utilizes a linear conjugate gradient method to solve a self-covariance matrix of the grid pseudo-points, reduces the operation amount of self-covariance matrix inversion and the calculation amount of self-covariance matrix decomposition in a simulation process, and can greatly improve the generation efficiency of a Gaussian random field. The large-scale non-homogeneous material space variation characteristic generation model of the Gaussian random field based on grid pseudo-point approximation constructed by the method can realize the generation of a large-scale Gaussian random field based on grid pseudo-points, utilizes an equivalent self-covariance function of a sparse power expectation propagation method to derive a Gaussian random field posterior mean function and a self-covariance function based on grid pseudo-point approximation, adopts Kronecker operation to realize fast decomposition of a posterior self-covariance matrix, and greatly improves the generation efficiency of a multi-dimensional large-scale random field.

Description

TECHNICAL FIELD

[0001] The present application belongs to the field of generating spatial variation characteristics of non-homogeneous materials, and more particularly relates to a large-scale spatial random variation characteristic generation method based on grid pseudo-point approximation. BACKGROUND

[0002] The random field model is commonly used to describe complex data distribution with spatial or structural dependence. In the field of civil engineering, the spatial variation characteristics of various materials in the engineering environment play a crucial role in understanding the mechanical properties, thermal conduction characteristics, and behavior in complex environments of the materials. However, natural materials and artificial composite materials often exhibit highly non-homogeneous characteristics, which show multi-scale, complex and unpredictable random distribution in space. Therefore, how to effectively generate and simulate the spatial variation characteristics of these non-homogeneous materials has become an important challenge in civil engineering research. Traditional material simulation methods often rely on a large amount of experimental data and empirical formulas, which not only consumes time and effort, but also is difficult to fully capture the subtle variations of materials in space. In recent years, with the rapid development of computer technology, especially the popularization of random field technology, it has become possible to generate material data with specific statistical properties through random field.

[0003] Gaussian random field has a flexible autocovariance function and is considered to be able to handle large data sets, but when the data set contains n data, it requires O(n 3 ) computation time and O(n 2 ) storage time, which makes Gaussian process only suitable for handling small data sets. When the data dimension is greater than 1, the computation, computation time and storage of Gaussian random field usually grow exponentially rather than linearly, which also hinders the use of Gaussian random field in large data sets. In the process of generating complex engineering material data, large data sets often need to be analyzed. For data sets subject to Gaussian random field, existing methods need to invert the covariance matrix calculated by the autocovariance function when calculating the posterior mean function and the autocovariance function. The size of this matrix is consistent with the length of the observed sample, so when the observed sample of engineering materials is large and the dimension is high, the matrix inversion calculation overhead is large. In addition, random field generation requires decomposition of the posterior covariance matrix, and the size of this matrix grows rapidly in multi-dimensional space, and the calculation overhead is large for high-density sampling with long time, space history and high sampling frequency. Therefore, more efficient posterior mean function, autocovariance function and random field generation method are needed. SUMMARY

[0004] In view of the above defects or improvement needs of the prior art, the present application provides a large-scale spatial random variation characteristic generation method based on grid pseudo-point approximation to improve the generation efficiency of Gaussian random field.

[0005] To achieve the above object, according to the first aspect of the present application, a large-scale non-homogeneous material spatial variation characteristic generation model construction method based on grid pseudo-point approximation Gaussian random field is provided, comprising:

[0006] S1, based on the observation point set x f , the corresponding real value set f, the pseudo-point set x u , the corresponding real value set u and the sampling point set , the corresponding real value set f * , the equivalent autocovariance matrix of Gaussian process is obtained by sparse pseudo-point approximation;

[0007] Wherein, f = H(x f ) is the corresponding real value of x f , y = f + ε, y is the observation value of x f , the observation error ε ~ N(0, ∑ ε ); u = H(x u ) is the corresponding real value of x u , is the corresponding real value, y = f * + ε, y * is the observation value of x * , H() is a D-dimensional random field, D > 1; the maximum value of each dimension in x * is greater than the maximum value of the corresponding dimension in x u , the minimum value of each dimension is less than the minimum value of the corresponding dimension in x f , and the spacing between each pseudo-point is the same to form a grid pseudo-point; The maximum value of each dimension in x f is less than the maximum value of the corresponding dimension in x u , and the minimum value of each dimension is greater than the minimum value of the corresponding dimension in x u .

[0008] S2, the equivalent autocovariance matrix is used to replace the autocovariance matrix in the dense calculation f * prediction distribution, so that the diagonal matrix items in the equivalent autocovariance matrix and the known autocovariance matrix of observation error appear in the posterior mean function and the posterior autocovariance function in the replaced f * prediction distribution S -1 ; the sum is decomposed into the product form of upper triangular matrix L and lower triangular matrix L T S -1 = LL T , and the posterior mean function and the posterior autocovariance function in the replaced f * prediction distribution are brought in, and sparse matrix is used to calculate xu with x f or x f* approximated by the self-covariance matrix of x

[0009]

[0010] wherein, and W f are sparse matrices; K ab = K(x a ,x b ), K(x a ,x b ) is a self-covariance matrix obtained by substituting x a ,x b into a set prior self-covariance function k(x i ,x j ), and x a ,x b are x f* , x u or x f ; is a non-negative diagonal matrix, and α ∈ [0, 1];

[0011] S3, each term in Σ f* is sampled and calculated to obtain which is added to μ f* to obtain a sampling formula of large-scale Gaussian random inhomogeneous material spatial variation characteristics:

[0012]

[0013] wherein, b1, b2,..., b D are D vectors about A conjugate in D dimensions, v1, v2, v3 are standard independent Gaussian random vectors,

[0014] According to the second aspect of the present application, a large-scale inhomogeneous material spatial variation characteristics generation method based on grid pseudo-point approximation Gaussian random field is provided, comprising:

[0015] obtaining a real value set f corresponding to an observation point set x f , a real value set u corresponding to a pseudo-point set x u , and a sampling point set corresponding to a real value set f * ;

[0016] The large-scale non-homogeneous material spatial variation characteristic generation model based on the grid pseudo point approximate Gaussian random field is constructed based on the method of the first aspect The non-homogeneous material spatial variation characteristics of each point in the middle.

[0017] According to a third aspect of the present application, an electronic device is provided, comprising: a computer readable storage medium and a processor;

[0018] The computer readable storage medium is used to store executable instructions;

[0019] The processor is used to read the executable instructions stored in the computer readable storage medium, and execute the method of the first aspect.

[0020] According to a fourth aspect of the present application, a computer readable storage medium is provided, the computer readable storage medium stores computer instructions, the computer instructions are used to make the processor execute the method of the first aspect.

[0021] Overall, compared with the prior art, the above technical solutions conceived by the present application can achieve the following beneficial effects:

[0022] The method provided by the present application reduces the size of the covariance matrix by using pseudo points, and simultaneously solves the autocovariance matrix of the grid pseudo points by using the linear conjugate gradient method, thereby reducing the operation amount of the autocovariance matrix inversion and the calculation amount of the autocovariance matrix decomposition in the simulation process, and greatly improving the generation efficiency of the Gaussian random field; the large-scale non-homogeneous material spatial variation characteristic generation model based on the grid pseudo point approximate Gaussian random field constructed by the method provided by the present application can realize the generation of the large-scale Gaussian random field of the grid pseudo point, derive the grid pseudo point approximate Gaussian process posterior mean function and the autocovariance function by using the equivalent autocovariance function of the sparse power expectation propagation method, generate the large-scale multi-dimensional Gaussian random field based on the posterior autocovariance matrix eigenvalue decomposition and Kronecker operation, and greatly improve the generation efficiency of the random field based on the grid pseudo point. BRIEF DESCRIPTION OF DRAWINGS

[0023] Figure 1 A large-scale non-homogeneous material spatial variation characteristic generation method flowchart based on a grid pseudo point approximate Gaussian random field is provided for the embodiment of the present application;

[0024] Figure 2 A mean value diagram of the Gaussian random field generated based on the grid pseudo point is provided for the embodiment of the present application;

[0025] Figure 3 A mean value diagram of the Gaussian random field generated by dense calculation is provided for the embodiment of the present application;

[0026] Figure 4 A sampling result schematic diagram of generating a large-scale Gaussian random field based on a grid pseudo point is provided for an embodiment of the present application. DETAILED DESCRIPTION

[0027] In order to make the purpose, technical solutions and advantages of the present application clearer and more apparent, the present application is further described in detail below in combination with the drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present application and do not limit the present application. In addition, the technical features involved in each embodiment of the present application described below can be combined with each other as long as they do not conflict with each other.

[0028] An embodiment of the present application provides a large-scale non-homogeneous material spatial variation characteristic generation model construction method based on a grid pseudo point approximation Gaussian random field, as shown in Figure 1 , which comprises:

[0029] S1, based on an observation point set x f , a corresponding true value set f, a pseudo point set x u , a corresponding true value set u and a sampling point set , a corresponding true value set f * , and an equivalent autocovariance matrix of a Gaussian process is obtained by sparse pseudo point approximation;

[0030] S2, the equivalent autocovariance matrix is used to replace the autocovariance matrix in the dense calculation f * prediction distribution, so that the posterior mean function and the posterior autocovariance function in the f * prediction distribution after replacement appear the sum between the diagonal matrix items in the equivalent autocovariance matrix and the autocovariance matrix of the known observation error S -1 ; the sum is decomposed into the product form of an upper triangular matrix L and a lower triangular matrix L T S -1 =LL T , and the posterior mean function and the posterior autocovariance function in the f * prediction distribution after replacement are brought in, and the autocovariance matrix of x u and x f or is approximated by a sparse matrix to approximately calculate the posterior mean function and the posterior autocovariance function, so as to obtain the Gaussian process posterior mean function and the posterior autocovariance function based on the grid pseudo point approximation;

[0031] S3, each item in the Gaussian process posterior autocovariance function obtained by the approximate calculation is sampled and calculated to obtain , which is combined with μ f*By adding them together, a large-scale inhomogeneous material spatial variation feature generation model based on grid pseudo-point approximation of Gaussian random field is obtained.

[0032] Specifically, through steps S1-S2, the derivation of the posterior mean function and the autocovariance function of the multi-dimensional Gaussian random field based on grid pseudo-point approximation is completed.

[0033] Given a D-dimensional random field H(X) (D>1), the parameter space is located on a Cartesian grid, that is, X=X (1) ×…×X (D) , X (d) Represents the input coordinate vector along the d dimension, the operator × represents the Cartesian product between vectors, and each X (d) The length is G d .

[0034] The prior mean function is m(x), and the prior autocovariance function is where x i ,x j For any two D-dimensional inputs belonging to X, x i (d) ∈X (d) is the input x i The data on the d-th dimension, k d (·,·) is the prior autocovariance function defined on the d-th dimension of the scalar input space. Substituting the input x into the prior autocovariance function, the autocovariance matrix calculated can be written as the Kronecker product of D smaller autocovariance matrices,

[0035] Each smaller autocovariance matrix is ​​obtained by (d) Substitute the prior autocovariance function k d (·,·) is calculated, that is, remember but where K is the final autocovariance matrix,

[0036] Each K d All G d ×G d Matrix with nature.

[0037] K(i',j') represents the first i The element in the row and column of K d (i' (d) ,j' (d) ) is the d-th matrix K d The i'th (d) ,j' (d) elements, and Observation data set F = {(x l ,y l )|l=1,......,n}, n is the number of observation points, where y=[y1,y2,......,y n ] T is the output, i.e. the observed value of the spatial variation characteristics of each observation point of the inhomogeneous material, It is the coordinate of the observation point set, that is, the location information of each observation point.

[0038] f=H(x f ) is the set of observation points x f The corresponding true value, and y=f+ε, where the observation error ε~N(0,Σ ε ), that is, the observation error ε has a mean of zero and an autocovariance of Σ ε Gaussian noise;

[0039] The coordinates of the pseudo-point set (i.e., grid pseudo-points) with equal spacing are selected as u=H(x u ) is a pseudo point set x u The corresponding true value, m is the number of pseudo points, m<n, and the pseudo point set x u The maximum value of each dimension in is greater than the maximum value of the corresponding dimension in the observation point set x, and the pseudo point set x u The minimum value of each dimension in is smaller than the minimum value of the corresponding dimension in the observation point set x.

[0040] Assume the coordinates of the sampling point set are f * =H(x f* ) is the sampling point set x f* The corresponding true value, and y * =f * +ε. The number of sampling points in the sampling point set is arbitrary and can be set according to actual prediction requirements.

[0041] Existing dense computation f * The predicted distribution of is:

[0042]

[0043] The equivalent autocovariance function (i.e., equivalent autocovariance matrix) derived from the sparse power expectation propagation method is:

[0044]

[0045] where δ(x,x')=diag(K(x,x')-K(x,x u )K -1 (x u ,x u )K(xu ,x')) is a non-negative diagonal matrix, α∈[0,1].

[0046] Remember K ab =K(x a ,x b ), K(x a ,x b ) is x a ,x b Substitute the set prior autocovariance function k(x i ,x j ) calculated autocovariance matrix, and x a ,x b is the sampling point set x * Or a pseudo-point set x u Or the set of observation points x f .

[0047] Dense calculation f * The autocovariance matrix of the predictive distribution of is K(x f ,x f ), K(x f* ,x f ), replace the above autocovariance matrix with the equivalent autocovariance matrix to obtain:

[0048]

[0049] Since after replacing f * In the predictive distribution of , the diagonal matrix entries exist in αδ(x f ,x f ),Σ ε and In the case of αδ(x f ,x f ) and ∑ ε Always appear together, so let αδ(x f ,x f )+∑ ε =S -1 , for S -1 Decompose, the decomposition result is the product of the lower triangular matrix and the upper triangular matrix, that is, S -1 =LL T , and then define the intermediate variable K uf (L T ) -1 =K us ,L -1 y=y s ,K su =K us T, then the derived posterior mean function is:

[0050]

[0051] Define Q = K uu +K us K su , then the derived posterior mean function is:

[0052]

[0053] And K(x b ,x u ) matrix can be obtained by two distances x b The nearest pseudo point and x b The matrix formed is approximated, x b is the sampling point set Or the set of observation points x f Any element in is represented as:

[0054] K bu ≈W b K uu

[0055] Among them, W b Is a sparse matrix. If local cubic interpolation is used, W b Each row has only 4 non-zero non-negative elements; if inverse distance weighted interpolation is used, W b Each row has only 2 non-zero non-negative elements. Generally speaking, in order to obtain higher accuracy, local cubic interpolation can be used on a grid with equal spacing. b represents the data set used, which can be x f or

[0056] Then K fu =W f K uu ,K uf =K uu W f T , Then the posterior mean function is:

[0057]

[0058] αδ(x,x)+∑ ε =S -1 Substitute into the posterior autocovariance function, then:

[0059]

[0060] and Use the intermediate variable K uf (LT -1 us -1 s su us T The posterior autocovariance function is:

[0061]

[0062] K fu f K uu uf uu f T , Substituting the posterior autocovariance function, we get:

[0063]

[0064] In step S3, the posterior mean vector and the autocovariance matrix are quickly calculated using the pseudo-point grid structure and the Kronecker operation.

[0065] The sample point set, the observation point set, and the pseudo-point set x f , x u are substituted into the derived posterior mean function and the posterior autocovariance function to form the posterior mean vector and the autocovariance matrix. Considering that the covariance matrix is usually a high-dimensional, symmetric, and positive definite matrix, directly operating the high-dimensional matrix is very complex in calculation. Through eigenvalue decomposition, the originally complex covariance matrix problem can be transformed into the operation of a diagonal matrix, thereby simplifying the processing of the covariance matrix. Therefore, the posterior autocovariance matrix is sampled to ensure that the generated random field sample can meet the given mean and covariance structure, so that it can be used for simulation, analysis, and prediction of spatially correlated data.

[0066] Since the posterior autocovariance matrix is composed of two parts and , the sampling of these two parts is performed again, and the overall sampling result of the posterior autocovariance matrix is obtained. The sampling result of each part is obtained by multiplying the left half obtained by eigenvalue decomposition of each part with an independent Gaussian random vector.

[0067] When sampling , since is too complex to be inverted, first sample , and then obtain the sampling result of by the linear conjugate gradient method; sample ​​​​​​​​​​When sampling, since it is Add together, and both parts obey the normal distribution, so they can be sampled separately. The sampling results are respectively The result obtained by multiplying the left half obtained after eigenvalue decomposition with the independent Gaussian random vector. are mutually transposed, just need to and Multiplying the sampling results of The overall sampling results.

[0068] and is a non-negative diagonal matrix, and its eigenvalue decomposition is Perform square root operation.

[0069] First of all, Perform eigenvalue decomposition: Because K uu is a Kronecker structure, that is Among them, K md It means that the pseudo point set x u The data x on the d-th dimension u (d) Substitute the autocovariance function k in the d-th dimension d The autocovariance matrix obtained by (·,·) can be used to calculate K md Perform eigenvalue decomposition and combine the decomposition results with Kronecker product to obtain K uu The eigenvalue decomposition of K md =Q md V md Q md T , where Q md It's K md The eigenvector matrix, V md It's K md The eigenvalue matrix of , according to the properties of Kronecker product, is obtained:

[0070]

[0071] Then to W f T SW f Perform eigenvalue decomposition: Because αδ(x f ,x f )+Σ ε =S -1 , and S -1 =LL T , so S=(LL T ) -1 =(L T )-1 (L) -1 , and W f is a sparse matrix, then W f T SW f It is still a non-negative diagonal matrix, so its eigenvalue decomposition is:

[0072] W f T SW f =W f T (L T ) -1 (L) -1 W f

[0073] =W f T (L T ) -1 [W f T (L T ) -1 ] T

[0074] right Perform eigenvalue decomposition to obtain:

[0075]

[0076] Then, a large-scale multidimensional Gaussian random field is generated based on the eigenvalue decomposition of the posterior autocovariance matrix.

[0077] For the sake of Sampling is performed to obtain directly The sampling result r is The left half of the eigenvalue decomposition result and the multiplication form of the independent Gaussian random vector are first Sampling is performed because Obeying the normal distribution, we can sample separately and then add them, that is, sample separately:

[0078]

[0079] z~(0,W f T SW f )

[0080] because And V m Non-negative eigenvalue matrix, then:

[0081]

[0082] but and So z=W f T (L T ) - 1 v2, then Where v1 and v2 are two standard independent Gaussian random vectors; then the conjugate gradient method (LCG) is used to solve the equation Solve to get the sampling result r. Solve the equation Ar=t using the LCG method, that is, solve the function The minimum value of the objective function is expressed as In D dimensions, we can find D vectors b1, b2, ..., b that are conjugate to A. D , then r in the D dimension can be expressed as where a d is the vector b in dimension d d The corresponding variables, the objective function becomes:

[0083]

[0084] Because the vector b i 、b j Mutually conjugate, when i≠j, there is b i T Ab j =0, then the objective function is:

[0085]

[0086] Then for each item a i Take the derivative to get the minimum value, that is:

[0087] a i b i T Ab i -t T b i =0

[0088]

[0089] So we end up with:

[0090]

[0091] The large-scale multidimensional Gaussian random field is generated according to the following formula, which is to obtain the large-scale inhomogeneous material spatial variation feature generation model based on the grid pseudo-point approximation Gaussian random field:

[0092]

[0093] where v3 is a standard independent Gaussian random vector.

[0094] The method proposed by the present application is further described below with a specific example.

[0095] Now assume a two-dimensional grid, the original signal is f=x1+sinx2, x1 is the data set in the first dimension, x2 is the data set in the second dimension, x n1 is set to 11 points uniformly distributed between [-4, 4], x n2 is set to 21 points uniformly distributed between [-6, 6], x n1 and x n2 Generate a grid structure, and then randomly select points from it to form the observation data set x f , then x f is a 77x2 matrix, as shown in Table 1.

[0096] Table 1 Partial observation data points and their output quantities

[0097] x n1 ]]> x n2 ]]> y -1.6 -6 -0.83 0.8 -6 0.95 1.6 -6 1.61 3.2 -6 3.43 4 -6 3.59 -3.2 -5.4 -2.79 -4 -4.8 -2.06 -3.2 -4.8 -3.67 -0.8 -4.8 0.69 0 -4.8 0.41 …… …… …… 0.8 6 -0.01 2.4 6 2.69

[0098] Set the prior mean function as m(x)=0.5, and the prior autocovariance function as where σ1=σ2=1, but in specific operation, need to use and l1=l2=0.2, the initial ε noise level is ∑ ε =1.0I, then y=x1+sin x2+ε, the hyperparameters optimized by maximizing the likelihood function are shown in Table 2.

[0099] Table 2 Hyperparameters optimized by maximizing the likelihood function

[0100]

[0101]

[0102] x u Each dimension of the maximum / minimum value is greater / smaller than the maximum / minimum value of x f , Table 3 is part of the pseudo point data, where x u1 is the data in the first dimension of x u , there are 41, x u2 is the data in the second dimension of x u , there are 61.

[0103] Table 3 Partial pseudo point data

[0104] x u1 ]]> x u2 ]]> -4.98 -6.98 -4.73 -6.75 -4.48 -6.52 -4.23 -6.28 -3.98 -6.05 -3.73 -5.82 -3.48 -5.59 -3.23 -5.35 -2.99 -5.12 -2.73 -4.89 …… …… 4.73 6.75 4.98 6.98

[0105] Sampling data set A gridding structure is generated for 81 data points uniformly spaced between [-4, 4] and 121 data points uniformly spaced between [-6, 6], then x f* is a 9801 x 2 matrix, and Table 4 is a part of the sampling data points, where x *1 is the data in the first dimension of x , and x *2 is the data in the second dimension of x .

[0106] Table 4 Part of the sampling data points

[0107] x *1 ]]> x *2 ]]> -4 -6 -3.9 -6 -3.8 -6 -3.7 -6 -3.6 -6 -3.5 -6 -3.4 -6 …… …… 3.9 6 4 6

[0108] When a = 0.5, the equivalent kernel derived from the sparse power expectation propagation method is Let 0.5δ(x, x') + 0.5I = S -1 , then the posterior mean and the autocovariance function of the derived sparse pseudo points are and

[0109] Let K fu = W f K uu , K uf = K uu W f T , and W f is the interpolation weight matrix of the observation data set and the pseudo point data set x f , x u , is the interpolation weight matrix of the observation data set and the sampling data set x f , The posterior mean function is and the posterior covariance function is wherein, in step two, the posterior mean vector and the autocovariance matrix are quickly calculated by using the pseudo point gridding structure and the Kronecker operation, and the sampling point set, the observation point set, and the pseudo point set x , x f , x u are substituted into the derived posterior mean function and the posterior autocovariance function to form the posterior mean vector and the autocovariance matrix, and the eigenvalue decomposition of the posterior autocovariance matrix is required, and since the posterior autocovariance matrix is composed of two parts of and , the eigenvalue decomposition of the two parts can be performed first, and then added up, and since can be sampled first, and then The sampling results of Perform eigenvalue decomposition, and They all obey the normal distribution and can be sampled separately. The sampling result of is the left half of each eigenvalue decomposition multiplied by an independent Gaussian random vector; and is a non-negative diagonal matrix, and its eigenvalue decomposition is Perform square root operation.

[0110] First of all, Perform eigenvalue decomposition: Because K uu is a Kronecker structure, that is where K md It means that the pseudo point set x u The data x on the d-th dimension u (d) Substitute the autocovariance function k in the d-th dimension d The autocovariance matrix obtained by (·,·) can be used to calculate K md Perform eigenvalue decomposition and combine the decomposition results with Kronecker product to obtain K uu The eigenvalue decomposition of K md =Q md V md Q md T , where Q md It's K md The eigenvector matrix, V md It's K md The eigenvalue matrix of , according to the properties of Kronecker product, is obtained:

[0111]

[0112] K uu =Q m V m Q m T

[0113]

[0114] Then to W f T SW f Perform eigenvalue decomposition: Because αδ(x f ,x f )+∑ ε =S -1 , and S -1 =LL T , so S=(LL T ) -1 =(LT -1 (L) -1 and W f is a sparse matrix, then W f T SW f is still a non-negative diagonal matrix, then its eigenvalue decomposition is:

[0115] W f T SW f = W f T (L T ) -1 (L) -1 W f

[0116] = W f T (L T ) -1 [W f T (L T ) -1 ] T

[0117] Eigenvalue decomposition of a(x * , x * ) is performed:

[0118]

[0119] In step three, large-scale multi-dimensional Gaussian random field generation based on the eigenvalue decomposition of the posterior autocovariance matrix, in order to sample , so as to directly obtain the sampling result r of , that is The left half of the eigenvalue decomposition result and the multiplication form of the independent Gaussian random vector, first sample , because obeys the normal distribution, so it can be sampled respectively and added, that is, sample respectively:

[0120]

[0121] z ~ (0, W f T SW f )

[0122] Because and V m is a non-negative eigenvalue matrix, then:

[0123]

[0124] then​ And W f T SW f =W f T (L T ) -1 [W f T (L T ) -1 ] T , so z=W f T (L T ) -1 v2, then Where v1 and v2 are two standard independent Gaussian random vectors; then the conjugate gradient method (LCG) is used to solve the equation Solve to get the sampling result r. Solve the equation Ar=t using the LCG method, that is, solve the function The minimum value of the objective function is expressed as In D dimensions, we can find D vectors b1, b2, ..., b that are conjugate to A. D , then r in the D dimension can be expressed as where a d is the vector b in dimension d d The corresponding variables, the objective function becomes:

[0125]

[0126] Because the vector b i 、b j Mutually conjugate, when i≠j, there is b i T Ab j =0, then the objective function is:

[0127]

[0128] Then for each item a i Take the derivative to get the minimum value, that is:

[0129] a i b i T Ab i -t T b i =0

[0130]

[0131] So we end up with:

[0132]

[0133] The large-scale multi-dimensional Gaussian random field is generated according to the following formula:

[0134]

[0135] where v3 is a standard independent Gaussian random vector.

[0136] like Figures 2-3 As shown, the mean of the Gaussian random field generated by the grid pseudo points is the same as the mean of the random field generated by dense calculation, where the posterior mean function of dense calculation is The posterior autocovariance function is

[0137] Since the posterior autocovariance matrix does not need to be directly inverted during the calculation process, and the linear conjugate gradient method can be used for rapid solution, the calculation is fast. As shown in Table 5, the calculation time for generating the Gaussian random field based on the grid pseudo points is less than that of the dense calculation. t1 is the time required for generating the random field based on the grid pseudo points, and t2 is the time required for generating the random field based on the dense calculation.

[0138] Table 5 Computational time (in seconds) for generating Gaussian random fields using grid pseudo-points and gridded dense calculations

[0139] ​ [ t2 ] 0.0973 43.794

[0140] like Figure 4 As shown in FIG, ten sampling results of generating a large-scale Gaussian random field based on grid pseudo points are roughly the same as the image whose mean value of the Gaussian random field generated based on the grid pseudo points is within the range of observed data.

[0141] An embodiment of the present invention provides a method for generating spatial variation characteristics of large-scale inhomogeneous materials based on grid pseudo-point approximation of Gaussian random fields, comprising:

[0142] Get the observation point set x f The corresponding real value set f and pseudo point set x u The corresponding true value set u and sampling point set The corresponding real value set f * ;

[0143] The sampling point set is calculated based on the large-scale inhomogeneous material spatial variation feature generation model based on grid pseudo point approximation Gaussian random field constructed by the method of any of the above embodiments. Spatial variation characteristics of heterogeneous materials at each point in the image.

[0144] An embodiment of the present invention provides an electronic device, comprising: a computer-readable storage medium and a processor;

[0145] The computer readable storage medium is configured to store executable instructions.

[0146] The processor is configured to read the executable instructions stored in the computer readable storage medium, and execute the method according to any one of the above embodiments.

[0147] An embodiment of the present application provides a computer readable storage medium, which stores computer instructions, and the computer instructions are configured to make a processor execute the method according to any one of the above embodiments.

[0148] Those skilled in the art can easily understand that the above description is only the preferred embodiment of the present application, and is not used to limit the present application, and any modification, equivalent replacement and improvement made within the spirit and principle of the present application should be included in the protection scope of the present application.

Claims

1. A method for constructing a large-scale inhomogeneous material spatial variation feature generation model based on grid pseudo-point approximation of Gaussian random field, characterized by: include: S1, based on the observation point set x f The corresponding real value set f and pseudo point set x u The corresponding true value set u and sampling point set The corresponding real value set f * , the equivalent autocovariance matrix of the Gaussian process is obtained by sparse pseudo-point approximation; Where, f=H(x f ) is x f The corresponding true value, y = f + ε, y is x f The corresponding observation value, observation error ε~N(0,∑ ε ); u=H(x u ) is x u The corresponding true value, for The corresponding true value, y * =f * +ε,y * is x * The corresponding observation value, H() is a D-dimensional random field, D>1; x u The maximum value of each dimension is greater than x f The maximum value of the corresponding dimension in , the minimum value of each dimension is less than x f The minimum value of the corresponding dimension in , and the spacing between each pseudo point is the same to form a gridded pseudo point; x f* The maximum value of each dimension is less than x u The maximum value of the corresponding dimension in , the minimum value of each dimension is greater than x u The minimum value of the corresponding dimension in ; S2, using the equivalent autocovariance matrix to replace the dense calculation f * The autocovariance matrix in the predictive distribution is such that the replaced f * The sum S between the diagonal matrix items in the equivalent autocovariance matrix and the autocovariance matrix of the known observation error appears in both the posterior mean function and the posterior autocovariance function in the predicted distribution. -1 ; Decompose the sum into upper triangular matrix L and lower triangular matrix L T The product form S -1 =LL T , and bring in the replaced f * The posterior mean function and posterior autocovariance function in the predictive distribution are sparsely matrixed using x u with x f or The autocovariance matrix of is approximated to approximate the posterior mean function and the posterior autocovariance function, and the Gaussian process posterior mean function and the posterior autocovariance function based on the grid pseudo point approximation are obtained as follows: Where, and W f Both are sparse matrices; K ab =K(x a ,x b ), K(x a ,x b ) is x a ,x b Substitute the set prior autocovariance function k(x i ,x j ) calculated autocovariance matrix, and x a ,x b for x u or x f ; is a non-negative diagonal matrix, α∈[0,1]; S3, yes The sum of the items in the formula is sampled and calculated separately. Combine it with Adding them together, we get the large-scale inhomogeneous material spatial variation feature generation model based on grid pseudo-point approximation of Gaussian random field: Where, b1,b2,...,b D are D vectors in D dimensions that are conjugate with respect to A, v1, v2, and v3 are all standard independent Gaussian random vectors.

2. The method according to claim 1, wherein In step S1, the sparse power expectation propagation method is used to perform sparse pseudo-point approximation, and the Gaussian process equivalent autocovariance matrix is ​​obtained as follows: In the formula, δ(x,x')=diag(K(x,x')-K(x,x u )K -1 (x u ,x u )K(x u ,x')) is a non-negative diagonal matrix, α∈[0,1].

3. The method according to claim 1, wherein In step S2, S -1 =LL T Bring in the replaced f * Before the posterior mean function and posterior autocovariance function in the predictive distribution, it also includes: Define the intermediate variable K uf (L T ) -1 =K us ,L -1 y=y s ,K su =K us T , and substitute it into the replaced f * The posterior mean function and posterior autocovariance function in the predictive distribution are used to calculate the posterior mean function and the posterior autocovariance function in the predictive distribution, and the sparse matrix is ​​used to calculate the x u with x f or After the autocovariance matrix of is approximated, the intermediate variables are eliminated based on the definition.

4. The method according to claim 1, wherein In step S3, the Gaussian process posterior autocovariance function obtained by the approximate calculation is: The step of sequentially performing eigenvalue decomposition and sampling calculation on each sum item in the Gaussian process posterior autocovariance function obtained by the approximate calculation comprises: (1) Yes W f T SW f After performing eigendecomposition respectively, the left half of each eigenvalue decomposition is multiplied by an independent Gaussian random vector and then added to obtain The sampling results of The sampling results are calculated using the linear conjugate gradient method. The sampling result r, and then W f* Multiplying by r gives The sampling calculation results of (2) Yes Perform square root operation and then multiply by v3 to get the sampling calculation result 5. A method for generating spatial variation characteristics of large-scale inhomogeneous materials based on grid pseudo-point approximation of Gaussian random fields, characterized by: include: Get the observation point set x f The corresponding real value set f and pseudo point set x u The corresponding true value set u and sampling point set The corresponding real value set f * ; The sampling point set is calculated based on the large-scale inhomogeneous material spatial variation feature generation model based on grid pseudo point approximation Gaussian random field constructed by the method according to any one of claims 1 to 4. Spatial variation characteristics of heterogeneous materials at each point in the image.

6. An electronic device, characterized in that: include: Computer-readable storage medium and processor; The computer-readable storage medium is used to store executable instructions; The processor is configured to read the executable instructions stored in the computer-readable storage medium and execute the method according to any one of claims 1 to 5.

7. A computer-readable storage medium, characterized in that The computer-readable storage medium stores computer instructions, and the computer instructions are used to enable a processor to execute the method according to any one of claims 1 to 5.

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