Large-scale space random variation feature generation method based on grid false point approximation

Abstract

The invention discloses a large-scale space random variation feature generation method based on grid false point approximation, and belongs to the field of non-homogeneous material space variation feature generation. According to the method, false points are utilized to reduce the size of a covariance matrix, and meanwhile, a linear conjugate gradient method is utilized to solve an auto-covariance matrix of the grid false points; the calculation amount of auto-covariance matrix inversion and the calculation amount of auto-covariance matrix decomposition in the simulation process are reduced, and the generation efficiency of the Gaussian random field can be greatly improved; according to the large-scale non-homogeneous material spatial variation feature generation model based on the grid false point approximate Gaussian random field constructed by the method, the generation of the large-scale Gaussian random field based on the grid false point can be realized; an equivalent auto-covariance function of a sparse power expectation propagation method is used for deducing a Gaussian random field posterior mean value function and an auto-covariance function based on grid pseudo point approximation, rapid decomposition of a posterior auto-covariance matrix is achieved through Kronecker operation, and the generation efficiency of a multi-dimensional large-scale random field is greatly improved.

Description

Technical Field

[0001] The present invention belongs to the field of spatial variation feature generation of inhomogeneous materials, and more specifically, relates to a large-scale spatial random variation feature generation method based on grid pseudo-point approximation. Background Art

[0002] Random field models are often used to describe complex data distributions with spatial or structural dependencies. In the field of civil engineering, the spatial variation characteristics of various materials in engineering environments play a vital role in understanding the mechanical properties, thermal conductivity characteristics, and behavior of materials in complex environments. However, natural materials and artificial composite materials often exhibit highly inhomogeneous properties, which exhibit multi-scale, complex, and unpredictable random distributions in space. Therefore, how to effectively generate and simulate the spatial variation characteristics of these inhomogeneous 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 is not only time-consuming and labor-intensive, but also 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 fields.

[0003] The flexible autocovariance function of Gaussian random fields is considered to be able to handle large data sets, but when the data set contains n data, O(n 3 ) and the computation time is O(n 2 ) Storage time means that Gaussian process can only be used to process small data sets, and when the data dimension is greater than 1, the amount of calculation, calculation 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 complex engineering material data generation, large data sets are usually required to be analyzed. For data sets that obey Gaussian random fields, existing methods need to invert the covariance matrix calculated by the autocovariance function when calculating the posterior mean function and autocovariance function. The size of this matrix is ​​consistent with the length of the observed sample. Therefore, when the observed sample of engineering materials is large and has many dimensions, the inversion of the matrix has a large computational cost. In addition, random field generation requires decomposition of the posterior covariance matrix, and the size of this matrix will grow rapidly in multidimensional space. For high-density sampling with long time and space history and high sampling frequency, the computational cost is large. Therefore, more efficient posterior mean function, autocovariance function and random field generation methods are needed. Summary of the invention

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

[0005] To achieve the above-mentioned purpose, according to a first aspect of the present invention, a method for constructing a large-scale inhomogeneous material spatial variation feature generation model based on grid pseudo-point approximation of Gaussian random field is provided, comprising:

[0006] 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 through sparse pseudo-point approximation;

[0007] 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 * For 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 , and 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; The maximum value of each dimension is less than x u The maximum value of the corresponding dimension in , and the minimum value of each dimension is greater than x u The minimum value of the corresponding dimension in ;

[0008] 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 an upper triangular matrix L and a 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.u With x f or x f* 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:

[0009]

[0010] In the formula, and W f All are sparse matrices; K ab =K(x a ,x b ), K(x a ,x b ) is to convert x a ,x b Substitute the set prior autocovariance function k(x i ,x j ) calculated the autocovariance matrix, and x a ,x b For x f* 、x u or x f ; is a non-negative diagonal matrix, α∈[0,1];

[0011] S3, for Σ f* The sum of the items in the formula is sampled and calculated separately. Combine it with μ f* Adding together, the sampling formula for the spatial variation characteristics of large-scale Gaussian random inhomogeneous materials is obtained as follows:

[0012]

[0013] In the formula, 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.

[0014] According to a second aspect of the present invention, a method for generating spatial variation characteristics of large-scale inhomogeneous materials based on grid pseudo-point approximation of Gaussian random fields is provided, comprising:

[0015] 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 * ;

[0016] 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 the first aspect. Spatial variation characteristics of inhomogeneous materials at each point in the image.

[0017] According to a third aspect of the present invention, there is provided an electronic device, 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 as described in the first aspect.

[0020] According to a fourth aspect of the present invention, a computer-readable storage medium is provided, wherein the computer-readable storage medium stores computer instructions, and the computer instructions are used to cause a processor to execute the method as described in the first aspect.

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

[0022] The method provided by the present invention reduces the size of the covariance matrix by using pseudo points, and solves the autocovariance matrix of the grid pseudo points by using a linear conjugate gradient method, thereby reducing the amount of calculation for inverting the autocovariance matrix and the amount of calculation for decomposing the autocovariance matrix during the simulation process, and can greatly improve the generation efficiency of the Gaussian random field. The large-scale inhomogeneous material spatial variation feature generation model based on the grid pseudo point approximation Gaussian random field constructed by the method provided by the present invention can realize the generation of the large-scale Gaussian random field of the grid pseudo point, derive the posterior mean function and the autocovariance function of the Gaussian process approximated by the grid pseudo point by using the equivalent autocovariance function of the sparse power expectation propagation method, and generate the large-scale multidimensional Gaussian random field based on the eigenvalue decomposition of the posterior autocovariance matrix and the Kronecker operation, and greatly improve the generation efficiency of the random field based on the grid pseudo point. BRIEF DESCRIPTION OF THE DRAWINGS

[0023] Figure 1 A schematic flow chart of a method for generating spatial variation characteristics of large-scale inhomogeneous materials based on grid pseudo-point approximation of Gaussian random fields provided in an embodiment of the present invention;

[0024] Figure 2 A schematic diagram of the mean value of a Gaussian random field generated based on grid pseudo points provided in an embodiment of the present invention;

[0025] Figure 3 A schematic diagram of the mean value of a Gaussian random field generated by dense calculation provided by an embodiment of the present invention;

[0026] Figure 4 A schematic diagram of sampling results of generating a large-scale Gaussian random field based on grid pseudo points provided in an embodiment of the present invention. DETAILED DESCRIPTION

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

[0028] The embodiment of the present invention provides a method for constructing a large-scale inhomogeneous material spatial variation feature generation model based on grid pseudo-point approximation of Gaussian random field, such as Figure 1 As shown, including:

[0029] 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 through sparse pseudo-point approximation;

[0030] 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 an upper triangular matrix L and a 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. u With x f or The autocovariance matrix of the Gaussian process is approximated to approximate the posterior mean function and the posterior autocovariance function, and the posterior mean function and the posterior autocovariance function of the Gaussian process based on the grid pseudo point approximation are obtained;

[0031] S3, the Gaussian process posterior autocovariance function obtained by the approximate calculation The sum of the items in the formula is sampled and calculated separately. Combine it 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, and the operator × represents the Cartesian product between vectors. 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 dth dimension of the scalar input space. Substituting the input x into the prior autocovariance function to calculate the autocovariance matrix 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 , and has 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 coordinates 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. 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] Suppose 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 an arbitrary value 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 to convert x a ,x b Substitute the set prior autocovariance function k(x i ,x j ) calculated the 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 computation 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 equation, αδ(x f ,x f ) and ∑ ε Always appear together, so let αδ(x f ,x f )+∑ ε =S -1 , for S -1 Decompose it, and 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 set of sampling points 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] So 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 Substituting into the posterior autocovariance function, we get:

[0059]

[0060] and Using the intermediate variable K uf (LT ) -1 =K us ,L -1 y=y s ,K su =K us T , then the posterior autocovariance function is:

[0061]

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

[0063]

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

[0065] The sampling point set, observation point set, and pseudo point set x f 、x u Substitute the derived posterior mean function and posterior autocovariance function to form the posterior mean vector and autocovariance matrix. Considering that the covariance matrix is ​​usually a high-dimensional, symmetric and positive definite matrix, it is very computationally complex to directly operate the high-dimensional matrix. 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 samples can meet the given mean and covariance structure, so that they can be used to simulate, analyze and predict spatially related data.

[0066] Since the posterior autocovariance matrix is ​​given by and The two parts are added together, so the two parts are sampled separately and then added together to obtain the overall sampling result of the posterior autocovariance matrix. The sampling result of each part is obtained by multiplying the left half obtained after eigenvalue decomposition with an independent Gaussian random vector.

[0067] right When sampling, due to Since the inverse operation is too complicated, we first Sampling is performed and then the linear conjugate gradient method is used to obtain The sampling results of When sampling, since it is Added 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 an independent Gaussian random vector. The remaining are mutually transposed, and only need to and Multiply the sampling results by 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 dth dimension d The autocovariance matrix obtained by (·,·) can be used to calculate K md Perform eigenvalue decomposition, and then combine the decomposition results with Kronecker product to get 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 is obtained according to the properties of the Kronecker product:

[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 It obeys the normal distribution, so it can be sampled separately and then added, that is, sampled 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, i.e., the LCG method, 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 In D dimensions, we can find D vectors b1, b2, ..., b that are conjugate with respect 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, then the objective function becomes:

[0083]

[0084] Because the vector b i , b j conjugate with each other, 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 multi-dimensional Gaussian random field is generated according to the following formula, that is, the large-scale inhomogeneous material spatial variation feature generation model based on the grid pseudo-point approximation Gaussian random field is obtained:

[0092]

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

[0094] The method proposed in the present invention is further illustrated by taking a specific example as follows.

[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, n1 Set to 11 points evenly distributed between [-4, 4], x n2 Let there be 21 points evenly distributed between [-6, 6], and x n1 and x n2 Generate a grid structure and then randomly select The points form the observation data set x f , then x f It is a 77×2 matrix, as shown in Table 1.

[0096] Table 1 Some observation data points and their output

[0097] <![CDATA[x n1 ]]> <![CDATA[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] Assume that the prior mean function is m(x) = 0.5 and the prior autocovariance function is Among them, σ1=σ2=1, but in specific operations, it is necessary 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 after maximizing the likelihood function optimization

[0100]

[0101]

[0102] x u The maximum / minimum value of each dimension in is greater than x f The maximum / minimum value of is larger / smaller. Table 3 shows some pseudo-point data, where x u1 is x u There are 41 data in the first dimension, x u2 is x u There are 61 pieces of data in the second dimension.

[0103] Table 3 Partial pseudo point data

[0104] <![CDATA[x u1 ]]> <![CDATA[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 Dataset The grid structure generated for 81 data points evenly spaced between [-4,4] and 121 data points evenly spaced between [-6,6], then x f* is a 9801×2 matrix. Table 4 shows some sampled data points, where x *1 yes The data in the first dimension, x *2 yes The data in the second dimension of .

[0106] Table 4 Some sampling data points

[0107] <![CDATA[x *1 ]]> <![CDATA[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 α = 0.5, the equivalent kernel derived by the sparse power expectation propagation method is Let 0.5δ(x,x')+0.5I=S -1 , then the derived posterior mean and autocovariance functions of the sparse pseudo points are and

[0109] Let K fu =W f K uu ,K uf =K uu W f T , W f is the observation data set and pseudo point data set x f 、x u The interpolation weight matrix, is the observed data set and the sampled data set x f , The interpolation weight matrix and the posterior mean function are The posterior covariance function is Among them, in step 2, the pseudo-point grid structure and Kronecker operation are used to quickly calculate the posterior mean vector and the autocovariance matrix, and the sampling point set, the observation point set, and the pseudo-point set are x f 、x u Substitute the derived posterior mean function and posterior autocovariance function to form the posterior mean vector and autocovariance matrix. It is necessary to perform eigenvalue decomposition on the posterior autocovariance matrix. Since the posterior autocovariance matrix is ​​composed of and The two parts are composed of the sum of two parts. We can first perform eigenvalue decomposition on these two parts separately and then add them together. Sampling is performed and then the linear conjugate gradient method is used to obtain The sampling results are Perform eigenvalue decomposition, and All of them obey the normal distribution, so they can be sampled separately. The sampling result of is the left half of each eigenvalue decomposition multiplied by an independent Gaussian random vector; 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 dth dimension d The autocovariance matrix obtained by (·,·) can be used to calculate K md Perform eigenvalue decomposition, and then combine the decomposition results with Kronecker product to get 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 is obtained according to the properties of the Kronecker product:

[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 = (LLT ) -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:

[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] For αδ(x * ,x * ) to perform eigenvalue decomposition:

[0118]

[0119] In step 3, a large-scale multidimensional Gaussian random field is generated based on the eigenvalue decomposition of the posterior autocovariance matrix. 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 It obeys the normal distribution, so it can be sampled separately and then added, that is, sampled separately:

[0120]

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

[0122] because And V m Non-negative eigenvalue matrix, then:

[0123]

[0124] but 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, i.e., the LCG method, 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 In D dimensions, we can find D vectors b1, b2, ..., b that are conjugate with respect 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, then the objective function becomes:

[0125]

[0126] Because the vector b i , b j conjugate with each other, 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 Figure 2-3 As shown in Figure 2, 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 a Gaussian random field based on grid pseudo points is less than that for dense calculations. t1 is the time required for generating a random field based on grid pseudo points, and t2 is the time required for generating a random field based on dense calculations.

[0138] Table 5 Computation time (in seconds) for generating Gaussian random fields by grid pseudo-points and grid dense calculation

[0139] <![CDATA[t1]]> <![CDATA[t2]]> 0.0973 43.794

[0140] like Figure 4 As shown, 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] The 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 described in any of the above embodiments. Spatial variation characteristics of inhomogeneous materials at each point in the image.

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

[0145] The computer-readable storage medium is used to store executable instructions;

[0146] The processor is used to read the executable instructions stored in the computer-readable storage medium and execute the method described in any of the above embodiments.

[0147] An embodiment of the present invention provides a computer-readable storage medium, wherein the computer-readable storage medium stores computer instructions, and the computer instructions are used to enable a processor to execute the method described in any of the above embodiments.

[0148] It will be easily understood by those skilled in the art that the above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions and improvements made within the spirit and principles of the present invention should be included in the protection scope of the present invention.

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 in that: 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 through 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 * For 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 , and the minimum value of each dimension is less than x f The minimum value of the corresponding dimension in , and the spacing between the pseudo points 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 , and 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 an upper triangular matrix L and a 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. 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: In the formula, and W f All are sparse matrices; K ab =K(x a ,x b ), K(x a ,x b ) is to convert x a ,x b Substitute the set prior autocovariance function k(x i ,x j ) calculated the 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 together, the large-scale inhomogeneous material spatial variation feature generation model based on grid pseudo-point approximation Gaussian random field is obtained as follows: In the formula, 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, characterized in that 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, characterized in that 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 * Predict the posterior mean function and posterior autocovariance function in the distribution, and use sparse matrix to calculate 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, characterized in that In step S3, the Gaussian process posterior autocovariance function obtained by the approximate calculation is: The step of performing eigenvalue decomposition and sampling calculation on each of the added terms in the Gaussian process posterior autocovariance function obtained by the approximate calculation respectively comprises: (1) Yes W f T SW f After performing eigendecomposition respectively, multiply the left half of each eigenvalue decomposition by an independent Gaussian random vector and add them together to obtain The sampling results; according to 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 in that: 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 inhomogeneous materials at each point in the image.

6. An electronic device, characterized in that: include: A computer readable storage medium and a processor; The computer-readable storage medium is used to store executable instructions; The processor is used 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.

Images