A neutron spectrum unfolding method independent of prior information
By introducing the total variation (TV) penalty function and the generalized cross-validation (GCV) method, combined with the Lanczos algorithm, a neutron energy spectrum interpretation method that does not rely on prior information is constructed. This solves the problem of unstable neutron energy spectrum interpretation results in existing technologies and achieves neutron energy spectrum measurement with high accuracy and independence.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- NUCLEAR POWER INSTITUTE OF CHINA
- Filing Date
- 2024-12-06
- Publication Date
- 2026-06-09
AI Technical Summary
Existing neutron energy spectrum interpretation techniques rely on prior information, resulting in insufficient stability and independence of experimental results. This makes it difficult to accurately measure neutron energy spectra under conditions of large energy range, severe gamma background interference, and drastic statistical fluctuations in the radiation field.
By introducing the total variation (TV) as a penalty function and combining the generalized cross-validation (GCV) method and the Lanczos double diagonal algorithm, a regularized spectral decomposition method is constructed. This method utilizes the smoothness of the neutron spectrum to achieve neutron spectrum decomposition without relying on prior information.
It improves the independence and accuracy of neutron energy spectrum measurement results, overcomes the instability of the spectral results caused by statistical fluctuations in measurement values and ill-conditioned response matrix, and the number of energy groups in the output neutron energy spectrum is consistent with the energy group structure, covering the hot region up to 20 MeV.
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Figure CN122172258A_ABST
Abstract
Description
Technical Field
[0001] This invention belongs to the field of radiation detection technology, specifically relating to a neutron energy spectrum interpretation method that does not rely on prior information. Background Technology
[0002] Accurate neutron spectrum measurement has always been a key focus in neutron science, reactor physics, and shielding. Neutron spectrum measurement typically faces challenges such as a large energy range (from hot regions to tens of MeV), severe gamma background interference, and dramatic statistical fluctuations in the radiation field. Neutron spectra usually cannot be measured directly and must be obtained through spectrum interpretation in conjunction with the detector response function. Existing spectrum interpretation techniques often treat the neutron spectrum problem as an indeterminate problem, inevitably relying on prior information—that is, a certain degree of prediction of the shape of the measured neutron spectrum. This introduces artificial interference into the neutron spectrum measurement experiment, causing the experimental results to lose stability and independence to some extent.
[0003] The transport of neutrons in matter is a process that gradually slows down from high energy to low energy. Based on this physical characteristic, we have observed a certain correlation between the fluxes of different neutron groups, and the neutron energy spectrum typically exhibits a smooth distribution. Against this backdrop, this invention proposes to introduce the smoothness index—Total Variation (TV)—as a penalty function into the regularized unregulated spectral analysis method, establishing a novel method to achieve neutron energy spectrum solving independent of prior information based on the smoothness of the neutron energy spectrum. Summary of the Invention
[0004] The purpose of this invention is to provide a neutron energy spectrum interpretation method that does not rely on prior information. Based on the smooth distribution of the neutron energy spectrum, a smoothness index, TV, is introduced as a penalty function on the basis of the regularized interpretation method to establish a new interpretation method, thereby realizing the solution of the neutron energy spectrum without relying on prior information and improving the independence of the neutron energy spectrum measurement results.
[0005] To achieve the above objectives, the technical solution adopted by the present invention is as follows:
[0006] A neutron energy spectrum interpretation method that does not rely on prior information:
[0007] Step 1: Prepare the raw data required for spectrum interpretation, including neutron spectrometer measurement results and response matrix. The neutron spectrometer consists of m detectors. Divide the neutron energy group structure into n segments and obtain a response matrix R∈R with size m×n. m×n The element r in the i-th row and j-th column of matrix R i,j This represents the magnitude of the response of the i-th detector to the unit neutron fluence rate of the j-th energy group;
[0008] Step 2: Construct an orthonormal basis for the response matrix and measurement results in the Krylov subspace, based on the response matrix R and the measurement results M∈R from m detectors. m Construct the Krylov subspace K k (R T R,R T The orthonormal basis matrix W = [w1, ..., w1] of M is [w1, ..., w2]. k ]∈R n×k With Krylov subspace K k+1 (RR T The orthonormal basis matrix Z = [z1, ..., z) of M) k+1 ]∈R m×(k+1) It supports iterative regularization methods for solving problems;
[0009] Step 3: Determine the regularization parameter λ using the generalized cross-validation (GCV) method;
[0010] Step 4: Determine the objective function of the TV regularization method; let y be the projection vector of the neutron energy spectrum φ into the linear space W constructed in step 2, then φ = Wy. At this point, the least squares optimization objective function is... Equivalent to Adding a penalty function to the least squares optimization objective function yields the optimization objective function of the novel regularization method:
[0011] f(y)=||ν1e1-By||2+λ×TV(y)
[0012] in Let y be the total variation of vector y; i Let be the i-th term of vector y;
[0013] Step 5: Solve for the projection vector y;
[0014] Step 6: Solve for the neutron energy spectrum φ.
[0015] Step 1: Obtain the response matrix R∈R of size m×n through modeling, Monte Carlo calculation, and standard source calibration. m×n .
[0016] Step 2: Construct the response matrix and the orthonormal basis of the measurement results in the Krylov subspace using the Lanczos double diagonal algorithm; the construction steps are as follows: 1) Input the m-dimensional detector measurement result vector M and the detector response matrix R; 2) Initialize: v1=||M||2, z1=M / v1, w mid =R T z1,μ1=||w mid ||2,w1=w mid / μ1;3)For j=2,…,k+1:a)Calculate zmid =Rw j-1 -μ j-1 z j-1 b) Set v j =||z mid ||2,z j =z mid / v j c) Calculate w mid =R T z j -v j w j1 ;d) Set μ j =||w mid ||2,w j =w mid / μ j .
[0017] Step 3: The generalized cross-validation method is a numerical method to obtain the optimal regularization parameter λ by minimizing the GCV function. The steps are as follows:
[0018] a) Record Let Z be the pseudo-inverse, and define matrix B as follows:
[0019]
[0020] b) Performing SVD decomposition on B yields:
[0021]
[0022] c) Let The GCV function is defined as follows:
[0023]
[0024] Where [P] T e1] i Represents [P] T The i-th term of the k+1 dimensional vector [e1];
[0025] d) Calculate the regularization parameter λ that minimizes the value of the GCV function.
[0026] d) Calculate the regularization parameter λ that minimizes the value of the GCV function. The specific solution method is the conjugate gradient method.
[0027] d) Calculate the regularization parameter λ that minimizes the value of the GCV function. The specific solution method is a genetic algorithm.
[0028] Step 5: Solve for the projection vector y by calculating the value of y that minimizes the objective function f(y).
[0029] The specific solution method adopted is the conjugate gradient method.
[0030] The specific solution method uses a genetic algorithm.
[0031] Step 6: The projection vector of the neutron energy spectrum φ to be determined in the linear space W is y. Solve for the neutron energy spectrum φ by φ = Wy. The solution φ is a vector with the same number of elements as the number of energy group divisions n. The values at each position represent the neutron flux measurement results of the corresponding energy group. The sum of all elements is the total neutron flux measurement result.
[0032] The beneficial effects achieved by this invention are as follows:
[0033] This invention provides a neutron spectrum solution method that fully utilizes the smoothness physical property of the neutron spectrum and eliminates dependence on prior information. This method introduces the TV (transformation time) as a smoothness index as a penalty function in the regularized spectrum solution process, establishing a TV regularized spectrum solution method. This method differs from commonly used L1 and L2 regularization methods in the field of spectrum solution because it takes into account and utilizes the smoothness physical property of the neutron spectrum, enabling neutron spectrum solutions without prior information conditions and ensuring the independence of neutron spectrum measurement results.
[0034] This invention is applicable to underdetermined, well-determined, and overdetermined problems. By adding a regularization term to the objective function during the iteration process, it overcomes the solution disruption caused by statistical fluctuations in measured values and ill-conditioned response matrices. The regularization parameter is adaptively selected using the generalized cross-validation (GCV) method.
[0035] This invention enables the solution of neutron energy spectra without relying on prior information. By applying a smoothness penalty to the iterative spectrum solution process, the accuracy and independence of neutron energy measurement results are improved.
[0036] The number of neutron energy groups output by the method of the present invention is consistent with the neutron energy group structure, and the neutron energy group structure is consistent with the energy group structure of the detector response function. The neutron energy range can cover the hot region up to 20MeV.
[0037] This invention effectively overcomes the insufficient smoothness or even failure of spectral resolution results caused by statistical fluctuations in measured values and matrix ill-conditioning. Compared with existing GRAVEL and MLEM algorithms, this invention provides superior spectral resolution results. (See attached diagram) Figure 1 Appendix Figure 2 The spectral decomposition method provided by this invention can successfully solve the neutron energy spectrum without relying on prior information (see appendix). Figure 3 ). Attached Figure Description
[0038] Figure 1 For the traditional GRAVEL algorithm 241 Neutron energy spectrum interpretation results from the Am-Be source.
[0039] Figure 2 For traditional MLEM algorithms 241 Neutron energy spectrum interpretation results from the Am-Be source.
[0040] Figure 3 The method provided by the present invention is for 241 Neutron energy spectrum interpretation results from the Am-Be source. Detailed Implementation
[0041] The present invention will now be described in detail with reference to the accompanying drawings and specific embodiments.
[0042] A neutron energy spectrum interpretation method that does not rely on prior information:
[0043] Step 1: Prepare the raw data required for spectrum interpretation, including neutron spectrometer measurement results and response matrix. The neutron spectrometer consists of m detectors, and the neutron energy group structure is divided into n segments according to the spectrum interpretation requirements. A response matrix R∈Rm×n is obtained through modeling, Monte Carlo calculations, and standard source calibration. m×n The element r in the i-th row and j-th column of matrix R i,j This represents the magnitude of the response of the i-th detector to the unit neutron fluence rate of the j-th energy group;
[0044] Step 2: The Lanczos double-diagonal algorithm is used to construct an orthonormal basis for the response matrix and measurement results in the Krylov subspace, providing a foundation for calculating the regularization parameters using the GCV method. This is based on the response matrix R and the measurement results M∈R from m detectors. m Construct the Krylov subspace K k (R T R,R T The orthonormal basis matrix W = [w1, ..., w1] of M is [w1, ..., w2]. k ]∈R n×k With Krylov subspace The orthonormal basis matrix Z = [z1,…,z2] k+1 ]∈R m×(k+1) It supports iterative regularization methods for solving the problem, where the constant k can be specified empirically by the researchers. The construction steps are as follows:
[0045] 1) Input the m-dimensional detector measurement result vector M and the detector response matrix R;
[0046] 2) Initialization: v1 = ||M||2, z1 = M / v1, w mid =R T z1,μ1=||w mid ||2,w1=w mid / μ1;
[0047] 3) For j = 2, ..., k+1:
[0048] a) Calculate z mid =Rw j-1 -μ j-1 z j-1 ;
[0049] b) Set v j =||z mid ||2,z j =z mid / v j ;
[0050] c) Calculate w mid =R T z j -v j w j-1 ;
[0051] d) Set μ j =||w mid ||2,w j =w mid / μ j .
[0052] Step 3: Determine the regularization parameter λ using the Generalized Cross-Validation (GCV) method. The Generalized Cross-Validation method refers to a numerical method that obtains the optimal regularization parameter λ by minimizing the GCV function. The following steps complete the definition of the GCV function and the calculation of the optimal regularization parameter λ:
[0053] a) Record Let Z be the pseudo-inverse, and define matrix B as follows:
[0054]
[0055] b) Performing SVD decomposition on B yields:
[0056]
[0057] c) Let The GCV function is defined as follows:
[0058]
[0059] Where [P] T e1] i Represents [P] T The i-th term of the k+1 dimensional vector [e1].
[0060] d) Calculate the regularization parameter λ that minimizes the value of the GCV function. The specific solution method can be the iterative method such as conjugate gradient or the heuristic algorithm such as genetic algorithm, which will not be elaborated here.
[0061] Step 4: Determine the objective function of the TV regularization method. Let y be the projection vector of the neutron energy spectrum φ into the linear space W constructed in step 2, then φ = Wy. At this point, the least squares optimization objective function is... Equivalent to Adding a penalty function to the least squares optimization objective function yields the optimization objective function of the novel regularization method:
[0062] f(y)=||ν1e1-By||2+λ×TV(y)
[0063] in Let y be the total variation of vector y; i Let be the i-th term of vector y.
[0064] Step 5: Solve for the projection vector y. The solution method is to calculate y that minimizes the objective function f(y). Specific solutions can be obtained using iterative methods such as conjugate gradients, heuristic algorithms such as genetic algorithms, etc., which will not be elaborated here.
[0065] Step 6: Solve for the neutron energy spectrum φ. Since the projection vector of the neutron energy spectrum φ in the linear space W is y, the neutron energy spectrum φ can be solved by φ = Wy. The solution φ is a vector with the same number of elements as the number of energy group partitions n. The values at each position represent the neutron flux measurement results of the corresponding energy group, and the sum of all elements is the total neutron flux measurement result.
[0066] This invention provides a spectral decomposition method for neutron energy spectrum measurements that does not rely on prior information. By utilizing the smoothness of the neutron energy spectrum, the total variation (TV) is introduced as a penalty function for the regularization method, eliminating the dependence on prior information in the neutron energy spectrum solution process and improving the accuracy and stability of the neutron energy spectrum solution results. This invention enables neutron energy spectrum solutions that do not rely on prior information, improving the independence of measured neutron energy spectrum results. The number of energy groups in the output neutron energy spectrum is consistent with the number of energy group structure divisions, and the neutron energy group structure is consistent with the energy group structure of the detector response function. The neutron energy range can cover the hot region to 20 MeV. Compared with existing spectral decomposition algorithms, such as the GRAVEL algorithm and MLEM algorithm, the spectral decomposition method provided by this invention can successfully achieve neutron energy spectrum solutions that do not rely on prior information and exhibits higher accuracy in smooth energy spectra.
[0067] By fully utilizing the easily overlooked physical property of neutron spectrum smoothness through the TV regularization method, the accuracy of neutron spectrum interpretation is improved and the dependence of the interpretation process on prior information is eliminated. The smoothness index TV (Total Variation) is used as the penalty function of the regularized dispersive spectrum method. The regularization parameter of the TV regularization method is adaptively selected using the GCV method. The Lanczos double diagonal algorithm is used to calculate the orthogonal basis of the response matrix and the Krylov subspace of the measurement data.
Claims
1. A neutron energy spectrum interpretation method that does not rely on prior information, characterized in that: Step 1: Prepare the raw data required for spectrum interpretation, including neutron spectrometer measurement results and response matrix. The neutron spectrometer consists of m detectors. Divide the neutron energy group structure into n segments and obtain a response matrix R∈R with size m×n. m×n The element r in the i-th row and j-th column of matrix R i,j This represents the magnitude of the response of the i-th detector to the unit neutron fluence rate of the j-th energy group; Step 2: Construct an orthonormal basis for the response matrix and measurement results in the Krylov subspace, based on the response matrix R and the measurement results M∈R from m detectors. m Construct the Krylov subspace K k (R T R,R T The orthonormal basis matrix W = [w1,...,w1,...,w2,m2,m3,m4,m3,m4,m4,m4,m5,m6,m4,m6,m7 ... k ]∈R n×k With Krylov subspace K k+1 (RR T The orthonormal basis matrix Z = [z1, ..., z) of M) k+1 ]∈R m×(k+1) It supports iterative regularization methods for solving problems; Step 3: Determine the regularization parameter λ using the generalized cross-validation (GCV) method; Step 4: Determine the objective function of the TV regularization method; let y be the projection vector of the neutron energy spectrum φ into the linear space W constructed in step 2, then φ = Wy. At this point, the least squares optimization objective function is... Equivalent to Adding a penalty function to the least squares optimization objective function yields the optimization objective function of the novel regularization method: f(y)=||ν1e1-By||2+λ×TV(y) in Let y be the total variation of vector y; i Let be the i-th term of vector y; Step 5: Solve for the projection vector y; Step 6: Solve for the neutron energy spectrum φ.
2. The neutron energy spectrum interpretation method according to claim 1, which does not rely on prior information, is characterized in that: Step 1: Obtain the response matrix R∈R of size m×n through modeling, Monte Carlo calculation, and standard source calibration. m×n .
3. The neutron energy spectrum interpretation method according to claim 1, which does not rely on prior information, is characterized in that: Step 2: Construct the response matrix and the orthonormal basis of the measurement results in the Krylov subspace using the Lanczos double diagonal algorithm; the construction steps are as follows: 1) Input the m-dimensional detector measurement result vector M and the detector response matrix R; 2) Initialize: v1=||M||2, z1=M / v1, w mid =R T z1,μ1=||w mid ||2,w1=w mid / μ1;3)For j=2,…,k+1:a)Calculate z mid =Rw j-1 -μ j- 1z j-1 b) Set v j =||z mid ||2,z j =z mid / v j c) Calculate w mid =R T z j -v j w j-1 ;d) Set μ j =||w mid ||2,w j =w mid / μ j .
4. The neutron energy spectrum interpretation method according to claim 3, which does not rely on prior information, is characterized in that: Step 3: The generalized cross-validation method is a numerical method to obtain the optimal regularization parameter λ by minimizing the GCV function. The steps are as follows: a) Record Let Z be the pseudo-inverse, and define matrix B as follows: b) Performing SVD decomposition on B yields: c) Let The GCV function is defined as follows: Where [P] T e1] i Represents [P] T The i-th term of the k+1 dimensional vector [e1]; d) Calculate the regularization parameter λ that minimizes the value of the GCV function.
5. The neutron energy spectrum interpretation method according to claim 4, which does not rely on prior information, is characterized in that: d) Calculate the regularization parameter λ that minimizes the value of the GCV function. The specific solution method is the conjugate gradient method.
6. The neutron energy spectrum interpretation method according to claim 4, which does not rely on prior information, is characterized in that: d) Calculate the regularization parameter λ that minimizes the value of the GCV function. The specific solution method is a genetic algorithm.
7. The neutron energy spectrum interpretation method according to claim 1, which does not rely on prior information, is characterized in that: Step 5: Solve for the projection vector y by calculating the value of y that minimizes the objective function f(y).
8. The neutron energy spectrum decomposition method according to claim 7, which does not rely on prior information, is characterized in that: The specific solution method adopted is the conjugate gradient method.
9. The neutron energy spectrum interpretation method according to claim 7, which does not rely on prior information, is characterized in that: The specific solution method uses a genetic algorithm.
10. The neutron energy spectrum decomposition method according to claim 1, which does not rely on prior information, is characterized in that: Step 6: The projection vector of the neutron energy spectrum φ to be determined in the linear space W is y. Solve for the neutron energy spectrum φ by φ = Wy. The solution φ is a vector with the same number of elements as the number of energy group divisions n. The values at each position represent the neutron flux measurement results of the corresponding energy group. The sum of all elements is the total neutron flux measurement result.