An adaptive mixed-precision block jacobi preconditioning method incorporating filtering strategies
Patent Information
- Application Number
- CN202611018079.X
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2026-07-09
- Publication Date
- 2026-09-25
AI Technical Summary
然而，低精度计算过程的引入会影响Krylov子空间迭代法的收敛速度和收敛精度，需要针对应用问题设计合适的混合精度策略
（1）精度保证：相比于一致高精度算法，本发明提出的混合精度块Jacobi预条件算法保证和一致高精度算法基本相同的收敛精度。
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Figure CN122817601A_ABST
Abstract
Description
Technical Field
[0001] This invention belongs to the field of high-performance computing technology, and in particular relates to an adaptive mixed-precision block Jacobi preconditioning method that includes a filtering strategy. Background Technology
[0002] In high-energy-density physics research, simulating the ablation process of matter by radiation fields is a core scientific problem, providing key technical support for numerous fields such as spacecraft thermal protection material design, laser precision machining, and astrophysics. When extreme energy sources such as high-powered lasers act on matter, they generate high-intensity radiation fields, heating the material surface to a high-temperature, high-pressure plasma state in a very short time, thereby initiating ablation. This complex process involves the coupling of multiple physical processes, including radiation transport, heat conduction, and fluid dynamics. Among these, the numerical solution of the radiation transport equations, due to its computational complexity and high resource consumption, has become a key bottleneck in the entire simulation. Its computational efficiency directly determines the feasibility and practicality of large-scale, high-fidelity simulations.
[0003] The radiative transport equations are the fundamental governing equations describing the transport of radiative energy in a medium. In practical numerical simulations, the discrete ordinate method (SN method), upwind scheme, and multi-group approximation are commonly used to discretize the radiative transport equations spatially, angularly, and energetically, transforming the original equations into a large-scale, sparse linear system. Due to the high-dimensionality and multi-group characteristics of this linear system, efficient solution becomes significantly challenging. The efficiency of solving the radiative transport equations directly affects the simulation accuracy and computational scale of numerical simulations, thus limiting the accurate characterization of the interaction between the radiation field and matter. Therefore, improving the numerical solution efficiency of the radiative transport equations is essential for more reliable and efficient research on the ablation process of matter by the radiation field.
[0004] The preconditioning Krylov subspace iteration method is one of the most common methods for solving the above linear equations. The design of the preconditioning algorithm has a crucial impact on the efficiency of solving the linear equations.
[0005] Block Jacobi (BJAC) algorithms are a class of fundamental and widely applicable classical algorithms suitable for large-scale parallelism. They can be used independently as preconditioners or as algorithm components in multigrid and other preconditioning scenarios. When solving large-scale sparse linear systems, it is necessary to adjust the coefficient matrix... A Task partitioning typically involves dividing the matrix into several blocks by rows, denoted as... For matrix A No. i A diagonal piece, n This represents the total number of blocks. Based on this partitioning, a Jacobi preconditioner is constructed, which takes the following form:
[0006]
[0007] in, I As a unit array, It is the inverse of the diagonal block matrix. k is the number of iterations. The cost of finding an accurate solution is high, especially when the scale is large. Approximate inverse is usually used. To replace it. In particular, if the inverse of the block matrix is solved using Jacobi iteration, an approximation of the above block Jacobi preconditioner is obtained. It becomes the following form:
[0008] in, The block diagonal approximation inverse has the following form:
[0009] in, Indicates the first Diagonal block matrix diagonal matrix , This represents the number of iterations within the block. This represents the number of iterations outside the block. If Then the preconditioner degenerates into the simplest diagonal-scale preconditioner. By adjusting and Different preconditioners can be obtained. Generally, and The larger, right The higher the approximation, the better the acceleration effect on the Krylov subspace method, but at the same time, the overhead of the preconditioner is also greater.
[0010] Mixed-precision strategies have been increasingly applied in scientific numerical simulations in recent years, offering a novel approach to improving the performance of numerical algorithms. Since preconditioners are essentially approximations of the inverse of the coefficient matrix of a system of linear equations, low-precision preconditioning is a favorable way to reduce computational overhead. However, the introduction of low-precision computation can affect the convergence speed and accuracy of the Krylov subspace iteration method, necessitating the design of appropriate mixed-precision strategies tailored to the specific application problem.
[0011] To address the aforementioned issues, this invention proposes an adaptive mixed-precision block Jacobi preconditioning method incorporating a filtering strategy. Summary of the Invention
[0012] The purpose of this invention is to provide an adaptive mixed-precision block Jacobi preconditioning method incorporating a filtering strategy to solve the problems mentioned in the background art. This invention reduces computational overhead by using a filtering strategy to reduce the number of non-zero elements based on the determination of the size of sparse matrix elements; at the same time, it uses an adaptive strategy to adjust the algorithm parameters in the BJAC preconditioning to further improve computational efficiency.
[0013] To achieve the above objectives, the present invention employs the following: An adaptive mixed-precision block Jacobi preconditioning method incorporating a filtering strategy includes the following steps: S1. Based on the coefficient matrix of the linear equation system A Constructing the precondition matrix M 0, Filtered by filtering strategy M 0 non-diagonal elements ， Obtain the low-precision precondition matrix The coefficient matrix A The zero element in the middle represents the loss and accumulation of radiative energy, the transport of radiation between adjacent grids, and the scattering transfer between radiation intensities in different directions. S2. Enter the iterative solution process of the low-precision block Jacobi preconditional Krylov subspace method; S3, in the j In the next Krylov iteration, the residual vector at the current working precision is obtained. r j Then, the working precision (fp64) was changed to low precision (fp32), denoted as ; S4, using low-precision (fp32) residual vectors For the right-hand side, solve the system of equations using a low-precision scheme. Among them, the number of iterations within the block is used. t Number of iterations outside the block k Related block Jacobi iterations yield low-precision calculation results. , ; S5. Calculate the results using low precision (fp32). Converted to working precision (fp64) and denoted as ; S6. Perform the remaining calculations using the working accuracy (fp64); S7. After each Krylov iteration, the algorithm parameters are adaptively selected based on the convergence of the iteration. t and k ; S8. Repeat S3-S7 until the iteration converges; S9. Obtain the solution to the linear equation system. The solution vector is the radiation intensity, which describes the distribution and propagation state of radiation energy in the medium. It can be used to calculate key physical quantities such as radiation energy density, radiation energy flow, and radiation pressure. These physical quantities are related to the symmetry of the radiation driving field (which affects the spherical symmetry of the target implosion), the ablation pressure distribution (which drives the implosion compression), and the implosion dynamics evolution process. They are the core physical quantities for evaluating the feasibility of fusion ignition.
[0014] Preferably, S1 includes the following: S1.1 Setting the filter threshold f tol ; S1.2, using the coefficient matrix A Constructing the precondition matrix M 0; S1.3, Precondition Matrix M 0 The absolute value on the off-diagonal is less than f tol Filter the elements; S1.4. The filtered precondition matrix is stored using low precision (fp32), denoted as... .
[0015] Preferably, S7 specifically includes the following: S7.1 Obtain residual information and determine convergence: Obtain the residual vector of the previous Krylov iteration step. r j-1， residual in the current iteration step r j Initial residual vector r 0. Calculate the current relative residual. or j = || r j || / || r 0||, calculate the current residual decrease. r j = || r j || / || r j-1 ||;If satisfied: or j If ≤tol, then the current system of linear equations has been determined to have met the convergence condition, the iteration ends, and the parameters are no longer updated. t and k ;like or j >tol ， Then it is determined that the current system of linear equations has not yet converged, and the adaptive parameter selection process begins; S7.2 Obtaining Time Information: Obtain the cumulative computation time consumed in solving the current system of linear equations. T total Get the time taken for the current iteration step. T And obtain the target solution time. T opt ;in, T Take the computation time of the current Krylov iteration step, including the time of the low-precision block Jacobi preconditioning and the remaining working precision computation time in the Krylov method; T opt The preset target solution time is the target optimization time obtained based on historical solution results and the solution results of the previous linear equation system. S7.3 Obtain the allowable range of intra-block iteration count and out-of-block iteration count. t min , t max , k min , k max ; S7.4 Calculate the target residual descent threshold: First, estimate the maximum number of remaining allowed iterations within the target solution time. m : m = max ( 1, ( T opt - T total ) / T In order to remain m Make the current relative residual within the step or j Decrease to the convergence threshold tol, target residual decrease threshold d for: d m = tol / or j , d This represents the maximum residual reduction ratio expected to be achieved in each subsequent iteration under the current time constraint; S7.5 Obtaining the residuals of equations within a block: Obtaining the relative residuals of local equations within a low-precision Jacobi preconditioned block. x j , x j Set a lower limit for the intra-block relative residual threshold to represent the maximum value of the intra-block relative residuals for all blocks. e in This is used to determine whether the local equations within a block have reached the preset solution accuracy. S7.6, Update Algorithm Parameters t andk : ①If r j ≤ d , x j > e in Increase the number of iterations within the block. t= min( t+ 1, t max ), k = k ; ②If r j ≤ d , x j ≤ e in Reduce the number of iterations within a block. t= max( t -1 ,t min ), k = k ; ③If r j > d , x j > e in Increase the number of iterations within the block. t= min( t+ 1, t max ), k = k ; ④If r j > d , x j ≤ e in Increase the number of iterations outside the block. t = t , k= min( k+ 1, k max ); Using the adaptive strategy shown in S7, this invention does not simply increase the number of iterations within a block based on the residual decrease. t Number of iterations outside the block k Instead, it solves the time by targeting the objective. T opt Current cumulative time T total and current single-step time T Estimate the remaining number of available iterations. mThen, the residual descent threshold required to achieve convergence within the target time is calculated. d When the actual residual decreases r j Greater than d If the current parameters are insufficient to guarantee convergence within the target time, then it is advisable to prioritize increasing the number of intra-block iterations, which has lower computational cost. t ;When the intra-block residual x j The preset threshold lower limit has been reached. e in They believe that continuing to increase t The benefits are limited, so we increase the number of out-of-block iterations instead. k This achieves a balance between convergence speed and single-step computational overhead.
[0016] The present invention further protects a computer device comprising a processor and a memory, wherein the memory stores at least one instruction, at least one program, code set, or instruction set, the instruction, program, code set, or instruction set being loaded and executed by the processor to implement the aforementioned adaptive mixed precision block Jacobi preconditioning method including a filtering strategy.
[0017] The present invention further protects a computer-readable storage medium storing at least one instruction, at least one program, code set, or instruction set, wherein the instruction, program, code set, or instruction set is loaded and executed by a processor to implement the above-described adaptive mixed precision block Jacobi preconditioning method including a filtering strategy.
[0018] Compared with existing technologies, this invention provides an adaptive mixed-precision block Jacobi method incorporating a filtering strategy, which has the following advantages: (1) Accuracy guarantee: Compared with the consistent high-precision algorithm, the hybrid precision block Jacobi precondition algorithm proposed in this invention guarantees the same convergence accuracy as the consistent high-precision algorithm.
[0019] For example, numerical experiments have shown that, within the framework of the PGMRES algorithm, the mixed-precision block Jacobi preconditioning algorithm can converge to the order of 1E-11 when solving radiative transport examples (unstructured mesh, matrix size 9,104,256), which is the same order of magnitude as the consistent high-precision algorithm (approximately 1E-11).
[0020] (2) Computational efficiency advantage: Compared with the consistent high-precision algorithm, the present invention introduces low-precision calculation in the precondition part, which can reduce the computational overhead of the precondition and improve the solution efficiency of the iterative method; at the same time, the filtering strategy proposed in the present invention can further reduce the computational overhead of the precondition part. For example, numerical experiments show that within the PGMRES algorithm framework, the mixed-precision block Jacobi preconditioning algorithm achieves approximately 15% performance improvement compared to the consistent high-precision algorithm when solving radiative transport examples (unstructured mesh, matrix size 9,104,256); with the filtering strategy enabled, the mixed-precision block Jacobi preconditioning algorithm with the filtering strategy achieves approximately 49% performance improvement. (3) Advantages in application scenarios: Compared with existing technologies, the adaptive strategy proposed in this invention can automatically select the optimal parameters when solving a series of linear equations, thereby further improving computational efficiency; For example, numerical experiments show that within the PGMRES algorithm framework, the mixed-precision block Jacobi preconditioning algorithm, employing an adaptive strategy, achieves approximately 27% performance improvement when solving radiative transport examples (unstructured mesh, matrix size 9,104,256) compared to the fixed-parameter mixed-precision block Jacobi preconditioning algorithm. Attached Figure Description
[0021] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the accompanying drawings involved in the embodiments are now briefly described. Obviously, the drawings in the following description are merely illustrative of some embodiments of the present invention. For those skilled in the art, other forms of drawings can be constructed based on these drawings without creative effort.
[0022] Figure 1 This is a flowchart illustrating an adaptive mixed precision block Jacobi method incorporating a filtering strategy proposed in this invention.
[0023] Table 2 shows the test results of a block Jacobi preconditioning algorithm with a filtering strategy proposed in this invention.
[0024] Table 3 shows the test results of the adaptive strategy proposed in this invention applied to the radiative transport equation sequence. Detailed Implementation
[0025] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.
[0026] This invention proposes an adaptive mixed-precision block Jacobi method incorporating a filtering strategy, the key points of which are: Key point 1: Filtering strategy for non-zero elements in low-precision preconditioned matrices: When constructing the mixed-precision preconditioning matrix, the off-diagonal elements in the preconditioning matrix are first filtered. The filtered preconditioning matrix is then stored at a lower precision (fp32) instead of the working precision (fp64), denoted as . ( During the solution process, the block BJAC stores and computes preconditioners in a low-precision format to reduce the time overhead and data movement overhead per iteration. Except for the preconditioners, other parts are stored and computed in working precision (fp64).
[0027] Key Point 2: Adaptive tuning strategy for the block Jacobi preconditioning algorithm: In the block Jacobi preconditioning algorithm, the number of iterations within the block is included. t Number of iterations outside the block k Two adjustable parameters. Adjustment... t and k This allows us to obtain preconditioners with varying degrees of approximation. Typically, t and k The larger it is, the more likely it will lead to right The higher the approximation, the better the acceleration effect on the overall solution algorithm, but the higher the overhead of the preconditioner. Therefore, from the perspective of optimal overall performance, it is necessary to select appropriate parameters. t and k This invention proposes an adaptive strategy for optimizing the preconditioning parameters of the block Jacobi algorithm. The adaptive adjustment method is as follows: during the solution process, the number of iterations within the block is increased or decreased based on the convergence of the iterations. t Number of iterations outside the block k。
[0028] This invention provides an adaptive mixed-precision block Jacobi method incorporating a filtering strategy for solving the linear equation system formed after discretization of the radiation energy equation. This method reduces the number of non-zero elements in the preconditioning matrix through a filtering strategy, thereby reducing computational overhead; simultaneously, it employs an adaptive strategy to dynamically adjust the number of iterations *t* within the block and *k* outside the block in the Jacobi preconditioning algorithm, further improving computational efficiency. The adaptive mixed-precision block Jacobi method incorporating a filtering strategy proposed in this invention is described below with reference to the accompanying drawings and specific examples.
[0029] Example 1: This example proposes an adaptive mixed-precision block Jacobi method incorporating a filtering strategy, such as... Figure 1As shown, the linear equations for the radiation transport problem formed by unstructured mesh discretization are solved. The linear equations are formed by discretizing the radiation transport equations spatially, angularly, and through energy groups, and their form is as follows:
[0030] in A The coefficient matrix, x Let be the vector to be solved. b This is the vector of the right-hand side.
[0031] In this embodiment, the coefficient matrix has a size of 9,104,256, and tests were conducted at parallel scales of 56 processes and 112 processes, respectively.
[0032] In practice, the coefficient matrix is first used as a basis. A Constructing the precondition matrix M 0, and stored in single-precision floating-point format fp32; then set the filtering threshold. f tol For the precondition matrix M The absolute value of 0 on the non-diagonal line is less than f tol Elements are filtered to reduce the number of weakly coupled elements in the preconditioning matrix, thereby lowering the storage and computational overhead of the preconditioning stage. After filtering, a low-precision preconditioning matrix is obtained. .
[0033] In the linear iterative solution process, the residual vector of the current iteration step is first obtained. The residual vector is then converted to a low-precision format to obtain... Then solve at low precision. When solving this system of equations, the number of iterations within the block is used. t Number of iterations outside the block k The controlled block Jacobi iteration yields low-precision preconditioning results. , Then Convert to working precision format to obtain working precision vector. The algorithm employs working precision to perform subsequent iterative updates, residual calculations, and convergence checks. This mixed-precision calculation method executes the computationally intensive preconditioning process at low precision, while maintaining the remaining iterative processes that affect convergence accuracy and numerical stability at working precision. This approach improves overall computational efficiency while ensuring solution stability.
[0034] Furthermore, an adaptive parameter selection strategy is introduced during the iteration process. This strategy obtains the residual from the previous iteration step. r j-1 Current iteration step residualr j and initial residual r 0, Get the time taken for a single step of the Krylov iteration. T Iteration optimization time T opt Total calculation time T total And the convergence threshold tol, and based on this, the upper limit threshold for residual descent is calculated. d Simultaneously, obtain the relative residuals of the local equations within the block. x j and the lower limit of the relative residual threshold within the block e in Then, based on the current residual decrease ratio... r j = || r j || / || r j-1 || with d The relationship between tol and the relative residuals of local equations within the block. x j and e in The relationship between algorithm parameters t、k Dynamic adjustment is performed: After the parameter adjustment is completed, the updated intra-block iteration number t and extra-block iteration number k are used for the next low-precision block Jacobi preconditioning action, and the residual precision conversion, low-precision preconditioning action, working precision Krylov iteration update and adaptive parameter adjustment process are repeated until the residual meets the preset convergence threshold, and finally the solution of the linear equation system of the radiation transport problem is completed.
[0035] Table 1. Test results of a block Jacobi preconditioning algorithm incorporating a filtering strategy: computation time and number of iterations.
[0036] Table 2. Test results of a block Jacobi preconditioning algorithm incorporating a filtering strategy: computation time for the radiative transport equation sequence.
[0037] The test results are shown in the tables above. Table 1 shows the test results of the block Jacobi preconditioning algorithm with filtering strategy proposed in this invention, and Table 2 shows the test results of the adaptive strategy proposed in this invention applied to the radiative transport equation sequence. Analysis of the tables shows that when applying the block Jacobi preconditioning algorithm with filtering strategy proposed in this invention to handle the radiative transport problem, in solving a single linear equation system, the performance is improved by approximately 50% compared to the fixed double-precision preconditioning algorithm, without increasing the number of linear iterations. In solving a sequence of linear equations, compared to the mixed-precision BJAC preconditioning algorithm, the adaptive strategy improves performance by approximately 27% at a parallel scale of 56 processes and by approximately 40% at a parallel scale of 112 processes. Therefore, the filtering strategy proposed in this invention can effectively reduce the preconditioning computation cost without affecting the number of iterations to converge, and the adaptive strategy can dynamically adjust the algorithm parameters according to the iteration convergence state and computation time overhead, thereby further improving the overall parallel solution efficiency of the mixed-precision linear solver for the radiative transport problem.
[0038] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some of the technical features; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of the present invention.
Claims
1. An adaptive mixed-precision block Jacobi preconditioning method incorporating a filtering strategy, characterized in that, Includes the following steps: S1. Based on the coefficient matrix of the linear equation system Constructing the precondition matrix Filtering through filtering strategies By omitting the diagonal elements, a low-precision preconditioning matrix is obtained. The coefficient matrix The zero element in the middle represents the loss and accumulation of radiative energy, the transport of radiation between adjacent grids, and the scattering transfer between radiation intensities in different directions; S2. Enter the iterative solution process of the low-precision block Jacobi preconditional Krylov subspace method; S3, in the In the next Krylov iteration, the residual vector at the current working precision is obtained. Then, the working precision is changed to low precision, denoted as ; S4, using low-precision residual vectors For the right-hand side, solve the system of equations using a low-precision scheme: Among them, the number of iterations within the block is used. Number of iterations outside the block Related block Jacobi iterations yield low-precision calculation results. : ； S5. Transfer the low-precision calculation results Converted to working precision, denoted as ; S6. Perform the remaining calculations using the working precision. S7. After each Krylov iteration, the algorithm parameters are adaptively selected based on the convergence of the iteration. t and k ; S8. Repeat S3~S7 until the iteration converges; S9. Obtain the solution to the linear equation system. The solution vector is the radiation intensity, which is used to calculate the radiation energy density, radiation energy flux, and radiation pressure.
2. The method according to claim 1, characterized in that, Step S1 includes the following specific contents; S1.1 Setting the filter threshold ; S1.2, using the coefficient matrix Constructing the precondition matrix ; S1.3, Precondition Matrix The absolute value on the off-diagonal is less than Filter the elements; S1.
4. Store the filtered precondition matrix in a low-precision format, denoted as... .
3. The method according to claim 1, characterized in that, The working precision is double-precision floating-point number fp64, and the low precision is single-precision floating-point number fp32.
4. The method according to claim 1, characterized in that, Step S7 specifically includes the following: S7.1 Obtain residual information and determine convergence: Obtain the residual vector of the previous Krylov iteration step. Residual in the current iteration step and the initial residual vector r 0; Calculate the current relative residual and current residual decrease ; If the following conditions are met: If the current system of linear equations has met the convergence condition, the iteration ends and the parameters are no longer updated. t and k ;like ， Then it is determined that the current system of linear equations has not yet converged, and the adaptive parameter selection process begins; where, Indicates the convergence threshold; S7.2 Obtaining Time Information: Obtain the cumulative computation time consumed in solving the current system of linear equations. Get the time taken for the current iteration step. And obtain the target solution time. ;in, T Take the computation time of the current Krylov iteration step, including the time of the low-precision block Jacobi preconditioning and the remaining working precision computation time in the Krylov method; T opt The target solution time is the preset target solution time, while the target optimization time is the time obtained based on historical solution results and the solution results of the previous linear equation system. S7.3 Obtain the allowable range of intra-block iteration count and out-of-block iteration count. , , , ; S7.4 Calculate the target residual descent threshold: Estimate the maximum number of remaining allowed iterations within the target solution time. : ； In order to remain Make the current relative residual within the step Decrease to the convergence threshold Target residual reduction threshold for: in, This represents the maximum residual reduction ratio expected to be achieved in each subsequent iteration under the current time constraint; S7.5 Obtaining the residuals of equations within a block: Obtaining the relative residuals of local equations within a low-precision Jacobi preconditioned block. Set the lower limit of the relative residual threshold within the block. This is used to determine whether the local equations within a block have reached the preset solution accuracy. S7.6, Update Algorithm Parameters and : ①If Increase the number of iterations within the block. ; ②If Reduce the number of iterations within a block. ; ③If Increase the number of iterations within the block. ; ④If Increase the number of iterations outside the block. .
5. The method according to claim 4, characterized in that, In step S7.2, the This represents the computation time of the current Krylov iteration step, including the time for the low-precision block Jacobi preconditioning and the remaining working precision computation time in the Krylov method. The This represents the preset target solution time, which is the target optimization time obtained based on historical solution results and the solution results of the previous linear equation system.
6. The method according to claim 5, characterized in that, The relative residual in step S7.5 This represents the maximum value of the intra-block relative residuals for all blocks.
7. The method according to claim 1 or 6, characterized in that, Step S7 is defined by the target solution time. Current cumulative time and current single-step time Estimate the remaining number of available iterations. Then, the residual descent threshold required to achieve convergence within the target time is calculated. ; When the actual residual decreases Greater than If the current parameters are insufficient to guarantee convergence within the target time, then it is advisable to prioritize increasing the number of intra-block iterations, which has lower computational cost. ; When the relative residual within the block The preset threshold lower limit has been reached. They believe that continuing to increase The benefits are limited, so we increase the number of out-of-block iterations instead. This achieves a balance between convergence speed and single-step computational overhead.
8. A computer device, characterized in that, The computer device includes a processor and a memory, the memory storing at least one instruction, at least one program, code set, or instruction set, the instruction, program, code set, or instruction set being loaded and executed by the processor to implement the adaptive mixed precision block Jacobi preconditioning method including a filtering strategy as described in any one of claims 1-7.
9. A computer-readable storage medium, characterized in that, The computer-readable storage medium stores at least one instruction, at least one program, code set, or instruction set, which is loaded and executed by a processor to implement the adaptive mixed-precision block Jacobi preconditioning method including a filtering strategy as described in any one of claims 1-7.