Search

CN-121980126-A - Mixed aggregation algebraic multiple grid solving method for time series linear system

CN121980126ACN 121980126 ACN121980126 ACN 121980126ACN-121980126-A

Abstract

The invention provides a mixed aggregation algebraic multi-grid solving method for a time sequence linear system, which relates to the field of grid data and comprises the steps of setting hierarchical demarcation parameters for a first linear system of the time sequence, constructing a coarse layer matrix through Galerkin projection, then caching related topology and communication information, solving the first system by using a multi-grid combined with a Krylov iteration method, recording a reference convergence state, judging whether a matrix sparse structure is consistent with the cache, multiplexing the cache to update only a numerical value if the matrix sparse structure is consistent with the cache, and then judging whether the current convergence state reaches a degradation threshold value, and if the current convergence state is inconsistent or converged to reach the degradation threshold value, re-executing a first construction flow to update the cache and then solving, and outputting a complete solution sequence after all the systems are processed. The mixed aggregation algebraic multi-grid solving method for the time series linear system solves the problem that convergence robustness and low construction cost are difficult to be compatible when smooth aggregation and non-smooth aggregation are singly adopted, and can be compatible.

Inventors

  • ZHAO CHENGPENG

Assignees

  • 无锡恒鼎超级计算中心有限公司
  • 神工坊(安徽)智能科技有限责任公司

Dates

Publication Date
20260505
Application Date
20260120

Claims (9)

  1. 1. A mixed aggregation algebraic multiple grid solving method for a time series linear system is characterized by comprising the following steps: s1, for the first linear system in the time sequence, performing first algebraic multiple grid hierarchical structure construction, including S11, setting level demarcation parameters ; S12, for the first layer to the first layer The layer adopts a non-smooth aggregation strategy to construct an interlayer transfer operator ; S13 for The layer and the coarser layer adopt a smooth aggregation strategy to construct an interlayer transfer operator; s14, constructing each coarse layer matrix through Galerkin projection based on interlayer transfer operators Caching the sparse structure of the interlayer transfer operator, the topological structure of the coarse layer matrix and the parallel communication plan; s15, solving a first linear system to obtain a first solution, storing the first solution in a solution sequence, and recording the convergence state of the solution as a reference; s2, judging whether a linear system to be solved exists in the time sequence, if so, executing a step S3, otherwise, executing a step S8; s3, judging whether the matrix sparse structure of the current linear system to be solved is consistent with the sparse structure cached in the step S14; If the two values are consistent, executing a step S4; If not, executing the step S7; s4, multiplexing the sparse structure, the topological structure and the communication plan cached in the step S14, only updating the numerical value of each layer of matrix, and solving the current linear system based on the updated hierarchical structure to obtain the current solution; S5, monitoring the convergence state solved in the step S4, and comparing the convergence state with the reference convergence state recorded in the step S15; S6, judging whether the comparison result meets a preset degradation threshold value; if yes, executing step S7; If not, executing step S9; S7, executing the initialization construction step S1 to update a cache, and solving a current linear system by using the updated cache to obtain a current solution; s8, outputting a de-sequence; And S9, storing the current solution to the solution sequence, and returning to the step S2.
  2. 2. The method of claim 1, wherein in step S12, the non-smooth aggregation policy is embodied based on a layer one Matrix array Generating clusters of strong connections or adjacencies of (1) and constructing transient extension operators And directly will As a final continuation operator And is provided with To obtain a non-smooth P/R.
  3. 3. The method of claim 1, wherein in step S13, the smooth aggregation policy is implemented by first based on the first Matrix array Generating clusters of strong connections or adjacencies of (1) and constructing transient extension operators Then through matrix related smooth operator pair Correcting to obtain the final continuation operator The smooth operator is in the form of Wherein Is that Is arranged in a diagonal array of the pair, As a parameter of the damping it is possible to provide, The method comprises the steps of obtaining a system matrix after strong connection filtering; the limiting operator is the transposition of the continuation operator 。
  4. 4. The method of claim 1, wherein in step S14, the cached content includes: The sparse structure of (1) includes row pointers, column indexes; non-zero position mode of intermediate product in Galerkin projection process, wherein the intermediate product is Or (b) ; Communication plans in a parallel computing environment with a data packing index that includes a set of row block indices or column block indices that are swapped across a process.
  5. 5. The method of claim 1, wherein in step S3, judging whether the matrix sparse structure of the linear system to be solved is consistent with the sparse structure cached in step S14 or not comprises judging by a grid topology unchanged mark, a pattern version number provided by a matrix assembler or a fast sampling hash check mode; If the sparse structure of the current matrix is consistent with the cache structure, the sparse structure of the current matrix is judged to be consistent, and step S4 is executed.
  6. 6. The method of claim 1, wherein updating only the values of the layer matrices in step S4 comprises: recalculating coarse layer matrix based on topology of cache Is a non-zero element value of (2); if the smoother relies on the diagonal matrix or block diagonal matrix data, the derivative data of the diagonal matrix or block diagonal matrix is updated synchronously.
  7. 7. The method of claim 1, wherein the convergence status in step S15 and step S5 comprises at least one of the following: Krylov iteration number Relative reference step An increase ratio of (2); logarithmic residual delta attenuation for unit iteration ; The execution times of V-cycle or W-cycle in the algebraic multiple grid; degradation index of the coarse layer correction effect.
  8. 8. The method according to claim 1 or 7, wherein the preset degradation threshold in step S6 comprises at least one of: the iteration times threshold value is that the ratio of the iteration times of the current convergence state to the iteration times of the reference convergence state is more than 1.5; residual attenuation threshold value: unit iteration log residual attenuation of current convergence state Lower than a preset minimum value; Convergence factor threshold value-the magnitude of the deterioration of the convergence factor in the current convergence state exceeds a preset tolerance.
  9. 9. The method of claim 1, wherein the linear system is solved in step S15 and step S4, specifically using a V-cycle or W-cycle multiple grid loop as a preconditioner, in combination with Krylov iteration methods including conjugate gradient methods or stable bi-conjugate gradient methods.

Description

Mixed aggregation algebraic multiple grid solving method for time series linear system Technical Field The invention relates to the field of grid data, in particular to a mixed aggregation algebraic multiple grid solving method for a time series linear system. Background In Computational Fluid Dynamics (CFD) simulation, especially where incompressible flows are involved, a large pressure equation needs to be solved repeatedly at each calculation step. Algebraic Multiple Grid (AMG) methods are core techniques to accelerate the solution of such equations. Currently, AMGs mainly have two implementation paths: Smooth aggregation (SA-AMG) has good convergence and stable result, but the construction process has large calculation amount and low speed. Non-smooth aggregation (UA-AMG) is fast in construction and low in cost, but when complex problems are handled, situations of slow convergence or instability may occur. In CFD simulations that require long repeated solutions, the user has to trade off between converging reliable but slow building SA-AMG and building fast but possibly unreliable UA-AMG. Whichever is chosen, it is difficult to maintain the robustness of the solution at all times while ensuring computational efficiency. Disclosure of Invention Aiming at the problems, the mixed aggregation algebraic multiple grid solving method for the time series linear system provided by the invention can achieve convergence robustness and low construction cost. In order to achieve the above purpose, the technical scheme adopted by the invention is as follows: The invention provides a mixed aggregation algebraic multiple grid solving method for a time series linear system, which comprises the following steps: s1, for the first linear system in the time sequence, performing first algebraic multiple grid hierarchical structure construction, including S11, setting level demarcation parameters; S12, for the first layer to the first layerThe layer adopts a non-smooth aggregation strategy to construct an interlayer transfer operator; S13 forThe layer and the coarser layer adopt a smooth aggregation strategy to construct an interlayer transfer operator; s14, constructing each coarse layer matrix through Galerkin projection based on interlayer transfer operators Caching the sparse structure of the interlayer transfer operator, the topological structure of the coarse layer matrix and the parallel communication plan; s15, solving a first linear system to obtain a first solution, storing the first solution in a solution sequence, and recording the convergence state of the solution as a reference; s2, judging whether a linear system to be solved exists in the time sequence, if so, executing a step S3, otherwise, executing a step S8; s3, judging whether the matrix sparse structure of the current linear system to be solved is consistent with the sparse structure cached in the step S14; If the two values are consistent, executing a step S4; If not, executing the step S7; s4, multiplexing the sparse structure, the topological structure and the communication plan cached in the step S14, only updating the numerical value of each layer of matrix, and solving the current linear system based on the updated hierarchical structure to obtain the current solution; S5, monitoring the convergence state solved in the step S4, and comparing the convergence state with the reference convergence state recorded in the step S15; S6, judging whether the comparison result meets a preset degradation threshold value; if yes, executing step S7; If not, executing step S9; S7, executing the initialization construction step S1 to update a cache, and solving a current linear system by using the updated cache to obtain a current solution; s8, outputting a de-sequence; And S9, storing the current solution to the solution sequence, and returning to the step S2. The mixed aggregation algebraic multiple grid solving method for the time series linear system provided by the invention is characterized in that in the step S12, the non-smooth aggregation strategy is concretely realized based on a layerMatrix arrayGenerating clusters of strong connections or adjacencies of (1) and constructing transient extension operatorsAnd directly willAs a final continuation operatorAnd is provided withTo obtain a non-smooth P/R. The invention provides a mixed aggregation algebraic multiple grid solving method for a time series linear system, preferably in step S13, the smooth aggregation strategy is realized by firstly based on the first stepMatrix arrayGenerating clusters of strong connections or adjacencies of (1) and constructing transient extension operatorsThen through matrix related smooth operator pairCorrecting to obtain the final continuation operatorThe smooth operator is in the form ofWhereinIs thatIs arranged in a diagonal array of the pair,As a parameter of the damping it is possible to provide,The method comprises the steps of obtaining a system matrix after strong conn