Modified Gauss-Seidel Preconditioner for Parallel Physical Field Simulation

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Solution Overview

Problem

Existing domain decomposition methods for simulating physical fields face challenges in achieving parallel scalability and convergence rate due to the sequential nature of Gauss-Seidel preconditioners and the lack of parallelizability of Jacobi preconditioners, which limits the efficiency of solving large system matrices.

Innovation Solution

A modified Gauss-Seidel preconditioner is introduced, combining the advantages of Jacobi and Gauss-Seidel preconditioners by omitting higher-order terms, allowing for parallelization while maintaining convergence rates similar to unmodified Gauss-Seidel preconditioners.

Engineering Contradictions & Design Principles

VSEngineering Contradiction Analysis

1Speed

If Gauss-Seidel preconditioner is used, then convergence rate is improved, but parallel scalability deteriorates due to sequential processing requirements

Engineering Contradiction:

Improveconvergence rateVSAvoidparallel scalability

Core Design Contradiction:

SpeedVSProductivity

Solution Approach 1:

The patent segments the Gauss-Seidel preconditioning process by decomposing the system matrix into subdomains and applying preconditioning operations independently to each subdomain. This segmentation allows parallel execution of preconditioning steps while maintaining the essential convergence properties of Gauss-Seidel through careful ordering of subdomain processing.

Inventive Principle:

Principle #1Segmentation

Solution Approach 2:

The patent introduces a new dimensional approach by organizing subdomains in a hierarchical structure with multiple levels. This dimensional transformation enables simultaneous processing of independent subdomains at the same hierarchy level while preserving the sequential dependency relationships through the hierarchy structure, thereby achieving both parallel scalability and convergence.

Inventive Principle:

Principle #17Another dimension (Dimensionality change)

2Productivity

If Jacobi preconditioner is used, then parallelizability is improved, but convergence rate deteriorates due to disregard of domain ordering

Engineering Contradiction:

ImproveparallelizabilityVSAvoidconvergence rate

Core Design Contradiction:

ProductivityVSSpeed

Solution Approach 1:

The patent applies preliminary ordering of subdomains based on their spatial or logical relationships in the system matrix. By pre-establishing an optimal processing sequence for subdomains, the method prepares the structure in advance to capture directional information flow, which then guides the parallel preconditioning process to respect domain ordering and improve convergence.

Inventive Principle:

Principle #10Preliminary action

Solution Approach 2:

The patent introduces an intermediary hierarchical structure that mediates between the parallel Jacobi approach and the sequential Gauss-Seidel requirement. This intermediary layer organizes subdomains into levels and groups, allowing parallel processing within groups while maintaining ordered progression between groups, thus bridging the gap between parallelizability and convergence rate.

Inventive Principle:

Principle #24Intermediary (Mediator)

Data Source

PatentUS9063882B1Matrix preconditioners for simulations of physical fields

Publication Date: 2015.06.23 ANSYS INC

AI summary

In a system for simulations of fields related to physical objects by iteratively solving a system matrix equation, the system matrix is preconditioned using a modified Gauss-Seidel preconditioner so as to eliminate or reduce a sequential nature of the iterations, thus facilitating parallel solution of the conditioned system matrix.