Efficient Preconditioning for Linear Systems from Non-local Evolutionary Equations with Weakly Singular Kernels

Core Concepts

The paper proposes a single-sided preconditioning technique that is computationally more efficient than the existing two-sided preconditioning method for solving linear systems arising from non-local evolutionary equations with weakly singular kernels.

Abstract

The paper presents a new single-sided preconditioning technique for solving linear systems that arise from the discretization of non-local evolutionary equations with weakly singular kernels. Key highlights: The authors modify the existing two-sided preconditioning method by combining the left and right preconditioners into a single-sided preconditioner. This simplifies the structure of the preconditioned matrix and reduces the computational cost of matrix-vector multiplications. The authors show theoretically that the single-sided preconditioned GMRES method has a convergence rate no worse than the two-sided preconditioned GMRES method. This ensures that the single-sided preconditioning approach maintains the fast convergence properties of the two-sided method. The authors provide a fast implementation of the matrix-vector product for the single-sided preconditioned system, which further improves the computational efficiency. Numerical results demonstrate that the single-sided preconditioning method outperforms the two-sided preconditioning method in terms of computational time and iteration numbers. The paper presents a comprehensive theoretical analysis and practical implementation of the proposed single-sided preconditioning technique, which can be efficiently applied to solve linear systems arising from various spatial discretization schemes for non-local evolutionary equations with weakly singular kernels.

Stats

The linear system (2.3) arises from the discretization of the non-local evolutionary equation (1.1). The matrix A in the linear system (2.3) has a Kronecker product structure, where B is a positive definite symmetric matrix and T is a lower triangular Toeplitz matrix. The condition number of the preconditioned matrix P^-1 A P^-1 is uniformly bounded by 3, independent of the discretization step sizes.

Quotes

"Clearly, the matrix-vector multiplication of the single-sided preconditioning is faster to compute than that of the two-sided one, since the single-sided preconditioned matrix has a simpler structure." "More importantly, we show theoretically that the single-sided preconditioned generalized minimal residual (GMRES) method has a convergence rate no worse than the two-sided preconditioned one."

Key Insights Distilled From

A single-sided all-at-once preconditioning for linear system from a non-local evolutionary equation with weakly singular kernels

by Xuelei Lin,J... at arxiv.org 04-23-2024

https://arxiv.org/pdf/2404.13974.pdf

Deeper Inquiries

How can the proposed single-sided preconditioning technique be extended to handle more general non-local operators or nonlinear problems

The proposed single-sided preconditioning technique can be extended to handle more general non-local operators or nonlinear problems by adapting the concept of approximating the spatial matrix with a fast diagonalizable matrix. For more general non-local operators, the key would be to identify an appropriate approximation method that simplifies the spatial matrix while keeping the temporal matrix unchanged. This could involve utilizing different types of diagonalizable matrices or other spatial approximations that can effectively reduce the complexity of the system. In the case of nonlinear problems, the preconditioning technique may need to be adjusted to account for the nonlinearity in the system. This could involve incorporating iterative methods or nonlinear solvers within the preconditioning step to handle the nonlinear terms effectively. By carefully designing the preconditioning strategy to address the specific characteristics of the non-local operators or nonlinearities, the single-sided preconditioning technique can be extended to a broader range of problems.

What are the potential limitations or challenges in applying the single-sided preconditioning method to real-world applications

While the single-sided preconditioning method offers advantages such as faster computation and simpler structure compared to traditional two-sided preconditioning techniques, there are potential limitations and challenges in applying this method to real-world applications. One limitation could be the scalability of the method to larger and more complex systems. As the size of the linear system increases, the computational cost of the preconditioning step may also increase significantly, potentially impacting the overall efficiency of the solver. Additionally, the effectiveness of the single-sided preconditioning technique may vary depending on the specific characteristics of the problem, such as the distribution of singularities or the nature of the non-local operators. Another challenge could be the generalizability of the method to diverse real-world applications. Real-world problems often involve a combination of different types of operators, boundary conditions, and nonlinearities, which may require tailored preconditioning strategies. Adapting the single-sided preconditioning method to address the unique requirements of various applications may require extensive testing and optimization to ensure its effectiveness.

Can the ideas behind the single-sided preconditioning be applied to other types of linear systems beyond the non-local evolutionary equations considered in this paper

The ideas behind the single-sided preconditioning technique can be applied to other types of linear systems beyond the non-local evolutionary equations considered in the paper. The concept of simplifying the spatial matrix while keeping the temporal matrix unchanged can be generalized to various linear systems with structured matrices. For example, in systems with sparse matrices or specific patterns, the single-sided preconditioning method can be utilized to reduce the computational complexity and improve the efficiency of iterative solvers. By identifying the key characteristics of the linear system, such as Toeplitz-like structures or shift-invariant kernels, similar preconditioning techniques can be developed to enhance the convergence rate and reduce the computational cost. Furthermore, the principles of single-sided preconditioning, such as utilizing fast diagonalizable matrices and optimizing the matrix-vector multiplication, can be applied to a wide range of scientific and engineering problems. By customizing the preconditioning strategy to suit the specific properties of the linear system, the benefits of the single-sided approach can be leveraged in diverse applications, including computational fluid dynamics, structural analysis, and optimization problems.