[Paper Review] A residual concept for Krylov subspace evaluation of the 𝓁 matrix function.

Mike A. Botchev, Leonid Knizhnerman+1 more|arXiv (Cornell University)|Oct 16, 2020

Matrix Theory and Algorithms58 references4 citations

TL;DR

This paper proposes a novel Krylov subspace algorithm for efficiently computing actions of the $ϕ$ matrix function in large-scale problems, using a residual-based stopping criterion and an innovative restarting procedure. It guarantees convergence for matrices with numerical range in the stable complex half-plane and demonstrates high efficiency in solving time-dependent PDEs such as diffusion and convection-diffusion equations.

ABSTRACT

An efficient Krylov subspace algorithm for computing actions of the $\varphi$ matrix function for large matrices is proposed. This matrix function is widely used in exponential time integration, Markov chains and network analysis and many other applications. Our algorithm is based on a reliable residual based stopping criterion and a new efficient restarting procedure. For matrices with numerical range in the stable complex half plane, we analyze residual convergence and prove that the restarted method is guaranteed to converge for any Krylov subspace dimension. Numerical tests demonstrate efficiency of our approach for solving large scale evolution problems resulting from discretized in space time-dependent PDEs, in particular, diffusion and convection-diffusion problems.

Motivation & Objective

• To develop an efficient and reliable algorithm for computing actions of the $ϕ$ matrix function in large-scale applications.
• To address the challenge of numerical instability and slow convergence in existing Krylov methods for $ϕ$-functions.
• To ensure convergence for matrices with numerical range in the stable complex half-plane using a new restarting strategy.
• To provide a practical stopping criterion based on residual norms for improved computational efficiency.

Proposed method

• The method employs a Krylov subspace framework to approximate the action of the $ϕ$ matrix function on a vector.
• A residual-based stopping criterion is used to dynamically determine convergence, reducing unnecessary iterations.
• A new restarting procedure is introduced to maintain accuracy and stability over multiple iterations.
• The algorithm is designed for matrices whose numerical range lies in the stable complex half-plane, ensuring theoretical convergence.
• The method is applied to time integration of large-scale PDEs, particularly diffusion and convection-diffusion problems.
• Theoretical analysis confirms convergence of the restarted method under the stated stability condition.

Experimental results

Research questions

• RQ1 How can the action of the $ϕ$ matrix function be computed efficiently for large sparse matrices in time integration?
• RQ2 What stopping criterion ensures both accuracy and computational efficiency in Krylov-based $ϕ$-function evaluation?
• RQ3 Can a new restarting strategy improve convergence and stability in Krylov subspace methods for $ϕ$-functions?
• RQ4 Under what conditions is the restarted Krylov method guaranteed to converge for $ϕ$-functions?
• RQ5 How does the proposed method perform on real-world PDE problems like convection-diffusion and diffusion equations?

Key findings

• The proposed algorithm guarantees convergence for any Krylov subspace dimension when the matrix has numerical range in the stable complex half-plane.
• The residual-based stopping criterion effectively balances accuracy and computational cost.
• The new restarting procedure enhances stability and efficiency in long-term time integration of large-scale PDEs.
• Numerical experiments confirm high efficiency in solving time-dependent PDEs, including diffusion and convection-diffusion problems.
• The method outperforms existing approaches in terms of robustness and convergence behavior for large-scale problems.
• The theoretical analysis supports the reliability of the method under broad conditions, making it suitable for practical applications.