[Paper Review] Polynomial Preconditioned Arnoldi

Mark Embree, Jennifer Loe+1 more|arXiv (Cornell University)|Jun 21, 2018

Matrix Theory and Algorithms25 references3 citations

TL;DR

This paper introduces polynomial preconditioned Arnoldi methods that accelerate eigenvalue computation for large, difficult matrices by applying GMRES-generated polynomials to shift the spectrum, enabling efficient use of high-degree polynomial approximations without increasing Krylov subspace dimension. The method drastically reduces communication-intensive dot products—especially beneficial in parallel computing—while maintaining convergence through stability-enhancing techniques like double polynomial preconditioning and root duplication.

ABSTRACT

Polynomial preconditioning can improve the convergence of the Arnoldi method for computing eigenvalues. Such preconditioning significantly reduces the cost of orthogonalization; for difficult problems, it can also reduce the number of matrix-vector products. Parallel computations can particularly benefit from the reduction of communication-intensive operations. The GMRES algorithm provides a simple and effective way of generating the preconditioning polynomial. For some problems high degree polynomials are especially effective, but they can lead to stability problems that must be mitigated. A two-level "double polynomial preconditioning" strategy provides an effective way to generate high-degree preconditioners.

Motivation & Objective

• Address slow convergence of standard Arnoldi methods for large, difficult eigenvalue problems.
• Develop a practical, low-cost alternative to shift-invert spectral transformations that avoids matrix inversion.
• Reduce communication and orthogonalization costs in parallel computing by leveraging matrix-vector products over inner products.
• Ensure numerical stability when using high-degree polynomials in preconditioning.
• Enable efficient computation of interior eigenvalues through advanced preconditioning strategies.

Proposed method

• Apply the Arnoldi method to $\pi(A)$, where $\pi$ is a polynomial preconditioner derived from the GMRES minimum residual polynomial.
• Use the GMRES algorithm to generate a polynomial $\pi$ that maps desired eigenvalues of $A$ to large-magnitude eigenvalues of $\pi(A)$, improving convergence.
• Construct Krylov subspaces of the form $\mathcal{K}_m(\pi(A), v)$, which are subspaces of high-degree Krylov spaces $\mathcal{K}_{d(m-1)+1}(A,v)$, enabling high-accuracy approximations with low dimension.
• Implement double polynomial preconditioning by applying a second GMRES polynomial $\pi_2$ to $\tau(A) = 1 - \pi_1(A)$, forming a composite polynomial of degree $d_1 d_2$ for high-degree approximation without high-dimensional subspaces.
• Stabilize the method by adding duplicate roots to the preconditioning polynomial, particularly when eigenvalues are poorly separated.
• Use the MaxPof test to detect instability and guide root duplication for improved convergence.

Experimental results

Research questions

• RQ1 Can polynomial preconditioning via GMRES-generated polynomials significantly reduce the number of matrix-vector products and dot products in Arnoldi-based eigenvalue computation?
• RQ2 How can high-degree polynomial approximations be effectively used in eigenvalue problems without increasing Krylov subspace dimension or orthogonalization cost?
• RQ3 What strategies can stabilize polynomial preconditioning when eigenvalues are poorly separated or when high-degree polynomials lead to numerical instability?
• RQ4 Can double polynomial preconditioning enable the use of very high-degree polynomials efficiently, and how does it compare to single preconditioning in terms of computational cost and convergence?
• RQ5 How can the choice of GMRES starting vector and its damping influence the effectiveness of polynomial preconditioning in capturing desired eigenvalues?

Key findings

• For a convection-diffusion problem, double polynomial preconditioning with degree $25 \times 40 = 1000$ reduced dot products from 52,312 to 321.0, a tenfold decrease compared to single preconditioning.
• With $d_1 = 15$, $d_2 = 20$, double preconditioning achieved convergence in two Arnoldi cycles with only 2.1 dot products per cycle, compared to 9.9 for single preconditioning.
• For a matrix with a problematic eigenvalue near 20,000, a single polynomial preconditioner with $d_1 = 5$ failed to converge due to an outstanding eigenvalue in $\pi_2(\tau(A))$, but adding a double root enabled convergence in two cycles.
• Double polynomial preconditioning allowed Arnoldi(50,20) to converge in one cycle with composite degree 1000, demonstrating scalability for high-degree approximations.
• The MaxPof test successfully identified instability in preconditioners and guided effective root duplication to restore convergence.
• The method reduces communication costs significantly—critical for high-performance computing—by replacing expensive dot products with cheaper matrix-vector products.