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Efficient Nonlinear Eigensolvers Using Contour Integration and Infinite GMRES


Core Concepts
The core message of this paper is to employ the infinite GMRES algorithm in contour integral-based nonlinear eigensolvers to efficiently solve the linear parameterized systems that arise, avoiding the computation of costly factorizations at each quadrature node.
Abstract
The paper presents a method to solve nonlinear eigenvalue problems (NEPs) efficiently using contour integral-based algorithms combined with the infinite GMRES (infGMRES) method. Key highlights: NEPs are important in many applications but are challenging to solve due to their large scale and nonlinearity. Contour integral-based algorithms are well-suited for extracting eigenvalues in a region of interest, but require solving a series of linear systems which can be computationally expensive. The authors propose using infGMRES to solve these linear parameterized systems efficiently, avoiding the need for costly matrix factorizations at each quadrature node. They analyze the relationship between polynomial eigenvalue problems and their scaled linearizations, and provide a novel weighting strategy to accelerate the convergence of infGMRES in this context. The authors also adopt the TOAR technique to reduce the memory footprint of infGMRES. Theoretical analysis and numerical experiments demonstrate the efficiency of the proposed algorithm for large, sparse NEPs.
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Deeper Inquiries

How can the proposed weighting strategy be extended or generalized to handle a broader class of nonlinear eigenvalue problems beyond polynomial forms

The proposed weighting strategy can be extended or generalized to handle a broader class of nonlinear eigenvalue problems beyond polynomial forms by considering different types of parameter-dependent matrices. Instead of focusing solely on polynomial eigenvalue problems, we can apply the weighting strategy to other types of nonlinear eigenvalue problems, such as rational eigenvalue problems or transcendental eigenvalue problems. By appropriately selecting the weights based on the characteristics of the specific problem, we can guide the GMRES algorithm to converge more efficiently and accurately. This extension allows for the adaptation of the weighting strategy to a wider range of nonlinear eigenvalue problems, providing a more versatile and effective approach to solving such problems.

What are the limitations of the infinite GMRES approach, and under what conditions might alternative linear system solvers be more appropriate

The infinite GMRES approach has certain limitations that may make alternative linear system solvers more appropriate under specific conditions. One limitation is the potential sensitivity to the choice of expansion points or scaling factors, which can impact the convergence and accuracy of the method. Additionally, the memory requirements of infinite GMRES can become prohibitive for very large problem sizes, limiting its scalability. In cases where the eigenvalues are clustered or the problem structure is ill-conditioned, infinite GMRES may struggle to converge efficiently. Alternative linear system solvers, such as iterative methods like BiCGSTAB or direct solvers like LU decomposition, may be more suitable in these scenarios. These solvers can offer better stability, convergence properties, and memory efficiency for certain types of nonlinear eigenvalue problems.

Can the memory-efficient TOAR technique be further improved or combined with other compression methods to enable infGMRES to scale to even larger problem sizes

The memory-efficient TOAR technique can be further improved or combined with other compression methods to enable infGMRES to scale to even larger problem sizes. One approach to enhancing TOAR is to incorporate adaptive strategies for selecting the level of orthogonalization based on the problem characteristics. By dynamically adjusting the level of orthogonalization during the Arnoldi process, the memory usage can be optimized without sacrificing accuracy. Additionally, combining TOAR with advanced compression techniques, such as randomized algorithms or hierarchical matrix approximations, can further reduce the memory footprint of infGMRES. These hybrid approaches can enhance the scalability of infGMRES to handle extremely large nonlinear eigenvalue problems efficiently.
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