Vassilis Kalantzis

Vassilis Kalantzis, Yuanzhe Xi, Yousef Saad

This paper proposes a rational filtering domain decomposition technique for the solution of large and sparse symmetric generalized eigenvalue problems. The proposed technique is purely algebraic and decomposes the eigenvalue problem associated with each subdomain into two disjoint subproblems. The first subproblem is associated with the interface variables and accounts for the interaction among neighboring subdomains. To compute the solution of the original eigenvalue problem at the interface variables we leverage ideas from contour integral eigenvalue solvers. The second subproblem is associated with the interior variables in each subdomain and can be solved in parallel among the different subdomains using real arithmetic only. Compared to rational filtering projection methods applied to the original matrix pencil, the proposed technique integrates only a part of the matrix resolvent while it applies any orthogonalization necessary to vectors whose length is equal to the number of interface variables. In addition, no estimation of the number of eigenvalues lying inside the interval of interest is needed. Numerical experiments performed in distributed memory architectures illustrate the competitiveness of the proposed technique against rational filtering Krylov approaches.

Anton Lebedev, Won Kyung Lee, Soumyadip Ghosh et al.

Large, sparse linear systems are pervasive in modern science and engineering, and Krylov subspace solvers are an established means of solving them. Yet convergence can be slow for ill-conditioned matrices, so practical deployments usually require preconditioners. Markov chain Monte Carlo (MCMC)-based matrix inversion can generate such preconditioners and accelerate Krylov iterations, but its effectiveness depends on parameters whose optima vary across matrices; manual or grid search is costly. We present an AI-driven framework recommending MCMC parameters for a given linear system. A graph neural surrogate predicts preconditioning speed from $A$ and MCMC parameters. A Bayesian acquisition function then chooses the parameter sets most likely to minimise iterations. On a previously unseen ill-conditioned system, the framework achieves better preconditioning with 50\% of the search budget of conventional methods, yielding about a 10\% reduction in iterations to convergence. These results suggest a route for incorporating MCMC-based preconditioners into large-scale systems.

Vassilis Kalantzis, Georgios Kollias, Shashanka Ubaru et al.

This paper considers the problem of updating the rank-k truncated Singular Value Decomposition (SVD) of matrices subject to the addition of new rows and/or columns over time. Such matrix problems represent an important computational kernel in applications such as Latent Semantic Indexing and Recommender Systems. Nonetheless, the proposed framework is purely algebraic and targets general updating problems. The algorithm presented in this paper undertakes a projection view-point and focuses on building a pair of subspaces which approximate the linear span of the sought singular vectors of the updated matrix. We discuss and analyze two different choices to form the projection subspaces. Results on matrices from real applications suggest that the proposed algorithm can lead to higher accuracy, especially for the singular triplets associated with the largest modulus singular values. Several practical details and key differences with other approaches are also discussed.