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Ron Morgan (Baylor University)5/19/26, 2:00 PMPolynomials, Krylov Methods and ApplicationsMinisymposium Talk
The BiCG method for solving linear equations has a polynomial at its core. The new Twin BiCG method solves multiple right-hand systems using the same polynomial for each system. This polynomial is applied implicitly by using the parameters from solving the first right-hand side for all of the systems. Twin BiCG has automatic stability control from the extra copies of eigenvalues that are...
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Daniele Toni (Scuola Normale Superiore)5/19/26, 2:25 PMPolynomials, Krylov Methods and ApplicationsMinisymposium Talk
We present randomized algorithms for estimating the log-determinant of regularized symmetric positive semi-definite matrices. The algorithms access the matrix only through matrix vector products, and are based on the introduction of a preconditioner and stochastic trace estimator.
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We claim that preconditioning as much as we can and making a rough estimate of the residual part with a small... -
Fabio Matti (EPFL)5/19/26, 2:50 PMPolynomials, Krylov Methods and ApplicationsMinisymposium Talk
Stochastic trace estimators are a family of widely used techniques for approximating traces of large matrices accessible only via matrix-vector products. These methods have been studied extensively when applied to constant matrices $B$. We analyze three standard stochastic trace estimators—the Girard-Hutchinson, Nyström, and Nyström++ estimators—when they are applied to parameter-dependent...
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Charbel Abi Younes (University of Washington)5/20/26, 10:45 AMPolynomials, Krylov Methods and ApplicationsMinisymposium Talk
We introduce a new approach for estimating the asymptotic spectral distribution (ASD) of a random matrix using a single, sufficiently high-dimensional sample, without computing the full spectrum. The method builds on the Lanczos algorithm, together with asymptotic analysis and perturbation theory for orthogonal polynomials, and enables efficient and accurate estimation of the ASD. We...
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Michele Rinelli (KU Leuven)5/20/26, 11:10 AMPolynomials, Krylov Methods and ApplicationsMinisymposium Talk
The deep connection between Krylov methods, scalar orthogonal polynomials, and moment matrices is well established, particularly for Hermitian and unitary matrices. In this talk, we extend this framework to block Krylov methods and orthogonal matrix polynomials.
By representing the elements of a block Krylov subspace via matrix polynomials, we consider the matrix-valued inner product...
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Robbe Vermeiren (KU Leuven)5/20/26, 11:35 AMPolynomials, Krylov Methods and ApplicationsMinisymposium Talk
Rational functions are fundamental to several non-linear approximation problems in, for example, model reduction, system identification, and PDE problems. Consequently, one is often interested in constructing an orthonormal basis of rational functions to ensure numerical stability and conditioning.
In this talk, we present a generalized framework for constructing such bases for rational...
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Andrea Baleani (Scuola Normale Superiore di Pisa)5/21/26, 11:00 AMPolynomials, Krylov Methods and ApplicationsMinisymposium Talk
Appearing in a wide variety of applications, often in the context of discretized (fractional) differential and integral operators, Hierarchically Semiseparable (HSS) matrices have a number of attractive properties facilitating the development of fast algorithms [6,4].
For HSS matrices, the rank-structure is numerically preserved if $f(z)$ is well-approximated by a rational function. The...
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Cooper Simpson (University of Washington)5/21/26, 11:25 AMPolynomials, Krylov Methods and ApplicationsMinisymposium Talk
The Lanczos process is a well-known Krylov subspace method for the orthogonal tridiagonalization of a hermitian matrix $\mathbf{Z}$. Equipped with a suitable function $f$, Lanczos function approximation (LFA) can be used as a powerful tool for approximating the matrix-function $f(\mathbf{Z})$ or matrix-function-vector products $f(\mathbf{Z})\mathbf{\omega}$.
We discuss an application of LFA...
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Robert Webber (University of California San Diego)5/21/26, 11:50 AMPolynomials, Krylov Methods and ApplicationsMinisymposium Talk
The partial pivoted Cholesky approximation accurately represents matrices that are close to being low-rank. Meanwhile, the Vecchia approximation accurately represents matrices with inverse Cholesky factors that are close to being sparse. What happens if a partial Cholesky approximation is combined with a Vecchia approximation of the residual? We show how the sum can be rewritten as a Vecchia...
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Chris Camaño (Caltech)5/22/26, 8:45 AMPolynomials, Krylov Methods and ApplicationsMinisymposium Talk
In recent years, tensor network methods have garnered increased attention for modeling high dimensional quantum many body systems and for representing high dimensional functions with structured correlations.
A basic unresolved question is whether standard numerical linear algebra primitives, such as Krylov based iterative solvers, can be realized in matrix product state and matrix...
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Alexander Hsu (University of Washington)5/22/26, 9:10 AMPolynomials, Krylov Methods and ApplicationsMinisymposium Talk
Computing the diagonal entries of a large linear operator is a common computational primitive in numerical linear algebra, with applications in uncertainty quantification, cross-validation, perturbation analysis, electronic structure calculation and more. However, estimating the diagonals of a matrix given only implicit matrix-vector access is challenging, as randomized algorithms suffer from...
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Raphael Meyer (UC Berkeley)5/22/26, 9:35 AMPolynomials, Krylov Methods and ApplicationsMinisymposium Talk
Solving linear systems Ax=b is a fundamental pillar of NLA. For over 60 years, iterative methods that access A only through matrix-vector products have been the standard approach for solving large linear systems. While lower bounds exist for many special cases, prior work has not shown that methods like GMRES and MINRES achieve an asymptotically optimal matrix-vector complexity for...
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