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Ilse Ipsen (North Carolina State University)5/18/26, 3:45 PMNew Directions and Challenges in Linear AlgebraMinisymposium Talk
It is well known that the problem of selecting k columns with maximal volume from a real matrix is NP-hard, and does not admit a polynomial time approximation scheme. However, one can show that the logarithm of the volume is a "submodular" function. Intuitively, this means: Adding a new column to a larger submatrix tends to be less beneficial than adding it to a smaller submatrix.
If subset...
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Cameron Musco (University of Massachusetts Amherst)5/18/26, 4:10 PMNew Directions and Challenges in Linear AlgebraMinisymposium Talk
In this talk, I will give an overview of recent progress on the problem of structured matrix approximation from matrix-vector products. Given a target matrix A that can only be accessed through a limited number of (possibly adaptively chosen) matrix-vector products, we seek to find a near-optimal approximation to A from some structured matrix class – e.g., a low-rank approximation, a...
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John Urschel (MIT)5/18/26, 4:35 PMNew Directions and Challenges in Linear AlgebraMinisymposium Talk
The discrete Fourier transform matrix, and sub-matrices of it, appear in a wide variety of applications. While the Fourier matrix itself is a scaled unitary matrix, its sub-matrices can be exponentially ill-conditioned. In this talk, we discuss applications, prior work, and we provide tight estimates for just how ill-conditioned such matrices can be.
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Stefan Güttel (The University of Manchester)5/18/26, 5:00 PMNew Directions and Challenges in Linear AlgebraMinisymposium Talk
Randomized sketching is a promising tool to reduce the number of inner products computed in Krylov methods for solving large systems of linear equations $Ax = b$, or more generally, when approximating the action of a matrix function on a vector, $f(A)b$. For the case of linear systems, it has recently been observed that one can often get away with completely inner product-free Krylov-based...
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Ioana Dumitriu (University of California, San Diego)5/19/26, 3:45 PMNew Directions and Challenges in Linear AlgebraMinisymposium Talk
The fastest method for diagonalizing a nonsymmetric matrix or matrix pencil is pseudospectral divide-and-conquer. This two-step algorithm diagonalizes a matrix/pencil by (1) randomly perturbing the input(s) and (2) running fast (and highly-parallel) spectral divide-and-conquer. The key to this approach is the random perturbation, which with high probability implies a guarantee of...
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Ryan Schneider (University of California Berkeley)5/19/26, 4:10 PMNew Directions and Challenges in Linear AlgebraMinisymposium Talk
While pseudospectral divide-and-conquer is optimal for nonsymmetric eigenvalue problems (in terms of both arithmetic and communication complexity) it is not currently implemented in any of our standard numerical linear algebra libraries. This is due to both the difficulty of translating the algorithm's technical pseudocode into something practical and to the challenge of preparing users for a...
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Diana Halikias (New York University)5/19/26, 4:35 PMNew Directions and Challenges in Linear AlgebraMinisymposium Talk
There is a mystery at the heart of operator learning: how can one recover a non-self-adjoint operator from data without probing the adjoint? Current practical approaches suggest that one can accurately recover an operator while only using data generated by the forward action of the operator without access to the adjoint. However, naively, it seems essential to sample the action of the adjoint....
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Heather Wilber (University of Washington)5/19/26, 5:00 PMNew Directions and Challenges in Linear AlgebraMinisymposium Talk
In low rank approximations to kernel matrices, skeletons consist of subsets of rows and columns from which CUR, ID, and related approximations can be formed. We consider applications in which parameter-dependent families of matrices with numerical low rank structures appear, such as in parameter dependent partial differential equations. We develop new techniques for analyzing and constructing...
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