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Yang Liu (Lawrence Berkeley National Laboratory)5/20/26, 10:45 AMHierarchical Low-Rank Approximations: Algorithms and ApplicationsMinisymposium Talk
The development of hierarchical matrix techniques has been essential for many modern scientific computing frameworks including fast direct solvers for PDEs and integral equations, scalable kernel methods and Gaussian processes, and second-order optimization and inverse problems, etc. These algorithms oftentimes lead to optimal computational and memory complexities. That said, when dealing with...
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Robin Armstrong (Cornell University)5/20/26, 11:10 AMHierarchical Low-Rank Approximations: Algorithms and ApplicationsMinisymposium Talk
Many algorithms in data assimilation and model order reduction rely on sample-based estimates for a covariance matrix associated with the trajectory of a high-dimensional dynamical system. The number of available samples is often far less than the dimension of the underlying state space because of computational constraints. Under these circumstances, extracting meaningful covariance...
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Abraham Khan (North Carolina State University)5/20/26, 11:35 AMHierarchical Low-Rank Approximations: Algorithms and ApplicationsMinisymposium Talk
Kernel matrices arising in applications such as Gaussian processes may not always admit a low-rank approximation. Important examples are kernel matrices induced by certain members of the Matérn family of covariance kernels, with smaller length scales and values of $\nu$. Still, they can often be approximated by a hierarchical matrix ($\mathcal{H}$-matrix or $\mathcal{H}^{2}$-matrix), which...
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Chenyang Cao (Purdue University)5/21/26, 11:00 AMHierarchical Low-Rank Approximations: Algorithms and ApplicationsMinisymposium Talk
This talk gives a superfast divide-and-conquer algorithm for computing the full singular value decomposition (SVD) of hierarchical rank-structured matrices with small off-diagonal ranks. The method achieves nearly linear complexity while delivering all singular values and singular vectors in structured forms. The structured representation of singular vectors enables near-linear operations with...
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Christopher Wang (Cornell University)5/21/26, 11:25 AMHierarchical Low-Rank Approximations: Algorithms and ApplicationsMinisymposium Talk
We describe a problem arising from operator learning for hyperbolic PDEs, in which one would like to recover an unknown, non-standard low-rank hierarchical partition of a linear operator using only input-output data, or, in the finite-dimensional case, matrix-vector products. We provide a solution to the operator learning problem by employing a continuous analogue of the randomized SVD (RSVD)...
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Chao Chen (NC State)5/22/26, 8:45 AMHierarchical Low-Rank Approximations: Algorithms and ApplicationsMinisymposium Talk
Dense matrices arise in many areas of computational science, and hierarchical approximation methods have been developed to reduce their storage and computational costs. However, many existing approaches require access to individual matrix entries, which may not be available or prohibitively expensive to compute in important applications. A prominent example is the Hessian in Bayesian inverse...
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Kate Wall (Tufts University)5/22/26, 9:10 AMHierarchical Low-Rank Approximations: Algorithms and ApplicationsMinisymposium Talk
A preconditioner for solving fractional partial differential equations (PDEs) is presented. In our method the fractional PDE is discretized on an adaptive grid, resulting in a Hierarchical matrix representation. The stiffness matrix has Toeplitz blocks along the diagonal and low-rank approximations off the diagonal. Our preconditioner expands on previously developed methods of conditioning...
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Ichitaro Yamazaki (Sandia Labs)5/22/26, 9:35 AMHierarchical Low-Rank Approximations: Algorithms and ApplicationsMinisymposium Talk
We discuss the adaptive coarse-space basis functions for the multi-level overlapping additive Schwarz preconditioners implemented in FROSch. The basis functions are formed based on the discrete Harmonic extensions of the local subdomain interface functions. The basis functions for the interface are composed of the eigenvectors corresponding to the small eigenvalues of the generalized...
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