arXiv subjects

Rudi Smith

Publications and source records attributed to Rudi Smith.

4 recordsLinked to original sources

arXiv2026

Randomized Tucker-Sketched GMRES

We address the problem of solving large-scale tensor-structured linear systems in the Tucker format. In this setting, standard iterative solvers such as GMRES face a fundamental bottleneck: the multilinear ranks of the Krylov basis vectors grow with the iteration count, leading to rapidly increasing tensor operation costs and memory requirements. To overcome these challenges, we propose two randomized algorithms within the sketched GMRES framework that replace full Arnoldi orthogonalization with short recurrences. The first, RHOSVD-Tucker sGMRES, uses randomized HOSVD with per-iteration rank selection, providing robustness across a wide range of problems. The second method, MLN-Tucker sGMRES, leverages the multilinear Nystr\"om approximation with a fixed rank, enabling streaming computations; the streamability of the approximation further allows, at no additional cost, a memory-efficient reconstruction of the solution from a compact sketched representation of the Krylov basis. Both methods outperform standard low-rank Tucker solvers in symmetric and non-symmetric settings. Applied to inverse problems, the low-rank Tucker constraint acts as an implicit regularizer; combined with adaptive projected Tikhonov penalization and automatic regularization parameter selection, the methods yield stable reconstructions.

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arXiv2026

Attention Mechanisms Through the Lens of Numerical Methods: Approximation Methods and Alternative Formulations

The attention mechanism is the computational core of modern Transformer architectures, but its quadratic complexity in the input sequence length is the bottleneck for large-scale inference. This has motivated a rapidly growing body of work aimed at accelerating attention through approximation and reformulation. In this survey, we revisit attention mechanisms through the lens of numerical analysis, with a particular emphasis on tools and perspectives from numerical linear algebra. Our goal is twofold: first, we aim to systematically review and classify fast approximation methods according to the numerical principles they exploit. These include sparsity and clustering approaches, low-rank and subspace projection techniques, randomized sketching methods, and tensor-based decompositions. We also discuss kernel-inspired reformulations of attention and recent architectural variants, such as Latent Attention, that modify the standard softmax formulation to improve efficiency. Second, by presenting these developments within a unified mathematical framework, we aim to bridge the gap between disciplines and highlight opportunities for further contributions from computational mathematics, particularly numerical linear algebra, to the design of scalable attention mechanisms.

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arXiv2026

Efficient Sketching-Based Summation of Tucker Tensors

We present efficient, sketching-based methods for the summation of tensors in Tucker format. Leveraging the algebraic structure of Khatri-Rao and Kronecker products, our approach enables compressed arithmetic on Tucker tensors while controlling rank growth and computational cost. The proposed sketching framework avoids the explicit formation of large intermediate tensors, instead operating directly on the factor matrices and core tensors to produce accurate low-rank approximations of tensor sums. Furthermore, we analyze the computational complexity and the theoretical approximation properties of the proposed methodology. Numerical experiments demonstrate the effectiveness of our approach on four problems: two synthetic test cases, a parameter-dependent elliptic equation (commonly referred to as the cookie problem) solved via GMRES, and a one-dimensional linear transport problem discretized via high-order discontinuous Galerkin methods, where repeated tensor summation arises as a core computational bottleneck. Across these examples, the sketching-based summation achieves substantial computational savings while preserving high accuracy relative to direct summation and re-compression.

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arXiv2025

A tangential low-rank ADI method for solving indefinite Lyapunov equations

Continuous-time algebraic Lyapunov equations have become an essential tool in various applications. In the case of large-scale sparse coefficient matrices and indefinite constant terms, indefinite low-rank factorizations have successfully been used to allow methods like the alternating direction implicit (ADI) iteration to efficiently compute accurate approximations to the solution of the Lyapunov equation. However, classical block-type approaches quickly increase in computational costs when the rank of the constant term grows. In this paper, we propose a novel tangential reformulation of the ADI iteration that allows for the efficient construction of low-rank approximations to the solution of Lyapunov equations with indefinite right-hand sides even in the case of constant terms with higher ranks. We provide adaptive methods for the selection of the corresponding ADI parameters, namely shifts and tangential directions, which allow for the automatic application of the method to any relevant problem setting. The effectiveness of the developed algorithms is illustrated by several numerical examples.

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