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1. Faster Kernel Matrix Algebra via Density Estimation.

   Backurs, Arturs; Indyk, Piotr; Musco, Cameron; Wagner, Tal. (January 2021, International Conference on Machine Learning (ICML))

   We study fast algorithms for computing fundamental properties of a positive semidefinite kernel matrix K∈ R^{n*n} corresponding to n points x1,…,xn∈R^d. In particular, we consider estimating the sum of kernel matrix entries, along with its top eigenvalue and eigenvector. We show that the sum of matrix entries can be estimated to 1+ϵ relative error in time sublinear in n and linear in d for many popular kernels, including the Gaussian, exponential, and rational quadratic kernels. For these kernels, we also show that the top eigenvalue (and an approximate eigenvector) can be approximated to 1+ϵ relative error in time subquadratic in n and linear in d. Our algorithms represent significant advances in the best known runtimes for these problems. They leverage the positive definiteness of the kernel matrix, along with a recent line of work on efficient kernel density estimation. 

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2. A Chebyshev Locally Optimal Block Preconditioned Conjugate Gradient Method for Product and Standard Symmetric Eigenvalue Problems

   https://doi.org/10.1137/23M1566017  

   Zhang, Tianqi; Xue, Fei (November 2024, SIAM Journal on Matrix Analysis and Applications)

   The discretized Bethe-Salpeter eigenvalue (BSE) problem arises in many body physics and quantum chemistry. Discretization leads to an {\color{black}algebraic} eigenvalue problem involving a matrix $$H\in \mathbb{C}^{2n\times 2n}$$ with a Hamiltonian-like structure. With proper transformations, {\color{black}the real BSE eigenproblem of form I and the complex BSE eigenproblem of form II} can be transformed into real product eigenvalue problems of order $$n$$ and $2n$, respectively. We propose a new variant of the Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) {\color{black}enhanced with polynomial filters} to improve the robustness and effectiveness of a few well-known algorithms for computing the lowest eigenvalues of the product eigenproblems. {\color{black}Furthermore, our proposed method can be easily employed to solve large sparse standard symmetric eigenvalue problems.} We show that our ideal locally optimal algorithm delivers Rayleigh quotient approximation to the desired lowest eigenvalue that satisfies a global quasi-optimality, which is similar to the global optimality of the preconditioned conjugate gradient method for the iterative solution of a symmetric positive definite linear system. The robustness and efficiency of {\color{black}our proposed method} is illustrated by numerical experiments. 

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3. Behavior of absorbing and generating $p$ -Robin eigenvalues in bounded and exterior domains

   https://doi.org/10.1016/j.na.2025.113943  

   Bundrock, Lukas; Giorgi, Tiziana; Smits, Robert (January 2026, Nonlinear Analysis)

   We establish rigorous quantitative inequalities for the first eigenvalue of the generalized 𝑝-Robin problem, for both the classical diffusion absorption case, where the Robin boundary parameter 𝛼 is positive, and the superconducting generation regime (𝛼 < 0), where the boundary acts as a source. In bounded domains, we use a unified approach to derive a precise asymptotic behavior for all 𝑝 and all small real 𝛼, improving existing results in various directions, including requiring weaker boundary regularity for the case of the classical 2-Robin problem, studied in the fundamental work by René Sperb. In exterior domains, we characterize the existence of eigenvalues, establish general inequalities and asymptotics as 𝛼 → 0 for the first eigenvalue of the exterior of a ball, and obtain some sharp geometric inequalities for convex domains in two dimensions. 

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4. The Kloosterman circle method and weighted representation numbers of quadratic forms

   https://doi.org/10.1007/s40687-025-00544-4  

   Jones, Edna (August 2025, Research in the Mathematical Sciences)

   We develop a version of the Kloosterman circle method with a bump function that is used to provide asymptotics for weighted representation numbers of nonsingular integral quadratic forms. Unlike many applications of the Kloosterman circle method, we explicitly state some constants in the error terms that depend on the quadratic form. This version of the Kloosterman circle method uses Gauss sums, Kloosterman sums, Salié sums, and a principle of nonstationary phase. We briefly discuss a potential application of this version of the Kloosterman circle method to a proof of a strong asymptotic local–global principle for certain Kleinian sphere packings. 

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5. Asymptotic Expansions for the Transmission Eigenvalues of Periodic Scatterers of Bounded Support

   https://doi.org/10.1177/09217134251407492  

   Cakoni, Fioralba; Moskow, Shari (January 2026, Asymptotic Analysis)

   We consider the transmission eigenvalues for a bounded scatterer with a periodically varying index of refraction, and derive the first-order corrections to the limiting transmission eigenvalues. We assume the scatterer contrast to be of one sign, in which case the transmission eigenvalue problem can be written in terms of operators corresponding to a fourth-order partial differential equation with periodic coefficients. We perform two scale asymptotics for this biharmonic-type homogenization problem and show convergence estimates, which require a boundary corrector function, and this boundary corrector function appears in the formula for the transmission eigenvalues correction. 

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