More Like this

1. Computational approaches for extremal geometric eigenvalue problems

   C.-Y. Kao; B. Osting; and E ́. Oudet (January 2023, Handbook of numerical analysis)

   In an extremal eigenvalue problem, one considers a family of eigenvalue problems, each with discrete spectra, and extremizes a chosen eigenvalue over the family. In this chapter, we consider eigenvalue problems defined on Riemannian manifolds and extremize over the metric structure. For example, we consider the problem of maximizing the principal Laplace–Beltrami eigenvalue over a family of closed surfaces of fixed volume. Computational approaches to such extremal geometric eigenvalue problems present new computational challenges and require novel numerical tools, such as the parameterization of conformal classes and the development of accurate and efficient methods to solve eigenvalue problems on domains with nontrivial genus and boundary. We highlight recent progress on computational approaches for extremal geometric eigenvalue problems, including (i) maximizing Laplace–Beltrami eigenvalues on closed surfaces and (ii) maximizing Steklov eigenvalues on surfaces with boundary. 

   more » « less
2. Regularity of the free boundary for the two-phase Bernoulli problem

   https://doi.org/10.1007/s00222-021-01031-7  

   De Philippis, Guido; Spolaor, Luca; Velichkov, Bozhidar (August 2021, Inventiones mathematicae)

   null (Ed.)

   Abstract We prove a regularity theorem for the free boundary of minimizers of the two-phase Bernoulli problem, completing the analysis started by Alt, Caffarelli and Friedman in the 80s. As a consequence, we also show regularity of minimizers of the multiphase spectral optimization problem for the principal eigenvalue of the Dirichlet Laplacian. 

   more » « less
3. Computing Large Deviation Rate Functions of Entropy Production for Diffusion Processes by an Interacting Particle Method

   https://doi.org/10.1137/24M1720172  

   Wu, Zhizhang; Raquépas, Renaud; Xin, Jack; Zhang, Zhiwen (December 2025, SIAM Journal on Scientific Computing)

   We develop an interacting particle method (IPM) for computing the large deviation rate function of entropy production for diffusion processes, with emphasis on the vanishing-noise limit and high dimensions. The crucial ingredient for obtaining the rate function is the computation of the principal eigenvalue \lambda of elliptic, non-self-adjoint operators. We show that this principal eigenvalue can be approximated in terms of the spectral radius of a discretized evolution operator, which is obtained from an operator splitting scheme and an Euler--Maruyama scheme with a small time step size. We also show that this spectral radius can be accessed through a large number of iterations of this discretized semigroup, which is suitable for computation using the IPM. The IPM applies naturally to problems in unbounded domains and scales easily to high dimensions. We show numerical examples of dimensions up to 16, and the results show that our numerical approximation of \lambda converges to the analytical vanishing-noise limit within visual tolerance with a fixed number of particles and a fixed time step size. It is numerically shown that the IPM can adapt to singular behaviors in the vanishing-noise limit. We also apply the IPM to explore situations with no explicit formulas of the vanishing-noise limit. Our paper appears to be the first to obtain numerical results of principal eigenvalue problems for non-self-adjoint operators in such high dimensions. 

   more » « less
4. 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. 

   more » « less
5. Computing the Tracy-Widom Distribution for Arbitrary $$\beta>0$$

   https://doi.org/10.3842/SIGMA.2024.005  

   Trogdon, Thomas; Zhang, Yiting (January 2024, Symmetry, Integrability and Geometry: Methods and Applications)

   We compute the Tracy-Widom distribution describing the asymptotic distribution of the largest eigenvalue of a large random matrix by solving a boundary-value problem posed by Bloemendal in his Ph.D. Thesis (2011). The distribution is computed in two ways. The first method is a second-order finite-difference method and the second is a highly accurate Fourier spectral method. Since $$\beta$$ is simply a parameter in the boundary-value problem, any $$\beta> 0$$ can be used, in principle. The limiting distribution of the $$n$$th largest eigenvalue can also be computed. Our methods are available in the Julia package TracyWidomBeta.jl. 

   more » « less