Foundations of Computational Mathematics最新文献

In this work, we develop new regularity conditions in nonsmooth analysis that parallel the stratification conditions of Whitney, Kuo, and Verdier. They quantify how subgradients interact with a certain “active manifold” that captures the nonsmooth activity of the function. Based on these new conditions, we show that several subgradient-based methods converge only to local minimizers when applied to generic Lipschitz and subdifferentially regular functions that are definable in an o-minimal structure. At a high level, our argument is appealingly transparent: we interpret the nonsmooth dynamics as an approximate Riemannian gradient method on the active manifold. As a by-product, we extend the stochastic processes techniques of Pemantle.

We prove optimal convergence rates for eigenvalues and eigenvectors of the graph Laplacian on Poisson point clouds. Our results are valid down to the critical percolation threshold, yielding error estimates for relatively sparse graphs.

The criticality problem in nuclear engineering asks for the principal eigenpair of a Boltzmann operator describing neutron transport in a reactor core. Being able to reliably design, and control such reactors requires assessing these quantities within quantifiable accuracy tolerances. In this paper, we propose a paradigm that deviates from the common practice of approximately solving the corresponding spectral problem with a fixed, presumably sufficiently fine discretization. Instead, the present approach is based on first contriving iterative schemes, formulated in function space, that are shown to converge at a quantitative rate without assuming any a priori excess regularity properties, and that exploit only properties of the optical parameters in the underlying radiative transfer model. We develop the analytical and numerical tools for approximately realizing each iteration step within judiciously chosen accuracy tolerances, verified by a posteriori estimates, so as to still warrant quantifiable convergence to the exact eigenpair. This is carried out in full first for a Newton scheme. Since this is only locally convergent we analyze in addition the convergence of a power iteration in function space to produce sufficiently accurate initial guesses. Here we have to deal with intrinsic difficulties posed by compact but unsymmetric operators preventing standard arguments used in the finite dimensional case. Our main point is that we can avoid any condition on an initial guess to be already in a small neighborhood of the exact solution. We close with a discussion of remaining intrinsic obstructions to a certifiable numerical implementation, mainly related to not knowing the gap between the principal eigenvalue and the next smaller one in modulus.

We develop Conley’s theory for multivalued maps on finite topological spaces. More precisely, for discrete-time dynamical systems generated by the iteration of a multivalued map which satisfies appropriate regularity conditions, we establish the notions of isolated invariant sets and index pairs, and use them to introduce a well-defined Conley index. In addition, we verify some of its fundamental properties such as the Ważewski property and continuation.

We present a randomized, inverse-free algorithm for producing an approximate diagonalization of any (n times n) matrix pencil (A, B). The bulk of the algorithm rests on a randomized divide-and-conquer eigensolver for the generalized eigenvalue problem originally proposed by Ballard, Demmel and Dumitriu (Technical Report 2010). We demonstrate that this divide-and-conquer approach can be formulated to succeed with high probability provided the input pencil is sufficiently well-behaved, which is accomplished by generalizing the recent pseudospectral shattering work of Banks, Garza-Vargas, Kulkarni and Srivastava (Foundations of Computational Mathematics 2023). In particular, we show that perturbing and scaling (A, B) regularizes its pseudospectra, allowing divide-and-conquer to run over a simple random grid and in turn producing an accurate diagonalization of (A, B) in the backward error sense. The main result of the paper states the existence of a randomized algorithm that with high probability (and in exact arithmetic) produces invertible S, T and diagonal D such that (||A - SDT^{-1}||_2 le varepsilon ) and (||B - ST^{-1}||_2 le varepsilon ) in at most (O left( log ^2 left( frac{n}{varepsilon } right) T_{text {MM}}(n) right) ) operations, where (T_{text {MM}}(n)) is the asymptotic complexity of matrix multiplication. This not only provides a new set of guarantees for highly parallel generalized eigenvalue solvers but also establishes nearly matrix multiplication time as an upper bound on the complexity of inverse-free, exact-arithmetic matrix pencil diagonalization.

Given a domain (Omega subset mathbb {R}^n), the de Rham complex of differential forms arises naturally in the study of problems in electromagnetism and fluid mechanics defined on (Omega ), and its discretization helps build stable numerical methods for such problems. For constructing such stable methods, one critical requirement is ensuring that the discrete subcomplex is cohomologically equivalent to the continuous complex. When (Omega ) is a hypercube, we thus require that the discrete subcomplex be exact. Focusing on such (Omega ), we theoretically analyze the discrete de Rham complex built from hierarchical B-spline differential forms, i.e., the discrete differential forms are smooth splines and support adaptive refinements—these properties are key to enabling accurate and efficient numerical simulations. We provide locally-verifiable sufficient conditions that ensure that the discrete spline complex is exact. Numerical tests are presented to support the theoretical results, and the examples discussed include complexes that satisfy our prescribed conditions as well as those that violate them.

We analyze constrained and unconstrained minimization problems on patches of tetrahedra sharing a common vertex with discontinuous piecewise polynomial data of degree p. We show that the discrete minimizers in the spaces of piecewise polynomials of degree p conforming in the (H^1), ({varvec{H}}(textbf{curl})), or ({varvec{H}}({text {div}})) spaces are as good as the minimizers in these entire (infinite-dimensional) Sobolev spaces, up to a constant that is independent of p. These results are useful in the analysis and design of finite element methods, namely for devising stable local commuting projectors and establishing local-best–global-best equivalences in a priori analysis and in the context of a posteriori error estimation. Unconstrained minimization in (H^1) and constrained minimization in ({varvec{H}}({text {div}})) have been previously treated in the literature. Along with improvement of the results in the (H^1) and ({varvec{H}}({text {div}})) cases, our key contribution is the treatment of the ({varvec{H}}(textbf{curl})) framework. This enables us to cover the whole de Rham diagram in three space dimensions in a single setting.

The proximal Galerkin finite element method is a high-order, low iteration complexity, nonlinear numerical method that preserves the geometric and algebraic structure of pointwise bound constraints in infinite-dimensional function spaces. This paper introduces the proximal Galerkin method and applies it to solve free boundary problems, enforce discrete maximum principles, and develop a scalable, mesh-independent algorithm for optimal design with pointwise bound constraints. This paper also introduces the latent variable proximal point (LVPP) algorithm, from which the proximal Galerkin method derives. When analyzing the classical obstacle problem, we discover that the underlying variational inequality can be replaced by a sequence of second-order partial differential equations (PDEs) that are readily discretized and solved with, e.g., the proximal Galerkin method. Throughout this work, we arrive at several contributions that may be of independent interest. These include (1) a semilinear PDE we refer to as the entropic Poisson equation; (2) an algebraic/geometric connection between high-order positivity-preserving discretizations and certain infinite-dimensional Lie groups; and (3) a gradient-based, bound-preserving algorithm for two-field, density-based topology optimization. The complete proximal Galerkin methodology combines ideas from nonlinear programming, functional analysis, tropical algebra, and differential geometry and can potentially lead to new synergies among these areas as well as within variational and numerical analysis. Open-source implementations of our methods accompany this work to facilitate reproduction and broader adoption.

The theory of group classifications has undergone significant changes in the past 25 years. New methods have been introduced, some difficult problems have been solved and group classifications have become widely available through computer algebra systems. This survey describes the state of the art of the group classification problem, its history, its recent advances and some open problems.