Foundations of Computational Mathematics最新文献

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.

We present a discretization of the dynamic optimal transport problem for which we can obtain the convergence rate for the value of the transport cost to its continuous value when the temporal and spatial stepsize vanish. This convergence result does not require any regularity assumption on the measures, though experiments suggest that the rate is not sharp. Via an analysis of the duality gap we also obtain the convergence rates for the gradient of the optimal potentials and the velocity field under mild regularity assumptions. To obtain such rates, we discretize the dual formulation of the dynamic optimal transport problem and use the mature literature related to the error due to discretizing the Hamilton–Jacobi equation.

We address the noncommutative version of the Edmonds’ problem, which asks to determine the inner rank of a matrix in noncommuting variables. We provide an algorithm for the calculation of this inner rank by relating the problem with the distribution of a basic object in free probability theory, namely operator-valued semicircular elements. We have to solve a matrix-valued quadratic equation, for which we provide precise analytical and numerical control on the fixed point algorithm for solving the equation. Numerical examples show the efficiency of the algorithm.

We consider the problem of reconstructing a function (fin L^2({mathbb R})) given phase-less samples of its Gabor transform, which is defined by

More precisely, given sampling positions (Omega subseteq {mathbb R}^2) the task is to reconstruct f (up to global phase) from measurements ({|{mathcal {G}}f(omega )|: ,omega in Omega }). This non-linear inverse problem is known to suffer from severe ill-posedness. As for any other phase retrieval problem, constructive recovery is a notoriously delicate affair due to the lack of convexity. One of the fundamental insights in this line of research is that the connectivity of the measurements is both necessary and sufficient for reconstruction of phase information to be theoretically possible. In this article we propose a reconstruction algorithm which is based on solving two convex problems and, as such, amenable to numerical analysis. We show, empirically as well as analytically, that the scheme accurately reconstructs from noisy data within the connected regime. Moreover, to emphasize the practicability of the algorithm we argue that both convex problems can actually be reformulated as semi-definite programs for which efficient solvers are readily available. The approach is based on ideas from complex analysis, Gabor frame theory as well as matrix completion. As a byproduct, we also obtain improved truncation error for Gabor expensions with Gaussian generators.

The NEPv approach has been increasingly used lately for optimization on the Stiefel manifold arising from machine learning. General speaking, the approach first turns the first order optimality condition into a nonlinear eigenvalue problem with eigenvector dependency (NEPv) and then solve the nonlinear problem via some variations of the self-consistent-field (SCF) iteration. The difficulty, however, lies in designing a proper SCF iteration so that a maximizer is found at the end. Currently, each use of the approach is very much individualized, especially in its convergence analysis phase to show that the approach does work or otherwise. Related, the NPDo approach is recently proposed for the sum of coupled traces and it seeks to turn the first order optimality condition into a nonlinear polar decomposition with orthogonal factor dependency (NPDo). In this paper, two unifying frameworks are established, one for each approach. Each framework is built upon a basic assumption, under which globally convergence to a stationary point is guaranteed and during the SCF iterative process that leads to the stationary point, the objective function increases monotonically. Also the notion of atomic function for each approach is proposed, and the atomic functions include commonly used matrix traces of linear and quadratic forms as special ones. It is shown that the basic assumptions of the approaches are satisfied by their respective atomic functions and, more importantly, by convex compositions of their respective atomic functions. Together they provide a large collection of objectives for which either one of approaches or both are guaranteed to work, respectively.

In this paper, we propose a general approach for explicit a posteriori error representation for convex minimization problems using basic convex duality relations. Exploiting discrete orthogonality relations in the space of element-wise constant vector fields as well as a discrete integration-by-parts formula between the Crouzeix–Raviart and the Raviart–Thomas element, all convex duality relations are transferred to a discrete level, making the explicit a posteriori error representation –initially based on continuous arguments only– practicable from a numerical point of view. In addition, we provide a generalized Marini formula that determines a discrete primal solution in terms of a given discrete dual solution. We benchmark all these concepts via the Rudin–Osher–Fatemi model. This leads to an adaptive algorithm that yields a (quasi-optimal) linear convergence rate.