Mathematics of Computation最新文献

We describe an elementary algorithm for recursively constructing diagonal approximations on those finite regular CW-complexes for which the closure of each cell can be explicitly collapsed to a point. The algorithm is based on the standard proof of the acyclic carrier theorem, made constructive through the use of explicit contracting homotopies. It can be used as a theoretical tool for constructing diagonal approximations on families of polytopes in situations where the diagonals are required to satisfy certain coherence conditions. We compare its output to existing diagonal approximations for the families of simplices, cubes, associahedra and permutahedra. The algorithm yields a new explanation of a magical formula for the associahedron derived by Markl and Shnider [Trans. Amer. Math. Soc. 358 (2006), pp. 2353–2372] and Masuda, Thomas, Tonks, and Vallette [J. Éc. polytech. Math. 8 (2021), pp. 121–146] and Theorem 4.1 provides a magical formula for other polytopes. We also describe a computer implementation of the algorithm and illustrate it on a range of practical examples including the computation of cohomology rings for some low-dimensional manifolds. To achieve some of these examples the paper includes two approaches to generating a regular CW-complex structure on closed compact 3 3 -manifolds, one using an implementation of Dehn surgery on links and the other using an implementation of pairwise identifications of faces in a tessellated boundary of the 3 3 -ball. The latter is illustrated in Proposition 8.1 with a topological classification of all closed orientable 3 3 -manifolds arising from pairwise identifications of faces of the cube.

Hierarchical matrices approximate a given matrix by a decomposition into low-rank submatrices that can be handled efficiently in factorized form. H 2 mathcal {H}^2 -matrices refine this representation following the ideas of fast multipole methods in order to achieve linear, i.e., optimal complexity for a variety of important algorithms.

The matrix multiplication, a key component of many more advanced numerical algorithms, has been a challenge: the only linear-time algorithms known so far either require the very special structure of HSS-matrices or need to know a suitable basis for all submatrices in advance.

In this article, a new and fairly general algorithm for multiplying H 2 mathcal {H}^2 -matrices in linear complexity with adaptively constructed bases is presented. The algorithm consists of two phases: first an intermediate representation with a generalized block structure is constructed, then this representation is re-compressed in order to match the structure prescribed by the application.

The complexity and accuracy are analyzed and numerical experiments indicate that the new algorithm can indeed be significantly faster than previous attempts.

The recovered gradient, using the polynomial preserving recovery (PPR), is constructed for the finite volume element method (FVEM) under simplex meshes. Regarding the main results of this paper, there are two aspects. Firstly, we investigate the supercloseness property of the FVEM, specifically examining the quadratic FVEM under tetrahedral meshes. Secondly, we present several guidelines for selecting computing nodes such that the least-squares fitting procedure of the PPR admits a unique solution. Numerical experiments demonstrate that the recovered gradient by the PPR exhibits superconvergence.

We study a class of nonlinear nonlocal conservation laws with discontinuous flux, modeling crowd dynamics and traffic flow. The discontinuous coefficient of the flux function is assumed to be of bounded variation (BV) and bounded away from zero, and hence the spatial discontinuities of the flux function can be infinitely many with possible accumulation points. Strong compactness of the Godunov and Lax-Friedrichs type approximations is proved, providing the existence of entropy solutions. A proof of the uniqueness of the adapted entropy solutions is provided, establishing the convergence of the entire sequence of finite volume approximations to the adapted entropy solution. As per the current literature, this is the first well-posedness result for the aforesaid class and connects the theory of nonlocal conservation laws (with discontinuous flux), with its local counterpart in a generic setup. Some numerical examples are presented to display the performance of the schemes and explore the limiting behavior of these nonlocal conservation laws to their local counterparts.

We present a wavenumber-explicit convergence analysis of the h p hp Finite Element Method applied to a class of heterogeneous Helmholtz problems with piecewise analytic coefficients at large wavenumber k k . Our analysis covers the heterogeneous Helmholtz equation with Robin, exact Dirichlet-to-Neumann, and second order absorbing boundary conditions, as well as perfectly matched layers.

The recently introduced Genetic Column Generation (GenCol) algorithm has been numerically observed to efficiently and accurately compute high-dimensional optimal transport (OT) plans for general multi-marginal problems, but theoretical results on the algorithm have hitherto been lacking. The algorithm solves the OT linear program on a dynamically updated low-dimensional submanifold consisting of sparse plans. The submanifold dimension exceeds the sparse support of optimal plans only by a fixed factor β beta . Here we prove that for β ≥ 2 beta geq 2 and in the two-marginal case, GenCol always converges to an exact solution, for arbitrary costs and marginals. The proof relies on the concept of c-cyclical monotonicity. As an offshoot, GenCol rigorously reduces the data complexity of numerically solving two-marginal OT problems from O ( ℓ 2 ) O(ell ^2) to O ( ℓ ) O(ell ) without any loss in accuracy, where ℓ ell is the number of discretization points for a single marginal. At the end of the paper we also present some insights into the convergence behavior in the multi-marginal case.

We prove that the Cohn–Elkies linear programming bound for sphere packing is not sharp in dimension 6. The proof uses duality and optimization over a space of modular forms, generalizing a construction of Cohn–Triantafillou [Math. Comp. 91 (2021), pp. 491–508] to the case of odd weight and non-trivial character.

In this paper we study the nonconvex constrained composition optimization, in which the objective contains a composition of two expected-value functions whose accurate information is normally expensive to calculate. We propose a STochastic nEsted Primal-dual (STEP) method for such problems. In each iteration, with an auxiliary variable introduced to track the inner layer function values we compute stochastic gradients of the nested function using a subsampling strategy. To alleviate difficulties caused by possibly nonconvex constraints, we construct a stochastic approximation to the linearized augmented Lagrangian function to update the primal variable, which further motivates to update the dual variable in a weighted-average way. Moreover, to better understand the asymptotic dynamics of the update schemes we consider a deterministic continuous-time system from the perspective of ordinary differential equation (ODE). We analyze the Karush-Kuhn-Tucker measure at the output by the STEP method with constant parameters and establish its iteration and sample complexities to find an ϵ epsilon -stationary point, ensuring that expected stationarity, feasibility as well as complementary slackness are below accuracy ϵ epsilon . To leverage the benefit of the (near) initial feasibility in the STEP method, we propose a two-stage framework incorporating a feasibility-seeking phase, aiming to locate a nearly feasible initial point. Moreover, to enhance the adaptivity of the STEP algorithm, we propose an adaptive variant by adaptively adjusting its parameters, along with a complexity analysis. Numerical results on a risk-averse portfolio optimization problem and an orthogonal nonnegative matrix decomposition reveal the effectiveness of the proposed algorithms.