Communications on Pure and Applied Mathematics最新文献

Perfectly matched layers are extensively used to compute approximate solutions for Maxwell's equations in $�R^{�1�+�3�}$ using a bounded computational domain, usually a rectangular solid. A smaller rectangular domain of interest is surrounded by layers designed to absorb outgoing waves in perfectly reflectionless manner. On the boundary of the computational domain, an absorbing boundary condition is imposed that is necessarily imperfect. The method replaces the Maxwell equations by a larger system, and introduces absorption coefficients positive in the layers. Well posedness of the resulting initial boundary value problem is proved here for the first time. The Laplace transform of a resulting Helmholtz system is studied. For positive real values of the transform variable $�\tau$, the Helmholtz system has a unique solution from a variational form that yields limited regularity for rectangular domains. When $�\tau$ is not real the complex variational form loses positivity. We smooth the domain and, in spite of this loss, construct $�H^2$ solutions with uniform $�L^2$ estimates. Using the $�H^2$ regularity, we deduce Maxwell from Helmholtz, then remove the smoothing. The boundary condition at the smoothed boundary must be carefully chosen. A method of Jerison-Kenig-Mitrea is extended to compensate the nonpositivity of the flux.

This article provides the first mathematical analysis of the Density Matrix Embedding Theory (DMET) method. We prove that, under certain assumptions, (i) the exact ground-state density matrix is a fixed-point of the DMET map for non-interacting systems, (ii) there exists a unique physical solution in the weakly-interacting regime, and (iii) DMET is exact up to first order in the coupling parameter. We provide numerical simulations to support our results and comment on the physical meaning of the assumptions under which they hold true. We show that the violation of these assumptions may yield multiple solutions to the DMET equations. We moreover introduce and discuss a specific $�N$-representability problem inherent to DMET.

We construct special Lagrangian pair of pants in general dimensions, inside the cotangent bundle of $�T^n$ with the Euclidean structure, building upon earlier topological ideas of Matessi. The construction uses a combination of PDE and geometric measure theory.

We consider the global approximate controllability of the two-dimensional incompressible Navier–Stokes system driven by a physically localized and degenerate force. In other words, the fluid is regulated via four scalar controls that depend only on time and appear as coefficients in an effectively constructed driving force supported in a given subdomain. Our idea consists of squeezing low mode controls into a small region, essentially by tracking their actions along the characteristic curves of a linearized vorticity equation. In this way, through explicit constructions and by connecting Coron's return method with recent concepts from geometric control, the original problem for the nonlinear Navier–Stokes system is reduced to one for a linear transport equation steered by a global force. This article can be viewed as an attempt to tackle a well-known open problem due to Agrachev.

In this work, we study the co-dimensional one interface limit and geometric motions of parabolic Ginzburg–Landau systems with potentials of high-dimensional wells. The main result generalizes the one by Lin et al. (Comm. Pure Appl. Math. 65 (2012), no. 6, 833–888) to a dynamical case. In particular combining modulated energy methods and weak convergence methods, we derive the limiting harmonic heat flows in the inner and outer bulk regions segregated by the sharp interface, and a non-standard boundary condition for them. These results are valid provided that the initial datum of the system is well-prepared under natural energy assumptions.

We study a stochastic Laplacian growth model, where a set $��U�\subseteq �R^d�$ grows according to a reflecting Brownian motion in $�U$ stopped at level sets of its boundary local time. We derive a scaling limit for the leading-order behavior of the growing boundary (i.e., “interface”). It is given by a geometric flow-type pde. It is obtained by an averaging principle for the reflecting Brownian motion. We also show that this geometric flow-type pde is locally well-posed, and its blow-up times correspond to changes in the diffeomorphism class of the growth model. Our results extend those of Dembo et al., which restricts to star-shaped growth domains and radially outwards growth, so that in polar coordinates, the geometric flow transforms into a simple ode with infinite lifetime. Also, we remove the “separation of scales” assumption that was taken in Dembo et al.; this forces us to understand the local geometry of the growing interface.

We introduce a new class of multilevel, adaptive, dual-space methods for computing fast convolutional transformations. These methods can be applied to a broad class of kernels, from the Green's functions for classical partial differential equations (PDEs) to power functions and radial basis functions such as those used in statistics and machine learning. The DMK (dual-space multilevel kernel-splitting) framework uses a hierarchy of grids, computing a smoothed interaction at the coarsest level, followed by a sequence of corrections at finer and finer scales until the problem is entirely local, at which point direct summation is applied. Unlike earlier multilevel summation schemes, DMK exploits the fact that the interaction at each scale is diagonalized by a short Fourier transform, permitting the use of separation of variables, but without relying on the FFT. This requires careful attention to the discretization of the Fourier transform at each spatial scale. Like multilevel summation, we make use of a recursive (telescoping) decomposition of the original kernel into the sum of a smooth far-field kernel, a sequence of difference kernels, and a residual kernel, which plays a role only in leaf boxes in the adaptive tree. At all higher levels in the grid hierarchy, the interaction kernels are designed to be smooth in both physical and Fourier space, admitting efficient Fourier spectral approximations. The DMK framework substantially simplifies the algorithmic structure of the fast multipole method (FMM) and unifies the FMM, Ewald summation, and multilevel summation, achieving speeds comparable to the FFT in work per gridpoint, even in a fully adaptive context. For continuous source distributions, the evaluation of local interactions is further accelerated by approximating the kernel at the finest level as a sum of Gaussians (SOG) with a highly localized remainder. The Gaussian convolutions are calculated using tensor product transforms, and the remainder term is calculated using asymptotic methods. We illustrate the performance of DMK for both continuous and discrete sources with extensive numerical examples in two and three dimensions.

We prove that the Riemannian Penrose inequality holds for asymptotically flat 3-manifolds with nonnegative scalar curvature and connected horizon boundary, provided the optimal decay assumptions are met, which result in the $�ADM$ mass being a well-defined geometric invariant. Our proof builds on a novel interplay between the Hawking mass and a potential-theoretic version of it, recently introduced by Agostiniani, Oronzio, and the third named author. As a consequence, we establish the equality between $�ADM$ mass and Huisken's isoperimetric mass under the above sharp assumptions. Moreover, we establish a Riemannian Penrose inequality in terms of the isoperimetric mass on any 3-manifold with nonnegative scalar curvature, connected horizon boundary, and which supports a well-posed notion of weak inverse mean curvature flow (IMCF). In particular, such isoperimetric Riemannian Penrose inequality does not require the asymptotic flatness of the manifold. The argument is based on a new asymptotic comparison result involving Huisken's isoperimetric mass and the Hawking mass.

The randomly pivoted Cholesky algorithm (RPCholesky) computes a factorized rank-$�k$ approximation of an $��N�\times �N�$ positive-semidefinite (psd) matrix. RPCholesky requires only $��(�k�+�1�)�N�$ entry evaluations and $��O�\left(�k^2�N�\right)�$ additional arithmetic operations, and it can be implemented with just a few lines of code. The method is particularly useful for approximating a kernel matrix. This paper offers a thorough new investigation of the empirical and theoretical behavior of this fundamental algorithm. For matrix approximation problems that arise in scientific machine learning, experiments show that RPCholesky matches or beats the performance of alternative algorithms. Moreover, RPCholesky provably returns low-rank approximations that are nearly optimal. The simplicity, effectiveness, and robustness of RPCholesky strongly support its use in scientific computing and machine learning applications.