Related papers: Dimensional estimates and rectifiability for measu…
Compressed sensing seeks to invert an underdetermined linear system by exploiting additional knowledge of the true solution. Over the last decade, several instances of compressed sensing have been studied for various applications, and for…
In this paper, the convergence of the solutions for a discretized linear state-based static peridynamic system to the corresponding continuous solution is analytically proven. To obtain an implementable model, we further apply…
In this work we consider the regularization of a supervised learning problem by partial differential equations (PDEs) and derive error bounds for the obtained approximation in terms of a PDE error term and a data error term. Assuming that…
The adaptive BDDC method is extended to the selection of face constraints in three dimensions. A new implementation of the BDDC method is presented based on a global formulation without an explicit coarse problem, with massive parallelism…
Partial differential equations (PDEs) play a central role in describing many physical phenomena. Various scientific and engineering applications demand a versatile and differentiable PDE solver that can quickly generate solutions with…
We consider the problem of under and over-approximating the image of general vector-valued functions over bounded sets, and apply the proposed solution to the estimation of reachable sets of uncertain non-linear discrete-time dynamical…
High-dimensional statistical inference deals with models in which the the number of parameters p is comparable to or larger than the sample size n. Since it is usually impossible to obtain consistent procedures unless $p/n\rightarrow0$, a…
Generalized equations are problems emerging in contexts of modern variational analysis as an adequate formalism to treat such issues as constraint systems, optimality and equilibrium conditions, variational inequalities, differential…
It is expected that conformal symmetry is an emergent property of many systems at their critical point. This imposes strong constraints on the critical behavior of a given system. Taking them into account in theoretical approaches can lead…
In this article, we will consider second order uniformly elliptic operators of divergence form defined on R^n with measurable coefficients. Mainly, we will give estimates on the dimension of space of solutions that grow at most polynomially…
Dimensionality reduction is an effective method for learning high-dimensional data, which can provide better understanding of decision boundaries in human-readable low-dimensional subspace. Linear methods, such as principal component…
We introduce the notion of \delta-viscosity solutions for fully nonlinear uniformly parabolic PDE on bounded domains. We prove that \delta-viscosity solutions are uniformly close to the actual viscosity solution. As a consequence we obtain…
We characterize $n$-rectifiable metric measure spaces as those spaces that admit a countable Borel decomposition so that each piece has positive and finite $n$-densities and one of the following: is an $n$-dimensional Lipschitz…
We derive globally reliable a posteriori error estimators for a PDE-constrained optimization problem involving linear models in fluid dynamics as state equation; control constraints are also considered. The corresponding local error…
We consider a ramification of the deep BSDE loss functional designed to apply for BSDEs on bounded domains, i.e. with random (unbounded) time horizons. We derive a general convergence rate of the loss functional; precisely for a class of…
In a complete metric space that is equipped with a doubling measure and supports a Poincar\'e inequality, we show that functions of bounded variation (BV functions) can be approximated in the strict sense and pointwise uniformly by special…
The aim of this paper is to present and analyze a new direct method for solving the linear elasticity inverse problem. Given measurements of some displacement fields inside a medium, we show that a stable reconstruction of elastic…
Many classical geometric inequalities on functionals of convex bodies depend on the dimension of the ambient space. We show that this dimension dependence may often be replaced (totally or partially) by different symmetry measures of the…
How can one meaningfully make a measurement, if the meter does not conform to any standard and its scale expands or shrinks depending on what is measured? In the present work it is argued that current evaluation practices for…
There has been a lot of interest in sufficient dimension reduction (SDR) methodologies as well as nonlinear extensions in the statistics literature. In this note, we use classical results regarding metric spaces and positive definite…