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In this work, we present, analyze, and implement a class of Multi-Level Markov chain Monte Carlo (ML-MCMC) algorithms based on independent Metropolis-Hastings proposals for Bayesian inverse problems. In this context, the likelihood function…
The Airy process is characterized by its finite-dimensional distribution functions. We show that each finite-dimensional distribution function is expressible in terms of a solution to a system of differential equations.
We propose an adaptive multigrid preconditioning technology for solving linear systems arising from Discontinuous Petrov-Galerkin (DPG) discretizations. Unlike standard multigrid techniques, this preconditioner involves only trace spaces…
In this work, we present a multiple-scale perturbation technique suitable for the study of open quantum systems, which is easy to implement and in few iterative steps allows us to find excellent approximate solutions. For any time-local…
In this article, we prove that small localized data yield solutions to Kawahara type equation which have linear dispersive decay on a finite time. We use the similar method used to derive the dispersive decay bound of the solutions to the…
New methods are developed for the stabilization of a linear system with general time-varying distributed delays existing at the system's states, inputs and outputs. In contrast to most existing literature where the function of time-varying…
A diffuse interface (phase field) model for an electrochemical system is developed. We describe the minimal set of components needed to model an electrochemical interface and present a variational derivation of the governing equations. With…
We consider a particle system on $\mathbb{Z}^d$ with real state space and interactions of infinite range. Assuming that the rate of change is continuous we obtain a Kalikow-type decomposition of the infinite range change rates as a mixture…
The prediction of the quasi-static response of industrial laminate structures requires to use fine descriptions of the material, especially when debonding is involved. Even when modeled at the mesoscale, the computation of these structures…
In this work, we analytically investigate a degenerating PDE system for phase separation and complete damage processes considered on a nonsmooth time-dependent domain with mixed boundary conditions. The evolution of the system is described…
The Faddeev equation for three-body scattering at arbitrary energies is formulated in momentum space and directly solved in terms of momentum vectors without employing a partial wave decomposition. For identical bosons this results in a…
Markov chains are a natural and well understood tool for describing one-dimensional patterns in time or space. We show how to infer $k$-th order Markov chains, for arbitrary $k$, from finite data by applying Bayesian methods to both…
A flexible treatment for gas- and aerosol-phase chemical processes has been developed for models of diverse scale, from box models up to global models. At the core of this novel framework is an "abstracted aerosol representation" that…
Many problems in sequential decision making and stochastic control often have natural multiscale structure: sub-tasks are assembled together to accomplish complex goals. Systematically inferring and leveraging hierarchical structure,…
This manuscript presents an efficient solver for the linear system that arises from the Hierarchical Poincar\'e-Steklov (HPS) discretization of three dimensional variable coefficient Helmholtz problems. Previous work on the HPS method has…
A multi-step extended maximum residual Kaczmarz method is presented for the solution of the large inconsistent linear system of equations by using the multi-step iterations technique. Theoretical analysis proves the proposed method is…
Nonlinear systems of partial differential equations (PDEs) may permit several distinct solutions. The typical current approach to finding distinct solutions is to start Newton's method with many different initial guesses, hoping to find…
We introduce a method for solving the "inverse" phase equilibria problem: How should the interactions among a collection of molecular species be designed in order to achieve a target phase diagram? Using techniques from convex optimization…
A class of abstract nonlinear time-periodic evolution problems is considered which arise in electrical engineering and other scientific disciplines. An efficient solver is proposed for the systems arising after discretization in time based…
In this paper, we first show the existence of solutions to the following system of nonlinear equations \begin{eqnarray*}\left\{\begin{array}{l} a_{11}x_1+a_{12}x_2+a_{13}x_3+\cdots+a_{1n}x_{n} =…