Related papers: Error analysis for learning fractional stochastic …
The state-of-the art proof of a global inf-sup condition on mixed finite element schemes does not allow for an analysis of truly indefinite, second-order linear elliptic PDEs. This paper, therefore, first analyses a nonconforming finite…
The reconstruction and inference of stochastic dynamical systems from data is a fundamental task in inverse problems and statistical learning. While surrogate modeling advances computational methods to approximate these dynamics, standard…
Training nonlinear parametrizations such as deep neural networks to numerically approximate solutions of partial differential equations is often based on minimizing a loss that includes the residual, which is analytically available in…
This paper is focused on the convergence analysis of an adaptive stochastic collocation algorithm for the stationary diffusion equation with parametric coefficient. The algorithm employs sparse grid collocation in the parameter domain…
We study the use of Temporal-Difference learning for estimating the structural parameters in dynamic discrete choice models. Our algorithms are based on the conditional choice probability approach but use functional approximations to…
We consider the problem of learning the parameters of a $N$-dimensional stochastic linear dynamics under both full and partial observations from a single trajectory of time $T$. We introduce and analyze a new estimator that achieves a small…
The effectiveness of non-parametric, kernel-based methods for function estimation comes at the price of high computational complexity, which hinders their applicability in adaptive, model-based control. Motivated by approximation techniques…
We consider Poisson's equation with a finite number of weighted Dirac masses as a source term, together with its discretization by means of conforming finite elements. For the error in fractional Sobolev spaces, we propose residual-type a…
The stochastic interpolant framework offers a powerful approach for constructing generative models based on ordinary differential equations (ODEs) or stochastic differential equations (SDEs) to transform arbitrary data distributions.…
We consider a linear elliptic partial differential equation (PDE) with a generic uniformly bounded parametric coefficient. The solution to this PDE problem is approximated in the framework of stochastic Galerkin finite element methods. We…
We consider a class of stochastic gradient optimization schemes. Assuming that the objective function is strongly convex, we prove weak error estimates which are uniform in time for the error between the solution of the numerical scheme,…
We derive computable error estimates for finite element approximations of linear elliptic partial differential equations (PDE) with rough stochastic coefficients. In this setting, the exact solutions contain high frequency content that…
We study the statistical inference of nonlinear stochastic approximation algorithms utilizing a single trajectory of Markovian data. Our methodology has practical applications in various scenarios, such as Stochastic Gradient Descent (SGD)…
First order optimization algorithms play a major role in large scale machine learning. A new class of methods, called adaptive algorithms, were recently introduced to adjust iteratively the learning rate for each coordinate. Despite great…
The success of deep learning has inspired recent interests in applying neural networks in statistical inference. In this paper, we investigate the use of deep neural networks for nonparametric regression with measurement errors. We propose…
We consider the problem of deriving from experimental data an approximation of an unknown function, whose derivatives also approximate the unknown function derivatives. Solving this problem is useful, for instance, in the context of…
There are many application papers that solve elliptic boundary value problems by meshless methods, and they use various forms of generalized stiffness matrices that approximate derivatives of functions from values at scattered nodes…
We consider a stochastic optimization problem involving two random variables: a context variable $X$ and a dependent variable $Y$. The objective is to minimize the expected value of a nonlinear loss functional applied to the conditional…
Fractional differential equations provide a tractable mathematical framework to describe anomalous behavior in complex physical systems, yet they introduce new sensitive model parameters, i.e. derivative orders, in addition to model…
We study efficiency of non-parametric estimation of diffusions (stochastic differential equations driven by Brownian motion) from long stationary trajectories. First, we introduce estimators based on conditional expectation which is…