Related papers: Gradient Schemes for Linear and Non-linear Elastic…
In this paper, we present a novel local and parallel two-grid finite element scheme for solving the Stokes equations, and rigorously establish its a priori error estimates. The scheme admits simultaneously small scales of subproblems and…
We introduce and study a notion of Asymptotic Preserving schemes, related to convergence in distribution, for a class of slow-fast Stochastic Differential Equations. In some examples, crude schemes fail to capture the correct limiting…
We analyze the mixed frame equations of radiation hydrodynamics under the approximations of flux-limited diffusion and a thermal radiation field, and derive the minimal set of evolution equations that includes all terms that are of leading…
Sparse grids are tailored to the approximation of smooth high-dimensional functions. On a $d$-dimensional tensor product space, the number of grid points is $N = \mathcal O(h^{-1} |\log h|^{d-1})$, where $h$ is a mesh parameter. The…
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,…
Numerically simulating deformations in thin elastic sheets is a challenging problem in computational mechanics due to destabilizing compressive stresses that result in wrinkling. Determining the location, structure, and evolution of…
Estimating the test performance of a model, possibly under distribution shift, without having access to the ground-truth labels is a challenging, yet very important problem for the safe deployment of machine learning algorithms in the wild.…
We construct a finite element like scheme for fully non-linear integro-partial differential equations arising in optimal control of jump-processes. Special cases of these equations include optimal portfolio and option pricing equations in…
Several convex formulation methods have been proposed previously for statistical estimation with structured sparsity as the prior. These methods often require a carefully tuned regularization parameter, often a cumbersome or heuristic…
Inverse problems have many applications in science and engineering. In Computer vision, several image restoration tasks such as inpainting, deblurring, and super-resolution can be formally modeled as inverse problems. Recently, methods have…
A convergence result for a discontinuous Galerkin multiscale method for a second order elliptic problem is presented. We consider a heterogeneous and highly varying diffusion coefficient in $L^\infty(\Omega,\mathbb{R}^{d\times d}_{sym})$…
In this work, an efficient approximation scheme has been proposed for getting accurate approximate solution of nonlinear partial differential equations with constant or variable coefficients satisfying initial conditions in a series of…
A space-time adaptive scheme is presented for solving advection equations in two space dimensions. The gradient-augmented level set method using a semi-Lagrangian formulation with backward time integration is coupled with a point value…
We propose a variant of the classical conditional gradient method for sparse inverse problems with differentiable measurement models. Such models arise in many practical problems including superresolution, time-series modeling, and matrix…
Solving large tensor linear systems poses significant challenges due to the high volume of data stored, and it only becomes more challenging when some of the data is missing. Recently, Ma et al. showed that this problem can be tackled using…
Over the past few decades, there has been substantial interest in evolution equations that involving a fractional-order derivative of order $\alpha\in(0,1)$ in time, due to their many successful applications in engineering, physics, biology…
Non linear regression models are a standard tool for modeling real phenomena, with several applications in machine learning, ecology, econometry... Estimating the parameters of the model has garnered a lot of attention during many years. We…
Recently, we proposed a method to estimate parameters of stochastic dynamics based on the linear response statistics. The method rests upon a nonlinear least-squares problem that takes into account the response properties that stem from the…
Diffusion models, which iteratively denoise data samples to synthesize high-quality outputs, have achieved empirical success across domains. However, optimizing these models for downstream tasks often involves nested bilevel structures,…
We introduce a class of unconditionally energy stable, high order accurate schemes for gradient flows in a very general setting. The new schemes are a high order analogue of the minimizing movements approach for generating a time discrete…