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Monotone finite difference methods provide stable convergent discretizations of a class of degenerate elliptic and parabolic Partial Differential Equations (PDEs). These methods are best suited to regular rectangular grids, which leads to…

Hidden Markov Models (HMMs) can be accurately approximated using co-occurrence frequencies of pairs and triples of observations by using a fast spectral method in contrast to the usual slow methods like EM or Gibbs sampling. We provide a…

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…

Embedded systems are becoming more in demand to work in dynamic and uncertain environments, and being confined to the strong requirements of real-time. Conventional static scheduling models usually cannot cope with runtime modification in…

Stiff hyperbolic balance laws exhibit large spectral gaps, especially if the relaxation term significantly varies in space. Using examples from rarefied gases and the general form of the underlying balance law model, we perform a detailed…

Gradient methods are widely used in optimization problems. In practice, while the smoothness parameter can be estimated utilizing techniques such as backtracking, estimating the strong convexity parameter remains a challenge; moreover, even…

Modern data analytics take advantage of ensemble learning and transfer learning approaches to tackle some of the most relevant issues in data analysis, such as lack of labeled data to use to train the analysis models, sparsity of the…

Spectral methods have emerged as a simple yet surprisingly effective approach for extracting information from massive, noisy and incomplete data. In a nutshell, spectral methods refer to a collection of algorithms built upon the eigenvalues…

Deep learning-based methods deliver state-of-the-art performance for solving inverse problems that arise in computational imaging. These methods can be broadly divided into two groups: (1) learn a network to map measurements to the signal…

We introduce a new numerical method for solving time-harmonic acoustic scattering problems. The main focus is on plane waves scattered by smoothly varying material inhomogeneities. The proposed method works for any frequency $\omega$, but…

High order accurate Hermite methods for the wave equation on curvilinear domains are presented. Boundaries are treated using centered compatibility conditions rather than more standard one-sided approximations. Both first-order-in-time…

This paper concerns a spectral estimation problem in which we want to find a spectral density function that is consistent with estimated second-order statistics. It is an inverse problem admitting multiple solutions, and selection of a…

Rapidly developing machine learning methods has stimulated research interest in computationally reconstructing differential equations (DEs) from observational data which may provide additional insight into underlying causative mechanisms.…

Solving parametric partial differential equations (PDEs) presents significant challenges for data-driven methods due to the sensitivity of spatio-temporal dynamics to variations in PDE parameters. Machine learning approaches often struggle…

This paper introduces an adaptive time splitting technique for the solution of stiff evolutionary PDEs that guarantees an effective error control of the simulation, independent of the fastest physical time scale for highly unsteady…

This paper aims to devise an adaptive neural network basis method for numerically solving a second-order semilinear partial differential equation (PDE) with low-regular solutions in two/three dimensions. The method is obtained by combining…

The insertion of deep learning in medical image analysis had lead to the development of state-of-the art strategies in several applications such a disease classification, as well as abnormality detection and segmentation. However, even the…

In this paper we consider estimation of sparse covariance matrices and propose a thresholding procedure which is adaptive to the variability of individual entries. The estimators are fully data driven and enjoy excellent performance both…

In this paper, we develop and analyze an adaptive multiscale approach for heterogeneous problems in perforated domains. In many applications, these problems have a multiscale nature arising because of the perforations, their geometries, the…

Many problems in engineering and sciences require the solution of large scale optimization constrained by partial differential equations (PDEs). Though PDE-constrained optimization is itself challenging, most applications pose additional…