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In this article, we prove convergence of the weakly penalized adaptive discontinuous Galerkin methods. Unlike other works, we derive the contraction property for various discontinuous Galerkin methods only assuming the stabilizing…

Multiscale mathematical models have shown great promise in computational brain electrophysiology but are still hindered by high computational costs due to fast dynamics and complex brain geometries, requiring very fine spatio-temporal…

Flexible and scalable decentralized learning solutions are fundamentally important in the application of multi-agent systems. While several recent approaches introduce (ensembles of) kernel machines in the distributed setting, Bayesian…

The aim of this paper is to propose diffusion strategies for distributed estimation over adaptive networks, assuming the presence of spatially correlated measurements distributed according to a Gaussian Markov random field (GMRF) model. The…

The online optimization of gasoline blending benefits refinery economies. However, the nonlinear blending mechanism, the oil property fluctuations, and the blending model mismatch bring difficulties to the optimization. To solve the above…

In this paper, we develop an efficient spectral-Galerkin-type search extension method (SGSEM) for finding multiple solutions to semilinear elliptic boundary value problems. This method constructs effective initial data for multiple…

This paper considers a convex optimization problem with cost and constraints that evolve over time. The function to be minimized is strongly convex and possibly non-differentiable, and variables are coupled through linear constraints. In…

Diffusion approximation provides weak approximation for stochastic gradient descent algorithms in a finite time horizon. In this paper, we introduce new tools motivated by the backward error analysis of numerical stochastic differential…

Near-optimal computational complexity of an adaptive stochastic Galerkin method with independently refined spatial meshes for elliptic partial differential equations is shown. The method takes advantage of multilevel structure in expansions…

We discuss an online decentralized decision making problem where the agents are coupled with affine inequality constraints. Alternating Direction Method of Multipliers (ADMM) is used as the computation engine and we discuss the convergence…

We propose a projection-based model order reduction method for the solution of parameter-dependent dynamical systems. The proposed method relies on the construction of time-dependent reduced spaces generated from evaluations of the solution…

We present a new algorithm for domain adaptation improving upon a discrepancy minimization algorithm previously shown to outperform a number of algorithms for this task. Unlike many previous algorithms for domain adaptation, our algorithm…

Subspace learning and matrix factorization problems have great many applications in science and engineering, and efficient algorithms are critical as dataset sizes continue to grow. Many relevant problem formulations are non-convex, and in…

The reduced basis method (RBM) empowers repeated and rapid evaluation of parametrized partial differential equations through an offline-online decomposition, a.k.a. a learning-execution process. A key feature of the method is a greedy…

This work is devoted to the study of a posteriori error estimation and adaptivity in parabolic problems with a particular focus on spatial discontinuous Galerkin (dG) discretisations. We begin by deriving an a posteriori error estimator for…

We consider the problem of learning multiple tasks in a continual learning setting in which data from different tasks is presented to the learner in a streaming fashion. A key challenge in this setting is the so-called "catastrophic…

In this work, we propose and develop an arbitrary-order adaptive discontinuous Petrov-Galerkin (DPG) method for the nonlinear Grad-Shafranov equation. An ultraweak formulation of the DPG scheme for the equation is given based on a minimal…

Expectation-Maximization (EM) is a prominent approach for parameter estimation of hidden (aka latent) variable models. Given the full batch of data, EM forms an upper-bound of the negative log-likelihood of the model at each iteration and…

With the rapidly growing model complexity and data volume, training deep generative models (DGMs) for better performance has becoming an increasingly more important challenge. Previous research on this problem has mainly focused on…

The aim of this work is to consider multiscale algorithms for solving PDEs with Galerkin methods on bounded domains. We provide results on convergence and condition numbers. We show how to handle PDEs with Dirichlet boundary conditions. We…