Related papers: Full state approximation by Galerkin projection re…
Polynomial chaos methods have been extensively used to analyze systems in uncertainty quantification. Furthermore, several approaches exist to determine a low-dimensional approximation (or sparse approximation) for some quantity of interest…
We develop a Petrov-Galerkin stabilization method for multiscale convection-diffusion transport systems. Existing stabilization techniques add a limited number of degrees of freedom in the form of bubble functions or a modified diffusion,…
In this paper, we study Riemannian zeroth-order optimization in settings where the underlying Riemannian metric $g$ is geodesically incomplete, and the goal is to approximate stationary points with respect to this incomplete metric. To…
Numerically solving high-dimensional random parametric PDEs poses a challenging computational problem. It is well-known that numerical methods can greatly benefit from adaptive refinement algorithms, in particular when functional…
The theory of nonlinear balanced truncation provides a system-theoretic framework for model reduction that preserves important properties such as stability, controllability, and observability. We present a scalable algorithm for computing…
Bilevel optimization problems are receiving increasing attention in machine learning as they provide a natural framework for hyperparameter optimization and meta-learning. A key step to tackle these problems is the efficient computation of…
This paper concerns the recently proposed quasi-balanced truncation model reduction method for linear quantum stochastic systems. It has previously been shown that the quasi-balanceable class of systems (i.e. systems that can be truncated…
In this work, we consider the $\mathcal{H}_2$ optimal model reduction of dynamical systems that are linear in the state equation and up to quadratic nonlinearity in the output equation. As our primary theoretical contributions, we derive…
This work analyzes the overall computational complexity of the stochastic Galerkin finite element method (SGFEM) for approximating the solution of parameterized elliptic partial differential equations with both affine and non-affine random…
In this paper, we study the almost sure boundedness and the convergence of the stochastic approximation (SA) algorithm. At present, most available convergence proofs are based on the ODE method, and the almost sure boundedness of the…
We present a mass lumping approach based on an isogeometric Petrov-Galerkin method that preserves higher-order spatial accuracy in explicit dynamics calculations irrespective of the polynomial degree of the spline approximation. To…
We study a class of fully-discrete schemes for the numerical approximation of solutions of stochastic Cahn--Hilliard equations with cubic nonlinearity and driven by additive noise. The spatial (resp. temporal) discretization is performed…
We present a multiscale continuous Galerkin (MSCG) method for the fast and accurate stochastic simulation and optimization of time-harmonic wave propagation through photonic crystals. The MSCG method exploits repeated patterns in the…
In this work, the authors introduce a generalized weak Galerkin (gWG) finite element method for the time-dependent Oseen equation. The generalized weak Galerkin method is based on a new framework for approximating the gradient operator.…
The proximal Galerkin finite element method is a high-order, low-iteration complexity, nonlinear numerical method that preserves the geometric and algebraic structure of point-wise bound constraints in infinite-dimensional function spaces.…
This work establishes a novel, unified theoretical framework for a class of high order embedded boundary methods, revealing that the Reconstruction for Off-site Data (ROD) treatment shares a fundamental structure with the recently developed…
This paper proposes a data-driven model reduction approach on the basis of noisy data. Firstly, the concept of data reduction is introduced. In particular, we show that the set of reduced-order models obtained by applying a Petrov-Galerkin…
Stochastic gradient descent (SGD) is a popular algorithm for minimizing objective functions that arise in machine learning. For constant step-sized SGD, the iterates form a Markov chain on a general state space. Focusing on a class of…
In the analysis of Markov chains and processes, it is sometimes convenient to replace an unbounded state space with a "truncated" bounded state space. When such a replacement is made, one often wants to know whether the equilibrium behavior…
This paper is concerned with generalized polynomial chaos (gPC) approximation for a general system of quasilinear hyperbolic conservation laws with uncertainty. The one-dimensional (1D) hyperbolic system is first symmetrized with the aid of…