A common strategy for the dimensionality reduction of nonlinear partial differential equations (PDEs) relies on use proper orthogonal decomposition (POD) to identify a reduced subspace and Galerkin projection evolving dynamics in this space. However, advection-dominated PDEs are represented poorly by methodology since process truncation discards important interactions between higher-order modes...

For initial boundary value problems of linear parabolic partial differential equations with random coefficients, we show analyticity of the solution with respect to the parameters and give an a priori error analysis for N-term generalized polynomial chaos approximations in a scale of Bochner spaces. The problem is reduced to a parametric family of deterministic initial boundary value problems o...

The this paper, we introduce a pair of Sobolev spaces with special Jacobi–Gegenbauer weights, in which the general boundary-value problem for class ordinary integro-differential equations characterized by positivity difference orders inner and outer differential operators is well-posed Hadamard sense. Based on result, justify polynomial projection method solving corresponding problem. An applic...

In the framework of abstract linear inverse problems in infinite-dimensional Hilbert space we discuss generic convergence behaviours approximate solutions determined by means general projection methods, namely outside standard assumptions Petrov–Galerkin truncation schemes. This includes a discussion mechanisms why error or residual generically fail to vanish norm, and identification practicall...

This paper derives state space error bounds for the solutions of reduced systems constructed using Proper Orthogonal Decomposition (POD) together with the Discrete Empirical Interpolation Method (DEIM) recently developed in [4] for nonlinear dynamical systems. The resulting error estimates are shown to be proportional to the sums of the singular values corresponding to neglected POD basis vecto...

This is a personal perspective on the development of numerical methods for solving Fredholm integral equations of the second kind, discussing work being done principally during the 1950s and 1960s. The principal types of numerical methods being studied were projection methods (Galerkin, collocation) and Nyström methods. During the 1950s and 1960s, functional analysis became the framework for th...

This paper is concerned with numerical solutions of large scale Sylvester equations AX −XB = C, Lyapunov equations as a special case in particular included, with C having very small rank. For stable Lyapunov equations, Penzl (2000) and Li andWhite (2002) demonstrated that the so called Cholesky factored ADI method with decent shift parameters can be very effective. In this paper we present a ge...

We consider the Galerkin finite element approximation of an elliptic Dirichlet boundary control model problem governed by the Laplacian operator. The functional theoretical setting of this problem uses L2 controls and a “very weak” formulation of the state equation. However, the corresponding finite element approximation uses standard continuous trial and test functions. For this approximation,...

We analyze the spatially semidiscrete piecewise linear finite element method for a nonlocal parabolic equation resulting from thermistor problem. Our approach is based on the properties of the elliptic projection defined by the bilinear form associated with the variational formulation of the finite element method. We assume minimal regularity of the exact solution that yields optimal order erro...

Slow Invariant Manifolds (SIM) are calculated for isothermal closed reaction-diffusion systems as a model reduction technique. Diffusion effects are examined using a Galerkin projection that rigorously accounts for the coupling of reaction and diffusion processes. This method reduces the infinite dimensional dynamical system by projecting it on a low dimensional approximate inertial manifold. A...