This paper proposes a simple and rigorous Fourier-matching method to study transverse-magnetic-polarized electro-magnetic resonances by a perfectly conducting slab with a finite number of subwavelength slits of width $h\ll 1$. Since variable separation is ...

Linear multistep methods (LMMs) are popular time discretization techniques for the numerical solution of differential equations. Traditionally they are applied to solve for the state given the dynamics (the forward problem), but here we consider their ...

In [IMA J. Numer. Anal., 29 (2009), pp. 24--42], Nielsen, Tveito, and Hackbusch study the operator generated by using the inverse of the Laplacian as the preconditioner for second order elliptic PDEs $-\nabla \cdot (k(x) \nabla u) = f$. They prove that ...

To provide mathematically rigorous eigenvalue bounds for the Steklov eigenvalue problem, an enhanced version of the eigenvalue estimation algorithm developed by the third author is proposed. Compared with the existing algorithm, which deals with ...

Motivated by studies on fully discrete numerical schemes for linear hyperbolic conservation laws, we present a framework on analyzing the strong stability of explicit Runge--Kutta (RK) time discretizations for seminegative autonomous linear systems. The ...

The problem of preconditioning the $p$-version mass matrix on meshes of (possibly curvilinear) triangular elements in two dimensions is considered. Through a judicious choice of hierarchical basis, it is shown that a preconditioner involving only diagonal ...

The focus of this work is on the construction and analysis of optimal-order multigrid preconditioners to be used in the Newton--Krylov method for a distributed optimal control problem constrained by the stationary Navier--Stokes equations. As in our ...

We prove that the a standard adaptive algorithm for the Taylor--Hood discretization of the stationary Stokes problem converges with optimal rate. This is done by developing an abstract framework for quite general problems, which allows us to prove ...

Dynamical low-rank approximation in the Tucker tensor format of given large time-dependent tensors and of tensor differential equations is the subject of this paper. In particular, a discrete time integration method for rank-constrained Tucker tensors ...

Various algebraic multigrid algorithms have been developed for solving problems in scientific and engineering computation over the past decades. They have been shown to be well-suited for solving discretized partial differential equations on unstructured ...

The notion of positive realness for linear time-invariant (LTI) dynamical systems, equivalent to passivity, is one of the oldest in system and control theory. In this paper, we consider the problem of finding the nearest positive real (PR) system to a non-...

In this work, we establish a new truncation error estimate of the singular value decomposition (SVD) for a class of Sobolev smooth bivariate functions $ \kappa {\,\in\,} L^2(\Omega,H^s(D))$, $s{\,\geq\,} 0$, and $\kappa\in L^2(\Omega,\dot{H}^s(D))$ with $D ...

In order to calculate the electronic properties of graphene structures a tight-binding approach can be used. The tight-binding formulation leads to linear systems of equations which are maximally indefinite, i.e., with equal number of positive and ...

In this paper, we present a two-level overlapping hybrid domain decomposition method for solving the large scale discrete elliptic eigenvalue problems. In order to eliminate the components in the orthogonal complement space of the eigenspace, we ...

The definition of partial differential equation models usually involves a set of parameters whose values may vary over a wide range. The solution of even a single set of parameter values may be quite expensive. In many cases, e.g., optimization, control, ...

We prove a universal limit theorem for the halting time, or iteration count, of the power/inverse power methods and the QR eigenvalue algorithm. Specifically, we analyze the required number of iterations to compute extreme eigenvalues of random, positive ...

In implicit solvation models, the electrostatic contribution to the solvation energy can be estimated by solving a system of elliptic partial differential equations modeling the reaction potential. The domain of definition of such elliptic equations is the ...

In this paper we study some properties of the classical Arnoldi-based methods for solving infinite dimensional linear equations involving compact operators. These problems are intrinsically ill-posed since a compact operator does not admit a bounded inverse. ...

The ensemble Kalman filter (EnKF) is a widely used methodology for state estimation in partially, noisily observed dynamical systems and for parameter estimation in inverse problems. Despite its widespread use in the geophysical sciences, and its gradual ...

We introduce four integral operators related to the Laplace equation in three dimensions on the circular unit disk. Two of them are related to the weakly singular operator and the other two are related to the hypersingular operator. We provide series ...