This is part II of our paper in which we propose and develop a dynamically bi-orthogonal method (DyBO) to study a class of time-dependent stochastic partial differential equations (SPDEs) whose solutions enjoy a low-dimensional structure. In part I of our paper [9], we derived the DyBO formulation and proposed numerical algorithms based on this formulation. Some important theoretical results regarding consistency and bi-orthogonality preservation were also established in the first part along with a range of numerical examples to illustrate the effectiveness of the DyBO method. In this paper, we focus on the computational complexity analysis and develop an effective adaptivity strategy to add or remove modes dynamically. Our complexity analysis shows that the ratio of computational complexities between the DyBO method and a generalized polynomial chaos method (gPC) is roughly of order O ((m/N p) 3) for a

Thetransmissioneigenvalueproblem,besidesitscriticalroleininversescattering problems, deserves special interest of its own due to the fact that the corresponding differen- tial operator is neither elliptic nor self-adjoint. In this paper, we provide a spectral analysis and propose a novel iterative algorithm for the computation of a few positive real eigen- values and the corresponding eigenfunctions of the transmission eigenvalue problem. Based on approximation using continuous finite elements, we first derive an associated symmetric quadratic eigenvalue problem (QEP) for the transmission eigenvalue problem to eliminate the nonphysical zero eigenvalues while preserve all nonzero ones. In addition, the derived QEP enables us to consider more refined discretization to overcome the limitation on the number of degrees of freedom. We then transform the QEP to a parameterized symmet- ric definite generalized eigenvalue problem (GEP) and develop a secant-type iteration for solving the resulting GEPs. Moreover, we carry out spectral analysis for various existence intervals of desired positive real eigenvalues, since a few lowest positive real transmission eigenvalues are of practical interest in the estimation and the reconstruction of the index of refraction. Numerical experiments show that the proposed method can find those desired smallest positive real transmission eigenvalues accurately, efficiently, and robustly.

Since the development of Yee scheme back in 1966, it has become one of the most popular simulation tools for modeling electromagnetic wave propagation in various situations.However, its rigorous error analysis on nonuniform rectangular type gridswas carried out until 1994 by Monk and Süli. They showed that theYee scheme is still second-order convergent on a nonuniform mesh even though the local truncation error is only of first order. In this paper, we extend their results to Maxwell’s equations in metamaterials by a simpler proof, and show the second-order superconvergence in space for the trueYee scheme instead of the only semi-discrete form discussed in Monk and Süli’s original work. Numerical results supporting our analysis are presented.

We propose a computational framework named iterative local adaptive majorize-minimization (I-LAMM) to simultaneously control algorithmic complexity and statistical error when fitting high dimensional models. I-LAMM is a two-stage algorithmic implementation of the local linear approximation to a family of folded concave penalized quasi-likelihood. The first stage solves a convex program with a crude precision tolerance to obtain a coarse initial estimator, which is further refined in the second stage by iteratively solving a sequence of convex programs with smaller precision tolerances. Theoretically, we establish a phase transition: the first stage has a sublinear iteration complexity, while the second stage achieves an improved linear rate of convergence. Though this framework is completely algorithmic, it provides solutions with optimal statistical performances and controlled algorithmic complexity for a large family

In this paper, we consider an online basis enrichment mixed generalized multiscale method with oversampling, for solving flow problems in highly heterogeneous porous media. This is an exten-sion of the online mixed generalized multiscale method [6]. The multiscale online basis functions are computed by solving a Neumann problem in an over-sampled domain, instead of a standard neighborhood of a coarse face. We are motivated by the restricted domain decomposition method. Extensive numerical experiments are presented to demonstrate the performance of our methods for both steady-state flow, and two-phase flow and transport problems.