Sparse modeling/approximation of images plays an important role in image restoration. Instead of using a fixed system to sparsely model any input image, a more promising approach is using a system that is adaptive to the input image. A non-convex variational model is proposed in [1] for constructing a tight frame that is optimized for the input image, and an alternating scheme is used to solve the resulting non-convex optimization problem. Although it showed good empirical performance in image denoising, the proposed alternating iteration lacks convergence analysis. This paper aims at providing the convergence analysis of the method proposed in [1]. We first established the sub-sequence convergence property of the iteration scheme proposed in [1], i.e., there exists at least one convergent sub-sequence and any convergent sub-sequence converges to a stationary point of the minimization problem. Moreover, we showed that the original method can be modified to have sequence convergence, i.e., the modified algorithm generates a sequence that converges to a stationary point of the minimization problem.

A new class of implicit high-order non-oscillatory time integration schemes is introduced in a method-of-lines framework. These schemes can be used in conjunction with an appropriate spatial discretization scheme for the numerical solution of time dependent conservation equations. The main concept behind these schemes is that the order of accuracy in time is dropped locally in regions where the time evolution of the solution is not smooth. By doing this, an attempt is made at locally satisfying monotonicity conditions, while maintaining a high order of accuracy in most of the solution domain. When a linear high order time integration scheme is used along with a high order spatial discretization, enforcement of monotonicity imposes severe time-step restrictions. We propose to apply limiters to these time-integration schemes, thus making them non-linear. When these new schemes are used with high order spatial discretizations, solutions remain non-oscillatory for much larger time-steps as compared to linear time integration schemes. Numerical results obtained on scalar conservation equations and systems of conservation equations are highly promising.

Numerical homogenization and multiscale finite element methods construct effective properties on a coarse grid by solving local problems and extracting the average effective properties from these local solutions. In some cases, the solutions of local problems can be expensive to compute due to scale disparity. In this setting, one can basically apply a homogenization or multiscale method reiteratively to solve for the local problems. This process is known as reiterated homogenization and has many variations in the numerical context. Though the process seems to be a straightforward extension of two-level process, it requires some careful implementation and the concept development for problems without scale separation and high contrast. In this paper, we consider the generalized multiscale finite element method (GMsFEM) and apply it iteratively to construct its multiscale basis functions. The main idea of the

In this paper, we present an adaptive, analysis of variance (ANOVA)-based data-driven stochastic method (ANOVA-DSM) to study the stochastic partial differential equations (SPDEs) in the multi-query setting. Our new method integrates the advantages of both the adaptive ANOVA decomposition technique and the data-driven stochastic method. To handle high-dimensional stochastic problems, we investigate the use of adaptive ANOVA decomposition in the stochastic space as an effective dimension-reduction technique. To improve the slow convergence of the generalized polynomial chaos (gPC) method or stochastic collocation (SC) method, we adopt the data-driven stochastic method (DSM) for speed up. An essential ingredient of the DSM is to construct a set of stochastic basis under which the stochastic solutions enjoy a compact representation for a broad range of forcing functions and/or boundary conditions

In this paper, we develop an adaptive generalized multiscale discontinuous Galerkin method (GMsDGM) for a class of high-contrast flow problems and derive a priori and a posteriori error estimates for the method. Based on the a posteriori error estimator, we develop an adaptive enrichment algorithm for our GMsDGM and prove its convergence. The adaptive enrichment algorithm gives an automatic way to enrich the approximation space in regions where the solution requires more basis functions, which are shown to perform well compared with a uniform enrichment. We also discuss an approach that adaptively selects multiscale basis functions by correlating the residual to multiscale basis functions (cf. [S. S. Chen, D. L. Donoho, and M. A. Saunders, <i>SIAM Rev.</i>, 43 (2001), pp. 129--159]). The proposed error indicators are L_2-based and can be inexpensively computed, which makes our approach efficient