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2012

Volume 50, Issue 3 (partial)


An Iterative Lagrange Multiplier Method for Constrained Total-Variation-Based Image Denoising

Jianping Zhang, Ke Chen, and Bo Yu

SIAM J. Numer. Anal. 50, pp. 983-1003 (21 pages)

Online Publication Date: May 03, 2012

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Various effective algorithms have been proposed in the past two decades for nonlinear PDEs arising from the unconstrained total-variation-based image denoising problem regularizing the total variation constrained minimization model. Such algorithms can be used to obtain a satisfactory result as long as a suitable regularization parameter balancing the trade-off between a good fit to the data and a regular solution is given. However, it is generally difficult to obtain a suitable regularization parameter without which restored images can be unsatisfactory: if it is too large, then the resulting solution is still contaminated by noise, while if too small, the solution is a poor approximation of the true noise-free solution. To provide an automatic method for the regularization parameter when the noise level is known a priori, one way is to address the coupled Karush–Kuhn–Tucker (KKT) systems from the constrained total variation optimization problem. So far much less work has been done on this problem. This paper presents an iterative update algorithm for a Lagrange multiplier to solve the KKT conditions, and our proposed method can adaptively deal with noisy images with different variances $\sigma^2$. Numerical experiments show that our model can effectively find a highly accurate solution and produce excellent restoration results in terms of image quality.

An Iterative Substructuring Algorithm for Two-Dimensional Problems in H(curl)

Clark R. Dohrmann and Olof B. Widlund

SIAM J. Numer. Anal. 50, pp. 1004-1028 (25 pages)

Online Publication Date: May 03, 2012

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A domain decomposition algorithm, similar to classical iterative substructuring algorithms, is presented for two-dimensional problems in the space $H_0(\mbox{curl};\Omega)$. It is defined in terms of a coarse space and local subspaces associated with individual edges of the subdomains into which the domain of the problem has been subdivided. The algorithm differs from others in three basic respects. First, it can be implemented in an algebraic manner that does not require access to individual subdomain matrices or a coarse discretization of the domain; this is in contrast to algorithms of the BDDC, FETI–DP, and classical two-level overlapping Schwarz families. Second, favorable condition number bounds can be established over a broader range of subdomain material properties than in previous studies. Third, we are able to develop theory for quite irregular subdomains and bounds for the condition number of our preconditioned conjugate gradient algorithm, which depend only on a few geometric parameters. The coarse space for the algorithm is based on simple energy minimization concepts, and its dimension equals the number of subdomain edges. Numerical results are presented which confirm the theory and demonstrate the usefulness of the algorithm for a variety of mesh decompositions and distributions of material properties.

An Adaptive Finite Element Eigenvalue Solver of Asymptotic Quasi-Optimal Computational Complexity

Carsten Carstensen and Joscha Gedicke

SIAM J. Numer. Anal. 50, pp. 1029-1057 (29 pages)

Online Publication Date: May 03, 2012

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This paper presents a combined adaptive finite element method with an iterative algebraic eigenvalue solver for a symmetric eigenvalue problem of asymptotic quasi-optimal computational complexity. The analysis is based on a direct approach for eigenvalue problems and allows the use of higher-order conforming finite element spaces with fixed polynomial degree. The asymptotic quasi-optimal adaptive finite element eigenvalue solver (AFEMES) involves a proper termination criterion for the algebraic eigenvalue solver and does not need any coarsening. Numerical evidence illustrates the asymptotic quasi-optimal computational complexity in 2 and 3 dimensions.

Stability Conditions for the Numerical Solution of Convection-Dominated Problems with Skew-Symmetric Discretizations

Erwan Deriaz

SIAM J. Numer. Anal. 50, pp. 1058-1085 (28 pages)

Online Publication Date: May 10, 2012

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This paper presents original and close to optimal stability conditions linking the time step and the space step, stronger than the CFL criterion, $\delta t\leq C\delta x^\alpha$ with $\alpha=\frac{2r}{2r-1}$, $r$ an integer for some numerical schemes we produce when solving convection-dominated problems. We test this condition numerically and prove that it applies to nonlinear equations under smoothness assumptions.

Material and Shape Derivative Method for Quasi-Linear Elliptic Systems with Applications in Inverse Electromagnetic Interface Problems

Ivan Cimrák

SIAM J. Numer. Anal. 50, pp. 1086-1110 (25 pages)

Online Publication Date: May 10, 2012

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We study a shape optimization problem for quasi-linear elliptic systems. The state equations describe an interface problem and the ultimate goal of our research is to determine the interface between two materials with different physical properties. The interface is identified by the minimization of the shape (or the cost) functional representing the misfit between the data and the simulations. For shape sensitivity of the shape functional we elaborate the material and the shape derivative method. In this concept a vector field is introduced that deforms the unknown shape toward the optimum. We characterize the elliptic interface problems whose solutions give the material and the shape derivatives. In particular, we show the existence of weak as well as strong material derivatives. Further, we employ the adjoint variable method to obtain an explicit expression for the gradient of the shape functional. This gradient is then used for the actual implementation of the minimization algorithm. In simulations we use the level set method for the representation of the interface. We present the simulation results showing the reconstructed voids in the nonlinear ferromagnetic material from the near-boundary measurements of magnetic induction.

Convergence of a Semi-Lagrangian Scheme for the BGK Model of the Boltzmann Equation

Giovanni Russo, Pietro Santagati, and Seok-Bae Yun

SIAM J. Numer. Anal. 50, pp. 1111-1135 (25 pages)

Online Publication Date: May 15, 2012

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Recently, a new class of semi-Lagrangian methods for the BGK model of the Boltzmann equation has been introduced [F. Filbet and G. Russo, Kinet. Relat. Models, 2 (2009), pp. 231–250; G. Russo and P. Santagati, A new class of large time step methods for the BGK models of the Boltzmann equation, arXiv:1103.5247; P. Santagati, High Order Semi-Lagrangian Methods for the BGK Model of the Boltzmann Equation, Ph.D. thesis, University of Catania, Italy, 2007]. These methods work in a satisfactory way either in a rarefied or a fluid regime. Moreover, because of the semi-Lagrangian feature, the stability property is not restricted by the CFL condition. These aspects make them very attractive for practical applications. In this paper, we prove that the discrete solution of the scheme converges in a weighted $L^1$ norm to the unique smooth solution by deriving an explicit error estimate.

On the Approximation of the Fokker–Planck Equation of the Finitely Extensible Nonlinear Elastic Dumbbell Model I: A New Weighted Formulation and an Optimal Spectral-Galerkin Algorithm in Two Dimensions

Jie Shen and Haijun Yu

SIAM J. Numer. Anal. 50, pp. 1136-1161 (26 pages)

Online Publication Date: May 15, 2012

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We propose a new weighted weak formulation for the Fokker–Planck equation of the finitely extensible nonlinear elastic dumbbell model and prove its well-posedness in weighted Sobolev spaces. We also propose simple and efficient semi-implicit time-discretization schemes which are unconditionally stable, i.e., the step size of time marching does not depend on the number of the bases used in configurational space. We then restrict ourselves to the two-dimensional case and construct two Fourier–Jacobi spectral-Galerkin algorithms which enjoy the following properties: (i) they are unconditionally stable, spectrally accurate, and of optimal computational complexity; (ii) they conserve the volume and provide accurate approximation to higher-order moments of the distribution function; and (iii) they can be easily extended to coupled nonhomogeneous systems. Numerical results are presented to show how to choose a proper weight to get the best numerical results of the distribution function and the polymer stress.

Total Variation Minimization with Finite Elements: Convergence and Iterative Solution

Sören Bartels

SIAM J. Numer. Anal. 50, pp. 1162-1180 (19 pages)

Online Publication Date: May 15, 2012

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The numerical solution of a convex minimization problem involving the nonsmooth total variation norm is analyzed. Consistent finite element discretizations that avoid regularizations lead to simple convergence proofs in the case of piecewise affine, globally continuous finite elements. For the approximation with piecewise constant finite elements it is proved that convergence to the exact solution cannot be expected in general. The iterative solution is based on a regularized $L^2$ flow of the energy functional, and convergence of the iteration to a stationary point is proved under a moderate constraint on the time-step size. The extension of the techniques to an energy functional that involves a negative order term is discussed. Numerical experiments that illustrate the theoretical results are presented.
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