James V. Lambers

Two new nonlinear nonlocal diffusions for noise reduction

Two new nonlocal nonlinear diffusion models for noise reduction are proposed, analyzed and implemented. They are both a close relative of the celebrated Perona-Malik equation. In a way, they can be viewed as a new regularization paradigm for Perona-Malik. They do preserve and enhance the most cherished features of Perona-Malik while delivering well-posed equations which admit a stable natural discretization. Unlike other regularizations, however, certain piecewise smooth functions are (meta)stable equilibria and, as a consequence, their dynamical behavior and that of their discrete implementations can be fully understood and do not lead to any “paradox”. The presence of nontrivial equilibria also explains why blurring is kept in check. One of the models has been proved to be well-posed. Numerical experiments are presented that illustrate the main features of the new models and that provide insight into their interesting dynamical behavior as well as demonstrate their effectiveness as a denoising tool.

Integration of local–global upscaling and grid adaptivity for simulation of subsurface flow in heterogeneous formations

We propose a methodology, called multilevel local–global (MLLG) upscaling, for generating accurate upscaled models of permeabilities or transmissibilities for flow simulation on adapted grids in heterogeneous subsurface formations. The method generates an initial adapted grid based on the given fine-scale reservoir heterogeneity and potential flow paths. It then applies local–global (LG) upscaling for permeability or transmissibility [7], along with adaptivity, in an iterative manner. In each iteration of MLLG, the grid can be adapted where needed to reduce flow solver and upscaling errors. The adaptivity is controlled with a flow-based indicator. The iterative process is continued until consistency between the global solve on the adapted grid and the local solves is obtained. While each application of LG upscaling is also an iterative process, this inner iteration generally takes only one or two iterations to converge. Furthermore, the number of outer iterations is bounded above, and hence, the computational costs of this approach are low. We design a new flow-based weighting of transmissibility values in LG upscaling that significantly improves the accuracy of LG and MLLG over traditional local transmissibility calculations. For highly heterogeneous (e.g., channelized) systems, the integration of grid adaptivity and LG upscaling is shown to consistently provide more accurate coarse-scale models for global flow, relative to reference fine-scale results, than do existing upscaling techniques applied to uniform grids of similar densities. Another attractive property of the integration of upscaling and adaptivity is that process dependency is strongly reduced, that is, the approach computes accurate global flow results also for flows driven by boundary conditions different from the generic boundary conditions used to compute the upscaled parameters. The method is demonstrated on Cartesian cell-based anisotropic refinement (CCAR) grids, but it can be applied to other adaptation strategies for structured grids and extended to unstructured grids.

Enhancement of Krylov Subspace Spectral Methods by Block Lanczos Iteration

This paper presents a modification of Krylov Subspace Spectral (KSS) Methods, which build on the work of Golub, Meurant and others pertaining to moments and Gaussian quadrature to produce high‐order accurate approximate solutions to variable‐coefficient time‐dependent PDE. Whereas KSS methods currently use Lanczos iteration to compute the needed quadrature rules, the modification uses block Lanczos iteration in order to avoid the need to compute two quadrature rules for each component of the solution, or use perturbations of quadrature rules. It will be shown that under reasonable assumptions on the coefficients of the problem, a 1‐node KSS method is unconditionally stable, and methods with more than one node are shown to possess favorable stability properties as well. Numerical results suggest that block KSS methods are significantly more accurate than their non‐block counterparts.

Practical Implementation of Krylov Subspace Spectral Methods

Abstract Krylov subspace spectral methods have been shown to be high-order accurate in time and more stable than explicit time-stepping methods, but also more difficult to implement efficiently. This paper describes how these methods can be fashioned into practical solvers by exploiting the simple structure of differential operators Numerical results concerning accuracy and efficiency are presented for parabolic problems in one and two space dimensions.

A Spectral Time-Domain Method for Computational Electrodynamics

We present a new approach to the numerical solution of Maxwell’s equations in the case of spatially‐varying electric permittivity and/or magnetic permeability, based on Krylov subspace spectral (KSS) methods. KSS methods for scalar equations compute each Fourier coefficient of the solution using techniques developed by Gene Golub and Gerard Meurant for approximating elements of functions of matrices by Gaussian quadrature in the spectral, rather than physical, domain. We show how they can be generalized to coupled systems of equations, such as Maxwell’s equations, by choosing appropriate basis functions that, while induced by this coupling, still allow efficient and robust computation of the Fourier coefficients of each spatial component of the electric and magnetic fields. We also discuss the implementation of appropriate boundary conditions for simulation on infinite computational domains, and how discontinuous coefficients can be handled.

Analysis of Wave Propagation in 1D Inhomogeneous Media

In this paper, we consider the one-dimensional inhomogeneous wave equation with particular focus on its spectral asymptotic properties and its numerical resolution. In the first part of the paper, we analyze the asymptotic nodal point distribution of high-frequency eigenfunctions, which, in turn, gives further information about the asymptotic behavior of eigenvalues and eigenfunctions. We then turn to the behavior of eigenfunctions in the high- and low-frequency limit. In the latter case, we derive a homogenization limit, whereas in the first we show that a sort of self-homogenization occurs at high frequencies. We also remark on the structure of the solution operator and its relation to desired properties of any numerical approximation. We subsequently shift our focus to the latter and present a Galerkin scheme based on a spectral integral representation of the propagator in combination with Gaussian quadrature in the spectral variable with a frequency-dependent measure. The proposed scheme yields accurate resolution of both high- and low-frequency components of the solution and as a result proves to be more accurate than available schemes at large time steps for both smooth and nonsmooth speeds of propagation.

Image Restoration with a New Class of Forward-Backward-Forward Diffusion Equations of Perona--Malik Type with Applications to Satellite Image Enhancement

A new class of anisotropic diffusion models is proposed for image processing which can be viewed either as a novel kind of regularization of the classical Perona--Malik model or, as advocated by the authors, as a new independent model. The models are diffusive in nature and are characterized by the presence of both forward and backward regimes. In contrast to the Perona--Malik model, in the proposed model the backward regime is confined to a bounded region, and gradients are only allowed to grow up to a large but tunable size, thus effectively preventing indiscriminate singularity formation, i.e., staircasing. Extensive numerical experiments demonstrate that the method is a viable denoising/deblurring tool. The method is significantly faster than competing state-of-the-art methods and appears to be particularly effective for simultaneous denoising and deblurring. An application to satellite image enhancement is also presented.

Krylov Subspace Spectral Methods for the Time-Dependent Schrodinger Equation with Non-Smooth Potentials

This paper presents modifications of Krylov Subspace Spectral (KSS) Methods, which build on the work of Gene Golub and others pertaining to moments and Gaussian quadrature to produce high-order accurate approximate solutions to the time-dependent Schrödinger equation in the case where either the potential energy or the initial data is not a smooth function. These modifications consist of using various symmetric perturbations to compute off-diagonal elements of functions of matrices. It is demonstrated through analytical and numerical results that KSS methods, with these modifications, achieve the same high-order accuracy and possess the same stability properties as they do when applied to parabolic problems, even though the solutions to the Schrödinger equation do not possess the same smoothness.

Eigenvalue Characterization and Computation for the Laplacian on General 2-D Domains

In this paper, we address the problem of determining and efficiently computing an approximation to the eigenvalues of the negative Laplacian − ▵ on a general domain Ω ⊂ ℝ2 subject to homogeneous Dirichlet or Neumann boundary conditions. The basic idea is to look for eigenfunctions as the superposition of generalized eigenfunctions of the corresponding free space operator, in the spirit of the classical method of particular solutions (MPS). The main novelties of the proposed approach are the possibility of targeting each eigenvalue independently without the need for extensive scanning of the positive real axis and the use of small matrices. This is made possible by iterative inclusion of more basis functions in the expansions and a projection idea that transforms the minimization problem associated with MPS and its variants into a relatively simple zero-finding problem, even for expansions with very few basis functions.

Solution of nonlinear time-dependent PDEs through componentwise approximation of matrix functions

Exponential propagation iterative (EPI) methods provide an efficient approach to the solution of large stiff systems of ODEs, compared to standard integrators. However, the bulk of the computational effort in these methods is due to products of matrix functions and vectors, which can become very costly at high resolution due to an increase in the number of Krylov projection steps needed to maintain accuracy. In this paper, it is proposed to modify EPI methods by using Krylov subspace spectral (KSS) methods, instead of standard Krylov projection methods, to compute products of matrix functions and vectors. Numerical experiments demonstrate that this modification causes the number of Krylov projection steps to become bounded independently of the grid size, thus dramatically improving efficiency and scalability. As a result, for each test problem featured, as the total number of grid points increases, the growth in computation time is just below linear, while other methods achieved this only on selected test problems or not at all.