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multigrid tomographic inversion with variable resolution data and image spaces

IEEE TRANSACTIONS ON IMAGE PROCESSING 2006年第9期15卷 2805-2819页

作者： Oh, Seungseok Bouman, Charles A. Webb, Kevin J. Fujifilm Software Calif Inc San Jose CA 95110 USA Purdue Univ Sch Elect & Comp Engn W Lafayette IN 47907 USA

A multigrid inversion approach that uses variable resolutions of both the data space and the image space is proposed. Since the computational complexity of inverse problems typically increases with a larger number of unknown image pixels and a larger number of measurements, the proposed algorithm further reduces the computation relative to conventional multigrid approaches, which change only the image space resolution at coarse scales. The advantage is particularly important for data-rich applications, where data resolutions may differ for different scales. Applications of the approach to Bayesian reconstruction algorithms in transmission and emission tomography with a generalized Gaussian Markov random field image prior are presented, both with a Poisson noise model and with a quadratic data term. Simulation results indicate that the proposed multigrid approach results in significant improvement in convergence speed compared to the fixed-grid iterative coordinate descent method and a multigrid method with fixed-data resolution.

关键词： computed tomography emission tomography image reconstruction inverse problems multigrid algorithms multiresolution transmission tomography

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Improvement of the rate of convergence estimates for multigrid algorithm

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APPLIED MATHEMATICS AND COMPUTATION 2007年第2期187卷 922-928页

作者： Tavakoli, Ali Kerayechian, Asghar Ferdowsi Univ Mashhad Sch Math Sci Dept Math Mashhad *** Iran

In this paper, we present a new convergence rate for both symmetric and nonsymmetric positive definite problems. In our theory, "regularity and approximation" assumption is used. A new rate of convergence estimate with alpha = 1/2 is used where a is a "regularity and approximation" parameter. Also, a new convergence rate is given by two-grid schemes. (C) 2006 Elsevier Inc. All rights reserved.

关键词： finite elements multigrid algorithms

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GRID APPLICATIONS ON DISTRIBUTED MEMORY ARCHITECTURES - IMPLEMENTATION AND EVALUATION

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PARALLEL COMPUTING 1988年第3期7卷 341-356页

作者： SOLCHENBACH, K SUPRENUM GmbH Hohestrasse 73 D-5300 Bonn Fed. Rep. Germany

It was shown in the paper of Solchenbach and Trottenberg (in this special issue) that grid algorithms are inherently parallel and that parallel grid algorithms for regular grids can be efficiently implemented on dm-mp systems using the concept of grid partitioning. In this paper, we demonstrate that grid applications can be implemented quite easily on dm-mp systems if a hardware-independent process system exists and convenient tools (such as the SUPRENUM mapping and communications library) are available. The evaluation of parallel grid algorithms shows that the multiprocessor speedup and efficiency for single grid applications depends on the communication/calculation performance ratio of the hardware, on the communication/calculation ratio of the algorithms, and on the process size. The efficiency of parallel multigrid algorithms additionally depends on the number of nodes.

关键词： Computational model performance analysis of grid applications multigrid algorithms implementation on distributed memory multiprocessors

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HP-multigrid as Smoother algorithm for higher order discontinuous Galerkin discretizations of advection dominated flows. Part II: Optimization of the Runge-Kutta smoother

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JOURNAL OF COMPUTATIONAL PHYSICS 2012年第22期231卷 7564-7583页

作者： van der Vegt, J. J. W. Rhebergen, S. Univ Twente Dept Appl Math NL-7500 AE Enschede Netherlands Univ Minnesota Sch Math Minneapolis MN 55455 USA

Using a detailed multilevel analysis of the complete hp-multigrid as Smoother algorithm accurate predictions are obtained of the spectral radius and operator norms of the multi-grid error transformation operator. This multilevel analysis is used to optimize the coefficients in the semi-implicit Runge-Kutta smoother, such that the spectral radius of the multigrid error transformation operator is minimal under properly chosen constraints. The Runge-Kutta coefficients for a wide range of cell Reynolds numbers and a detailed analysis of the performance of the hp-MGS algorithm are presented. In addition, the computational complexity of the hp-MGS algorithm is investigated. The hp-MGS algorithm is tested on a fourth order accurate space-time discontinuous Galerkin finite element discretization of the advection-diffusion equation for a number of model problems, which include thin boundary layers and highly stretched meshes, and a non-constant advection velocity. For all test cases excellent multigrid convergence is obtained. (C) 2012 Elsevier Inc. All rights reserved.

关键词： multigrid algorithms Discontinuous Galerkin methods Higher order accurate discretizations Space-time methods Runge-Kutta methods Fourier analysis Multilevel analysis

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A general framework for nonlinear multigrid inversion

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IEEE TRANSACTIONS ON IMAGE PROCESSING 2005年第1期14卷 125-140页

作者： Oh, S Milstein, AB Bouman, CA Webb, KJ Purdue Univ Sch Elect & Comp Engn W Lafayette IN 47907 USA

A variety of new imaging modalities, such as optical diffusion tomography, require the inversion of a forward problem that is modeled by the solution to a three-dimensional partial differential equation. For these applications, image reconstruction is particularly difficult because the forward problem is both nonlinear and computationally expensive to evaluate. In this paper, we propose a general framework for nonlinear multigrid inversion that is applicable to a wide variety of inverse problems. The multigrid inversion algorithm results from the application of recursive multigrid techniques to the solution of optimization problems arising from inverse problems. The method works by dynamically adjusting the cost functionals at different scales so that they are consistent with, and ultimately reduce, the finest scale cost functional. In this way, the multigrid inversion algorithm efficiently computes the solution to the desired fine-scale inversion problem. Importantly, the new algorithm can greatly reduce computation because both the forward and inverse problems are more coarsely discretized at lower resolutions. An application of our method to Bayesian optical diffusion tomography with a generalized Gaussian Markov random-field image prior model shows the potential for very large computational savings. Numerical data also indicates robust convergence with a range of initialization conditions for this nonconvex optimization problem.

关键词： inverse problems multigrid algorithms multiresolution optical diffusion tomography (ODT)

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A multilevel discontinuous Galerkin method

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NUMERISCHE MATHEMATIK 2003年第3期95卷 527-551页

作者： Gopalakrishnan, J Kanschat, G Univ Florida Dept Math Gainesville FL 32611 USA Univ Heidelberg Inst Angew Math D-69121 Heidelberg Germany

A variable V-cycle preconditioner for an interior penalty finite element discretization for elliptic problems is presented. An analysis under a mild regularity assumption shows that the preconditioner is uniform. The interior penalty method is then combined with a discontinuous Galerkin scheme to arrive at a discretization scheme for an advection-diffusion problem, for which an error estimate is proved. A multigrid algorithm for this method is presented, and numerical experiments indicating its robustness with respect to diffusion coefficient are reported.

关键词： Finite-element-method multigrid algorithms Elliptic problems Interiorpenalty Equations

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Application of Helmholtz/Hodge Decomposition to Finite Element Methods for Two-Dimensional Maxwell's Equations

Application of Helmholtz/Hodge Decomposition to Finite Eleme...

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作者： Nan, Zhe Louisiana State University

学位级别：博士

In this work we apply the two-dimensional Helmholtz/Hodge decomposition to develop new finite element schemes for two-dimensional Maxwell's equations. We begin with the introduction of Maxwell's equations and a brief survey of finite element methods for Maxwell's equations. Then we review the related fundamentals in Chapter 2. In Chapter 3, we discuss the related vector function spaces and the Helmholtz/Hodge decomposition which are used in Chapter 4 and 5. The new results in this dissertation are presented in Chapter 4 and Chapter 5. In Chapter 4, we propose a new numerical approach for two-dimensional Maxwell's equations that is based on the Helmholtz/Hodge decomposition for divergence-free vector fields. In this approach an approximate solution for Maxwell's equations can be obtained by solving standard second order scalar elliptic boundary value problems. This new approach is illustrated by a P1 finite element method. In Chapter 5, we further extend the new approach described in Chapter 4 to the interface problem for Maxwell's equations. We use the extraction formulas and multigrid method to overcome the low regularity of the solution for the Maxwell interface problem. The theoretical results obtained in this dissertation are confirmed by numerical experiments.

关键词： multigrid algorithms Helmholtz/Hodge decomposition Maxwell's equations interface problems extraction formulas finite element methods

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multigrid methods for Maxwell's equations

Multigrid methods for Maxwell's equations

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作者： Cui, Jintao Louisiana State University

学位级别：博士

In this work we study finite element methods for two-dimensional Maxwell's equations and their solutions by multigrid algorithms. We begin with a brief survey of finite element methods for Maxwell's equations. Then we review the related fundamentals, such as Sobolev spaces, elliptic regularity results, graded meshes, finite element methods for second order problems, and multigrid algorithms. In Chapter 3, we study two types of nonconforming finite element methods on graded meshes for a two-dimensional curl-curl and grad-div problem that appears in electromagnetics. The first method is based on a discretization using weakly continuous P1 vector fields. The second method uses discontinuous P1 vector fields. Optimal convergence rates (up to an arbitrary positive epsilon) in the energy norm and the L2 norm are established for both methods on graded meshes. In Chapter 4, we consider a class of symmetric discontinuous Galerkin methods for a model Poisson problem on graded meshes that share many techniques with the nonconforming methods in Chapter 3. Optimal order error estimates are derived in both the energy norm and the L2 norm. Then we establish the uniform convergence of W-cycle, V-cycle and F-cycle multigrid algorithms for the resulting discrete problems. In Chapter 5, we propose a new numerical approach for two-dimensional Maxwell's equations that is based on the Hodge decomposition for divergence-free vector fields. In this approach, an approximate solution for Maxwell's equations can be obtained by solving standard second order scalar elliptic boundary value problems. We illustrate this new approach by a P1 finite element method. In Chapter 6, we first report numerical results for multigrid algorithms applied to the discretized curl-curl and grad-div problem using nonconforming finite element methods. Then we present multigrid results for Maxwell's equations based on the approach introduced in Chapter 5. All the theoretical results obtained in this dissertation ar

关键词： Maxwell's equations nonconforming finite element methods discontinuous Galerkin methods graded meshes multigrid algorithms

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Optimized parametric inference for the inner loop of the multigrid Ensemble Kalman Filter

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JOURNAL OF COMPUTATIONAL PHYSICS 2022年 471卷

作者： Moldovan, G. Lehnasch, G. Cordier, L. Meldi, M. Univ Poitiers Inst Pprime CNRS ISAEENSMA 11 Bd Marie & Pierre CurieSite FuturoscopeTSA 41 F-86073 Poitiers 9 France

Essential features of the multigrid Ensemble Kalman Filter (Moldovan et al. (2021) [24]) recently proposed for Data Assimilation of fluid flows are investigated and assessed in this article. The analysis is focused on the improvement in performance due to the inner loop. In this step, data from solutions calculated on the higher resolution levels of the multigrid approach are used as surrogate observations to improve the model prediction on the coarsest levels of the grid. The latter represents the level of resolution used to run the ensemble members for global Data Assimilation. The method is tested over two classical one-dimensional problems, namely the linear advection problem and the Burgers' equation. The analyses encompass a number of different aspects such as different grid resolutions. The results indicate that the contribution of the inner loop is essential in obtaining accurate flow reconstruction and global parametric optimization. These findings open exciting perspectives of application to grid-dependent reduced-order models extensively used in fluid mechanics applications for complex flows, such as Large Eddy Simulation (LES).(c) 2022 Elsevier Inc. All rights reserved.

关键词： Kalman Filter Data Assimilation multigrid algorithms

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A semi-algebraic approach that enables the design of inter-grid operators to optimize multigrid convergence

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NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS 2008年第2-3期15卷 219-247页

作者： Michelinil, Pablo Navarrete Coyle, Edward J. Purdue Univ Ctr Wireless Syst & Applicat W Lafayette IN 47907 USA Georgia Inst Technol Sch Elect & Comp Engn Atlanta GA 30332 USA Univ Chile Dept Ingn Elect Av Tupper 2007 Santiago 8370451 Chile

We study the effect of inter-grid operators-the interpolation and restriction operators-on the convergence of two-grid algorithms for linear models. We show how a modal analysis of linear systems, along with some assumptions on the normal modes of the system, allows us to understand the role of inter-grid operators in the speed and accuracy of a full-multigrid step. We state an assumption that generalizes local Fourier analysis (LFA) by means of a precise description of aliasing effects on the system. This assumption condenses, in a single algebraic property called the harmonic aliasing property, all the information needed from the geometry of the discretization and the structure of the system's eigenvectors. We first state a harmonic aliasing property based on the standard coarsening strategies of ID problems. Then, we extend this property to a more aggressive coarsening typically used in 2D problems with the help of additional assumptions on the structure of the system matrix. Under our general assumptions, we determine the exact rates at which groups of modal components of the error evolve and interact. With this knowledge, we are then able to design inter-grid operators that optimize the two-grid algorithm convergence. By different choices of operators, we verify the classic heuristics based on Fourier harmonic analysis, show a trade-off between the rate of convergence and the number of computations required per iteration, and show how our analysis differs from LFA. Copyright (c) 2008 John Wiley & Sons, Ltd.

关键词： multigrid algorithms inter-grid operators convergence analysis modal analysis aliasing

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