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V. Girault - One of the best experts on this subject based on the ideXlab platform.
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hermite interpolation of nonsmooth functions preserving boundary conditions
Mathematics of Computation, 2002Co-Authors: V. Girault, L R ScottAbstract:This article is devoted to the construction of a Hermite-type Regularization Operator transforming functions that are not necessarily C 1 into globally C 1 finite-element functions that are piecewise polynomials. This Regularization Operator is a projection, it preserves appropriate first and second order polynomial traces, and it has approximation properties of optimal order. As an illustration, it is used to discretize a nonhomogeneous Navier-Stokes problem, with tangential boundary condition.
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A Local Regularization Operator for Triangular and Quadrilateral Finite Elements
SIAM Journal on Numerical Analysis, 1998Co-Authors: C. Bernardi, V. GiraultAbstract:This paper develops a local Regularization Operator on triangular or quadrilateral finite elements built on structured or unstructured meshes. This Operator is a variant of the Regularization Operator of Clément; however, ours is constructed via a local projection in a reference domain. We prove in this paper that it has the same optimal approximation properties as the standard interpolation Operator, and we present some applications. Introduction.
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A Local Regularization Operator for Triangular and Quadrilateral Finite Elements
SIAM Journal on Numerical Analysis, 1998Co-Authors: C. Bernardi, V. GiraultAbstract:This paper develops a local Regularization Operator on triangular or quadrilateral finite elements built on structured or unstructured meshes. This Operator is a variant of the Regularization Operator of Clement; however, ours is constructed via a local projection in a reference domain. We prove in this paper that it has the same optimal approximation properties as the standard interpolation Operator, and we present some applications.
Lothar Reichel - One of the best experts on this subject based on the ideXlab platform.
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Tikhonov Regularization based on generalized Krylov subspace methods
Applied Numerical Mathematics, 2012Co-Authors: Lothar Reichel, Fiorella SgallariAbstract:We consider Tikhonov Regularization of large linear discrete ill-posed problems with a Regularization Operator of general form and present an iterative scheme based on a generalized Krylov subspace method. This method simultaneously reduces both the matrix of the linear discrete ill-posed problem and the Regularization Operator. The reduced problem so obtained may be solved, e.g., with the aid of the singular value decomposition. Also, Tikhonov Regularization with several Regularization Operators is discussed.
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an iterative method for tikhonov Regularization with a general linear Regularization Operator
Journal of Integral Equations and Applications, 2010Co-Authors: Michiel E. Hochstenbach, Lothar ReichelAbstract:Tikhonov Regularization is one of the most popular approaches to solve discrete ill-posed problems with error-contaminated data. A Regularization Operator and a suitable value of a Regularization parameter have to be chosen. This paper describes an iterative method, based on Golub-Kahan bidiagonalization, for solving large-scale Tikhonov minimization problems with a linear Regularization Operator of general form. The Regularization parameter is determined by the discrepancy principle. Computed examples illustrate the performance of the method.
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an iterative method for tikhonov Regularization with a general linear Regularization Operator
CASA-report, 2010Co-Authors: Michiel E. Hochstenbach, Lothar ReichelAbstract:Tikhonov Regularization is one of the most popular approaches to solve discrete ill-posed problems with error-contaminated data. A Regularization Operator and a suitable value of a Regularization parameter have to be chosen. This paper describes an iterative method, based on Golub-Kahan bidiagonalization, for solving large-scale Tikhonov minimization problems with a linear Regularization Operator of general form. The Regularization parameter is determined by the discrepancy principle. Computed examples illustrate the performance of the method. Key words. Discrete ill-posed problem, iterative method, Tikhonov Regularization, general linear Regularization Operator, discrepancy principle.
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SIMPLE SQUARE SMOOTHING Regularization OperatorS
Electronic Transactions on Numerical Analysis, 2009Co-Authors: Lothar ReichelAbstract:Tikhonov Regularization of linear discrete ill-posed prob lems often is applied with a finite differ- ence Regularization Operator that approximates a low-order derivative. These Operators generally are represented by a banded rectangular matrix with fewer rows than columns. They therefore cannot be applied in iterative meth- ods that are based on the Arnoldi process, which requires the Regularization Operator to be represented by a square matrix. This paper discusses two approaches to circumvent this difficulty: zero-padding the rectangular matrices to make them square and extending the rectangular matrix to a square circulant. We also describe how to com- bine these Operators by weighted averaging and with orthogonal projection. Applications to Arnoldi and Lanczos bidiagonalization-based Tikhonov Regularization, as well as to truncated iteration with a range-restricted minimal residual method, are presented.
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L-Curve and Curvature Bounds for Tikhonov Regularization
Numerical Algorithms, 2004Co-Authors: Daniela Calvetti, Lothar Reichel, A. ShuibiAbstract:The L-curve is a popular aid for determining a suitable value of the Regularization parameter when solving linear discrete ill-posed problems by Tikhonov Regularization. However, the computational effort required to determine the L-curve and its curvature can be prohibitive for large-scale problems. Recently, inexpensively computable approximations of the L-curve and its curvature, referred to as the L-ribbon and the curvature-ribbon, respectively, were proposed for the case when the Regularization Operator is the identity matrix. This note discusses the computation and performance of the L- and curvature-ribbons when the Regularization Operator is an invertible matrix.
C. Bernardi - One of the best experts on this subject based on the ideXlab platform.
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Quelques propriétés d'approximation des éléments finis de Nédélec, application à l'analyse a posteriori
Elsevier, 2007Co-Authors: C. Bernardi, Frédéric HechtAbstract:We prove some approximation properties of nonsmooth functions in the space constructed from Nédélec finite elements of order 1. They rely either on the Nédélec Operator or on a Clément type Regularization Operator linked to these elements. The main application of these results is the a posteriori analysis of the error when the discretization involves this space, we present a basic example.
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A Local Regularization Operator for Triangular and Quadrilateral Finite Elements
SIAM Journal on Numerical Analysis, 1998Co-Authors: C. Bernardi, V. GiraultAbstract:This paper develops a local Regularization Operator on triangular or quadrilateral finite elements built on structured or unstructured meshes. This Operator is a variant of the Regularization Operator of Clément; however, ours is constructed via a local projection in a reference domain. We prove in this paper that it has the same optimal approximation properties as the standard interpolation Operator, and we present some applications. Introduction.
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A Local Regularization Operator for Triangular and Quadrilateral Finite Elements
SIAM Journal on Numerical Analysis, 1998Co-Authors: C. Bernardi, V. GiraultAbstract:This paper develops a local Regularization Operator on triangular or quadrilateral finite elements built on structured or unstructured meshes. This Operator is a variant of the Regularization Operator of Clement; however, ours is constructed via a local projection in a reference domain. We prove in this paper that it has the same optimal approximation properties as the standard interpolation Operator, and we present some applications.
Stefan Stoll - One of the best experts on this subject based on the ideXlab platform.
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optimal tikhonov Regularization for deer spectroscopy
Journal of Magnetic Resonance, 2018Co-Authors: Thomas H Edwards, Stefan StollAbstract:Abstract Tikhonov Regularization is the most commonly used method for extracting distance distributions from experimental double electron-electron resonance (DEER) spectroscopy data. This method requires the selection of a Regularization parameter, α , and a Regularization Operator, L. We analyze the performance of a large set of α selection methods and several Regularization Operators, using a test set of over half a million synthetic noisy DEER traces. These are generated from distance distributions obtained from in silico double labeling of a protein crystal structure of T4 lysozyme with the spin label MTSSL. We compare the methods and Operators based on their ability to recover the model distance distributions from the noisy time traces. The results indicate that several α selection methods perform quite well, among them the Akaike information criterion and the generalized cross validation method with either the first- or second-derivative Operator. They perform significantly better than currently utilized L-curve methods.
Samuel Forest - One of the best experts on this subject based on the ideXlab platform.
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Nonlinear Regularization Operators as derived from the micromorphic approach to gradient elasticity, viscoplasticity and damage
Proceedings of the Royal Society A: Mathematical Physical and Engineering Sciences, 2016Co-Authors: Samuel ForestAbstract:The construction of Regularization Operators presented in this work is based on the introduction of strain or damage micromorphic degrees of freedom in addition to the displacement vector and of their gradients into the Helmholtz free energy function of the constitutive material model. The combination of a new balance equation for generalized stresses and of the micromorphic constitutive equations generates the Regularization Operator. Within the small strain framework, the choice of a quadratic potential w.r.t. the gradient term provides the widely used Helmholtz Operator whose Regularization properties are well known: smoothing of discontinuities at interfaces and boundary layers in hardening materials, and finite width localization bands in softening materials. The objective is to review and propose nonlinear extensions of micromorphic and strain/damage gradient models along two lines: the first one introducing nonlinear relations between generalized stresses and strains; the second one envisaging several classes of finite deformation model formulations. The generic approach is applicable to a large class of elastoviscoplastic and damage models including anisothermal and multiphysics coupling. Two standard procedures of extension of classical constitutive laws to large strains are combined with the micromorphic approach: additive split of some Lagrangian strain measure or choice of a local objective rotating frame. Three distinct Operators are finally derived using the multiplicative decomposition of the deformation gradient. A new feature is that a free energy function depending solely on variables defined in the intermediate isoclinic configuration leads to the existence of additional kinematic hardening induced by the gradient of a scalar micromorphic variable.