Introducing regularization into the virtual fields method (VFM) to identify nonhomogeneous elastic property distributions

Extension of the virtual fields method to elasto-plastic material identification with cyclic loads and kinematic hardening

In this paper, we develop a non‐iterative way to solve biharmonic boundary value problems by using a radial basis meshless method. This is an original application of meshless method to solving inverse problems without any iteration, since traditional numerical methods for inverse boundary value problems mainly are iterative and hence very time‐consuming. Numerical examples are presented for inverse biharmonic boundary value problems and corresponding direct problems, since solving direct problems is a preliminary step for inverse problems. All our examples of direct and inverse problems are solved within seconds in CPU time on a standard PC, which makes our proposed technique a great potential candidate for wide‐spread applications to other inverse problems. Copyright © 2004 John Wiley & Sons, Ltd.

The Newsletter is a key element in further enhancing the value of the journal to the inverse problems community. So why not be a part of this exciting forum by sending to our Bristol office material suitable for inclusion under any of the categories mentioned above. Your contributions will be very welcome. Book review Introduction to Inverse Problems in Imaging M Bertero and P Boccacci 1998 Bristol: Institute of Physics 362 pp ISBN 0-7503-0435-9 (pbk) £25.00, $49.00 This book shows that several problems in imaging are in fact linear inverse problems. In general inverse problems are ill-posed in the sense that small perturbations in the (measured) data have a significant influence on the output. Consequently, numerical methods developed in a general setting of inverse problems which cure the ill-posedness (such methods are called regularization methods) can be applied to solve problems in imaging. This is an excellent textbook on the principle of linear inverse problems, methods of their numerical solution and practical applications in imaging. It introduces basic ideas and methods for the solution of inverse and ill-posed problems while avoiding the mathematics of functional treatment of operator equations. This goal is achieved by a bottom-up presentation: the authors focus on the two problems of image deconvolution and tomography (bottom) and develop regularization theory specifically for these two problems. Most other books which are concerned with the numerical solution of inverse problems present a top-down approach. That is, a general regularization theory (top) is developed which is then applied to particular problems. The distinguished presentation makes the book very useful to students in applied mathematics, practitioners in engineering science and image processing. This book has the best prerequisites to establish a close link between the mathematical fields of imaging and inverse problems. Mainly the book focuses on two topics: image deconvolution and linear imaging systems. 1. In image deconvolution the goal is to restore a space invariant blurred image. This part of the book contains mathematical tools in image deconvolution, examples of space invariant imaging systems, a comparison of image deconvolution techniques and low-pass filtering techniques, the analysis of the ill-posedness of the deconvolution problem, regularization methods for deconvolution such as constrained least squares regularization, Tikhonov regularization, iterative regularization and statistical methods. Since the authors focus on particular problems they are able to analyse various methods for deconvolution on a very concrete level. As an example I would like to mention that the authors analyse Fourier methods for deconvolution. Thus they provide an optimal starting point to the theory of inverse problems for people working in the area of signal and image processing. 2. The second part of the book considers linear imaging systems that are not of a convolution type such as tomography problems, diffraction problems and inverse scattering problems. Efficient numerical methods for the solution of this class of problems such as singular value decomposition, Tikhonov regularization methods and Fourier-based methods are presented. O Scherzer Universität Linz

This report demonstrates the use of a genetic algorithm search in the solution of an inverse problem. The genetic algorithm is used to solve the one-dimensional inverse heat conduction problem using numerical data generated by solution of the corresponding direct problem. Both "pure" and noisy data are considered. If used with regularization, the method is shown to yield reasonable results even in the face of noisy data. Although only a simple one-dimensional case is presented to demonstrate the method, the genetic algorithm approach is expected to excel in the solution of complicated multidimensional inverse problems.

Reconstruction of contaminant release history is crucial for subsurface remediation actions. This task amounts to a high-dimensional inverse problem, whose solution requires multiple forward solves of contaminant transport equations. It also must cope with both sparse observations of solute concentration and subsurface heterogeneity. The computational burden of solving this inverse problem can be reduced by deploying a surrogate model, e.g., neural networks (NNs), which provides a low-cost approximation of its expensive physics-based counterpart. However, to construct such NNs, a large amount of high-fidelity forward runs may be required to provide training data, and these computations might be as cost-prohibitive as the solution of the inverse problem. To address this issue, we generate multi-fidelity data by running simulations of the forward transport model on fine and coarse meshes. The resulting high- and low-fidelity temporal snapshots of solute concentration are subsequently used, with a Transfer Learning technique, to train a Convolutional NN to identify the initial contaminant source location. The training is divided into three phases. In the initial phase, the training exclusively employs low-fidelity data. In the subsequent two steps, the learning phase for the network is finalized with only a relatively small number of high-fidelity data. The obtained results demonstrate that the transfer-learning-based surrogate model is a promising tool to reduce the computational cost as well as to obtain accurate solutions of high dimensional inverse problems.

In elastic modulus imaging or quantitative elastography, tissue stiffness is inferred from an (ultrasonically) measured displacement field. Doing so requires the solution of an elastic inverse problem. We present several exact solutions of the compressible inverse elasticity problem: Given the (possibly transient) dispalcement field measured everywhere in an isotropic, compressible, linear elastic solid, and given density ρ, determine the Lamé parameters λ(x) and μ(x). The cases we treat are: (a) for μ(x) known a priori, find λ; (b) for λ(x) known a priori, find μ; (c) neither λ(x) nor μ is known a priori, but Poisson’s ratio ν is known; (d) neither λ(x) nor μ nor ν is known a priori. We present several example applications including 2D and 3D problems, quasistatic and time dependent displacement fields. Many of the results here are drawn from P.E. Barbone and A.A. Oberai [‘‘Elastic Modulus Imaging: Some exact solutions of the compressible elastography inverse problem,’’ Phys. Med. Biol. (in press)].

This study deals with an experimental methodology developed in order to identify the elastic properties of super- conducting ring-shaped windings, constituents of the main coil winding of a magnetic resonance imaging magnet (MRI). Mechanical tensile tests were conducted on real scale specimens associated to an optical full-field displacement measurement technique (stereo image correlation). Strain fields were then obtained from the measured displacement fields by numerical dif- ferentiation. Finally, the four in-plane orthotropic stiffnesses of the windings were determined using the Virtual Fields Method (VFM). Experimental set-up also allows detecting a possible occurrence of a delamination thanks to in-plane displacement fields.

This work is devoted to the development and implementation of a two-step method for solving the vector three-dimensional inverse diffraction problem on an inhomogeneous dielectric scatterer having the form of a hemisphere characterized by piecewise constant permittivity. The original boundary value problem for Maxwell’s equations is reduced to a system of integro-differential equations. An integral formulation of the vector inverse diffraction problem is proposed and the uniqueness of the solution of the first-kind integro-differential equation in special function classes is established. A two-step method for solving the vector inverse diffraction problem on the hemisphere is developed. Unlike traditional approaches, the two-step method for solving the inverse problem is non-iterative and does not require knowledge of the exact initial approximation. Consequently, there are no issues related to the convergence of the numerical method. The results of calculations of approximate solutions to the inverse problem are presented. It is shown that the two-step method is an efficient approach to solving vector problems in near-field tomography.

The resolution of many large-scale inverse problems using MCMC methods\nrequires a step of drawing samples from a high dimensional Gaussian\ndistribution. While direct Gaussian sampling techniques, such as those based on\nCholesky factorization, induce an excessive numerical complexity and memory\nrequirement, sequential coordinate sampling methods present a low rate of\nconvergence. Based on the reversible jump Markov chain framework, this paper\nproposes an efficient Gaussian sampling algorithm having a reduced computation\ncost and memory usage. The main feature of the algorithm is to perform an\napproximate resolution of a linear system with a truncation level adjusted\nusing a self-tuning adaptive scheme allowing to achieve the minimal computation\ncost. The connection between this algorithm and some existing strategies is\ndiscussed and its efficiency is illustrated on a linear inverse problem of\nimage resolution enhancement.\n

The paper presents consideration of the solution of the nonlinear inverse problem of gravity exploration in the grid class of field sources if there are anomalies in the section that have positive and negative effective densities. In this case, practical equivalence is very strong and the reliability of the results of quantitative interpretation decreases. Based on the assembly method the author consideres the possibility of decomposing the inverse problem into two separate computational procedures. It is implemented using pre-separation of the gravitational field into components caused by objects with different effective densities. The author used a system of equivalent sources that were successively plunged to a depth approximating the interpreted field. Experiments showed that with a certain depth of sources, it is possible to rather accurately restore the «positive» and «negative» components. For each of them, the solution of the inverse problem is separately carried out, then the obtained results are synthesized, i.e. the construction of a unified geodensity model, which makes it possible to provide complete automation of additive interpretation technologies. The paper includes model examples illustrating the proposed approach.

PreviousNext No AccessSEG Technical Program Expanded Abstracts 1995Two‐dimensional regularized inversion of magnetotelluric dataAuthors: Michael S. ZhdanovPatricia P. de LugaoOleg N. PortniaguineMichael S. ZhdanovUniversity of Utah, Patricia P. de LugaoUniversity of Utah, and Oleg N. PortniaguineUniversity of Utahhttps://doi.org/10.1190/1.1887564 SectionsAboutPDF/ePub ToolsAdd to favoritesDownload CitationsTrack CitationsPermissions ShareFacebookTwitterLinked InRedditEmail Permalink: https://doi.org/10.1190/1.1887564FiguresReferencesRelatedDetailsCited ByA Selection Algorithm of Survey Lines and the Interactive Mode of the Sounding Curve7 January 2015 SEG Technical Program Expanded Abstracts 1995ISSN (print):1052-3812 ISSN (online):1949-4645Copyright: 1995 Pages: 1566 publication data© 1995 Copyright © 1995 Society of Exploration GeophysicistsPublisher:Society of Exploration Geophysicists HistoryPublished: 10 Feb 2005 CITATION INFORMATION Michael S. Zhdanov, Patricia P. de Lugao, and Oleg N. Portniaguine, (1995), "Two‐dimensional regularized inversion of magnetotelluric data," SEG Technical Program Expanded Abstracts : 803-806. https://doi.org/10.1190/1.1887564 Plain-Language Summary PDF DownloadLoading ...

The solution of direct and inverse transient heat conduction problems with layered materials using exponential basis functions

The numerical solution of inverse source problem for time fractional diffusion equation was studied: the time fractional derivative was is discretized by L1 algorithm, and the space derivative was approximated by central difference. The L1-finite difference scheme of the inverse source problem was constructed with the accuracy of O(τ 2–α + h 2). By adding a random disturbance to the final value and combining with the discrete scheme of L1-finite difference method, the algorithm flow was given. The effectiveness of L1-finite difference method for solve the inverse problem of time fractional diffusion equation is verified by numerical examples. The method provides an effective reference scheme for the numerical solution of the inverse problem.

ForewordThis volume of Journal of Physics: Conference Series is dedicated to the scientific contributions presented during the 6th International Workshop on New Computational Methods for Inverse Problems, NCMIP 2016 (http://complement.farman.ens-cachan.fr/NCMIP 2016.html). This workshop took place at Ecole Normale Supérieure de Cachan, on May 20, 2016. The prior editions of NCMIP also took place in Cachan, France, firstly within the scope of ValueTools Conference, in May 2011, and secondly at the initiative of Institut Farman, in May 2012, May 2013, May 2014 and May 2015.The New Computational Methods for Inverse Problems (NCMIP) workshop focused on recent advances in the resolution of inverse problems. Indeed, inverse problems appear in numerous scientific areas such as geophysics, biological and medical imaging, material and structure characterization, electrical, mechanical and civil engineering, and finances. The resolution of inverse problems consists in estimating the parameters of the observed system or structure from data collected by an instrumental sensing or imaging device. Its success firstly requires the collection of relevant observation data. It also requires accurate models describing the physical interactions between the instrumental device and the observed system, as well as the intrinsic properties of the solution itself. Finally, it requires the design of robust, accurate and efficient inversion algorithms. Advanced sensor arrays and imaging devices provide high rate and high volume data; in this context, the efficient resolution of the inverse problem requires the joint development of new models and inversion methods, taking computational and implementation aspects into account. During this one- day workshop, researchers had the opportunity to bring to light and share new techniques and results in the field of inverse problems.The topics of the workshop were: algorithms and computational aspects of inversion, Bayesian estimation, Kernel methods, learning methods, convex optimization, free discontinuity problems, metamodels, proper orthogonal decomposition, reduced models for the inversion, non-linear inverse scattering, image reconstruction and restoration, and applications (bio-medical imaging, nondestructive evaluation...).NCMIP 2016 was a one-day workshop held in May 2016 which attracted around seventy attendees. Each of the submitted papers has been reviewed by two reviewers. There have been eleven accepted papers. In addition, three international speakers were invited to present a longer talk.The workshop was supported by Institut Farman (ENS Cachan, CNRS) and endorsed by the following French research networks: GDR ISIS, GDR MIA, GDR MOA, GDR Ondes. The program committee acknowledges the following research laboratories: CMLA, LMT, LURPA and SATIE.Eric Vourc'h and Thomas RodetWorkshop co-chairs:Eric Vourc'h, SATIE laboratory, Ecole Normale Supérieure de Cachan, CNRS, France Thomas Rodet, SATIE laboratory, Ecole Normale Supérieure de Cachan, CNRS, FranceTechnical program committee:Alexandre Baussard, ENSTA Bretagne, Lab-STICC, France Marc Bonnet, ENSTA, ParisTech, FranceLaure Blanc-Féraud, I3S laboratory and INRIA Nice Sophia-Antipolis, FranceAntonin Chambolle, CMAP, Ecole Polytechnique, CNRS, FranceOliver Dorn, School of Mathematics, University of Manchester, UKCécile Durieu, SATIE, ENS Cachan, CNRS, FranceLaurent Fribourg, LSV, ENS Cachan, CNRS, FranceJerôme Idier, IRCCyN Laboratory, Ecole Centrale de Nantes, FrancePierre-Yves Joubert, IEF, Paris-Sud University, CNRS, FranceMarc Lambert, Geeps Laboratory, CNRS, CentraleSupElec, Paris-Sud University, France Giacomo Oliveri, eledia research center/eledia@L2S group, University of Trento, Italy Dominique Lesselier, L2S Laboratory, CNRS, CentraleSupElec, Paris-Sud University, France Matteo Pastorino, DIBE, University of Genoa, ItalyGabriel Peyré, Ceremade laboratory, University of Paris Dauphine, France Anthony Quinn, Trinity College Dublin, Ireland Christian Rey, Safran Tech, FranceMarco Salucci, eledia research center/eledia@L2S group, University of Trento, Italy Jean-Yves Tourneret, IRIT laboratory, ENSEEIHT (Toulouse), FranceLocal chairs:Virginie Pauchont, CMLA Laboratory, ENS Cachan, Paris-Sud University, CNRS, France Sophie Abriet, SATIE Laboratory, ENS Cachan, Paris-Sud University, CNRS, France Béatrice Bacquet, SATIE Laboratory, ENS Cachan, Paris-Sud University, CNRS, France Aurore Gracia, SATIE Laboratory, ENS Cachan, Paris-Sud University, CNRS, FranceInvited speakersLaure Blanc-Féraud, I3S Laboratory, CNRS UMR 6070, Sophia Antipolis, France Michel Defrise, Department of Nuclear Medicine, Vrije Universiteit Brussel, Belgium Houssem Haddar, INRIA, CMAP, Ecole Polytechnique, Palaiseau, France.

Geophysical Inverse Theory and Regularization Problems