Search results for " inverse problem"
showing 10 items of 12 documents
Numerical methods for nonlinear inverse problems
1996
AbstractInverse problems of distributed parameter systems with applications to optimal control and identification are considered. Numerical methods and their numerical analysis for solving this kind of inverse problems are presented, main emphasis being on the estimates of the rate of convergence for various schemes. Finally, based on the given error estimates, a two-grid method and related algorithms are introduced, which can be used to solve nonlinear inverse problems effectively.
Jacobian of solutions to the conductivity equation in limited view
2022
Abstract The aim of hybrid inverse problems such as Acousto-Electric Tomography or Current Density Imaging is the reconstruction of the electrical conductivity in a domain that can only be accessed from its exterior. In the inversion procedure, the solutions to the conductivity equation play a central role. In particular, it is important that the Jacobian of the solutions is non-vanishing. In the present paper we address a two-dimensional limited view setting, where only a part of the boundary of the domain can be controlled by a non-zero Dirichlet condition, while on the remaining boundary there is a zero Dirichlet condition. For this setting, we propose sufficient conditions on the bounda…
Numerical modelling of electromagnetic sources by integral formulation
2012
Analysis of electromagnetic (EM) transients can be carried out by employing a eld approach in frequency domain, based on an appropriate integral equation. This approach is a powerful method for the analysis of EM antennas and scatterers. Recent work by the authors in modeling electromagnetic scattering in frequency domain are summarized. Thin-wire electric eld integral equation has been handled and possible application in obtaining sources localization information are discussed. Moments method (MoM) is used and time domain analysis is also carried out by discrete Fourier transform. Di erent approaches have been considered by using direct MoM formulation. Simulation results obtained both via…
Some Features of Modeling Ultrasound Propagation in Non-Destructive Control of Metal Structures Based on the Magnetostrictive Effect
2023
A method and mathematical models of direct and inverse problems of ultrasonic testing and diagnostics of complex metal structures for defects were developed and tested. A prototype of a system for magnetostrictive control of elements of the objects under study was manufactured and experimentally tested. Mathematical simulation of ultrasonic testing processes using MATLAB and the COMSOL Multiphysics software environment was carried out. The adequacy of the mathematical models was verified by the results of their comparison with real physical experiments. Information support and a methodology that implements it was developed, which ensure the functioning of the control facilities for these ob…
Learning, regularization and ill-posed inverse problems
2005
Many works have shown that strong connections relate learning from examples to regularization techniques for ill-posed inverse problems. Nevertheless by now there was no formal evidence neither that learning from examples could be seen as an inverse problem nor that theoretical results in learning theory could be independently derived using tools from regularization theory. In this paper we provide a positive answer to both questions. Indeed, considering the square loss, we translate the learning problem in the language of regularization theory and show that consistency results and optimal regularization parameter choice can be derived by the discretization of the corresponding inverse prob…
COMPUTATION OF LOCAL VOLATILITIES FROM REGULARIZED DUPIRE EQUATIONS
2005
We propose a new method to calibrate the local volatility function of an asset from observed option prices of the underlying. Our method is initialized with a preprocessing step in which the given data are smoothened using cubic splines before they are differentiated numerically. In a second step the Dupire equation is rewritten as a linear equation for a rational expression of the local volatility. This equation is solved with Tikhonov regularization, using some discrete gradient approximation as penalty term. We show that this procedure yields local volatilities which appear to be qualitatively correct.
Thresholding projection estimators in functional linear models
2008
We consider the problem of estimating the regression function in functional linear regression models by proposing a new type of projection estimators which combine dimension reduction and thresholding. The introduction of a threshold rule allows to get consistency under broad assumptions as well as minimax rates of convergence under additional regularity hypotheses. We also consider the particular case of Sobolev spaces generated by the trigonometric basis which permits to get easily mean squared error of prediction as well as estimators of the derivatives of the regression function. We prove these estimators are minimax and rates of convergence are given for some particular cases.
A New Procedure for the Evaluation of Non-Uniform Residual Stresses by the Hole Drilling Method Based on the Newton-Raphson Technique
2010
The hole drilling method is one of the most used semi-destructive techniques for the analysis of residual stresses in mechanical components. The non-uniform stresses are evaluated by solving an integral equation in which the strains relieved by drilling a hole are introduced. In this paper a new calculation procedure, based on the Newton-Raphson method for the determination of zeroes of functions, is presented. This technique allows the user to introduce complex and effective forms of stress functions for the solution of the problem. All the relationships needed for the evaluation of the stresses are obtained in explicit form, eliminating the need to use additional mathematical tools. The t…
A meshfree approach for brain activity source modeling
2015
Weak electrical currents in the brain flow as a consequence of acquisition, processing and transmission of information by neurons, giving raise to electric and magnetic fields, which are representable by means of quasi-stationary approximation of the Maxwell’s equations. Measurements of electric scalar potential differences at the scalp and magnetic fields near the head constitute the input data for, respectively, electroencephalography (EEG) and magnetoencepharography (MEG), which allow for reconstructing the cerebral electrical currents and thus investigating the neuronal activity in the human brain in a non-invasive way. This is a typical erectromagnetic inverse problem, since measuremen…
Inverse problem for tripotential measures in the study of buried cavities
1996
This paper presents a solution to the inverse electrical problem for the interpretation of apparent resistivity anomalies due to empty buried cavities of quasi-spherical shape when tripotential measures are carried out. The anomalies of the apparent resistivities ra,rb andrg,and the composed resistivitiesrmand rt were previously calculated for a sufficient class of spherical models of resistivity anomalies. Then, for the whole class of models, some functionals of spatial distribution of the apparent and composed resistivity were identified and analyzed. They represent the average characteristics of the anomalies and, depending in a simple way on the fundamental parameters of the sources of …