Search results for "Inverse problems"
showing 9 items of 39 documents
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…
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…
Inverse problems and invisibility cloaking for FEM models and resistor networks
2013
In this paper we consider inverse problems for resistor networks and for models obtained via the finite element method (FEM) for the conductivity equation. These correspond to discrete versions of the inverse conductivity problem of Calderón. We characterize FEM models corresponding to a given triangulation of the domain that are equivalent to certain resistor networks, and apply the results to study nonuniqueness of the discrete inverse problem. It turns out that the degree of nonuniqueness for the discrete problem is larger than the one for the partial differential equation. We also study invisibility cloaking for FEM models, and show how an arbitrary body can be surrounded with a layer …
Quantitative Runge Approximation and Inverse Problems
2017
In this short note we provide a quantitative version of the classical Runge approximation property for second order elliptic operators. This relies on quantitative unique continuation results and duality arguments. We show that these estimates are essentially optimal. As a model application we provide a new proof of the result from \cite{F07}, \cite{AK12} on stability for the Calder\'on problem with local data.
Refined instability estimates for some inverse problems
2022
Many inverse problems are known to be ill-posed. The ill-posedness can be manifested by an instability estimate of exponential type, first derived by Mandache [29]. In this work, based on Mandache's idea, we refine the instability estimates for two inverse problems, including the inverse inclusion problem and the inverse scattering problem. Our aim is to derive explicitly the dependence of the instability estimates on key parameters. The first result of this work is to show how the instability depends on the depth of the hidden inclusion and the conductivity of the background medium. This work can be regarded as a counterpart of the depth-dependent and conductivity-dependent stability estim…
On some partial data Calderón type problems with mixed boundary conditions
2021
In this article we consider the simultaneous recovery of bulk and boundary potentials in (degenerate) elliptic equations modelling (degenerate) conducting media with inaccessible boundaries. This connects local and nonlocal Calderón type problems. We prove two main results on these type of problems: On the one hand, we derive simultaneous bulk and boundary Runge approximation results. Building on these, we deduce uniqueness for localized bulk and boundary potentials. On the other hand, we construct a family of CGO solutions associated with the corresponding equations. These allow us to deduce uniqueness results for arbitrary bounded, not necessarily localized bulk and boundary potentials. T…
Partial Data Problems and Unique Continuation in Scalar and Vector Field Tomography
2022
AbstractWe prove that if P(D) is some constant coefficient partial differential operator and f is a scalar field such that P(D)f vanishes in a given open set, then the integrals of f over all lines intersecting that open set determine the scalar field uniquely everywhere. This is done by proving a unique continuation property of fractional Laplacians which implies uniqueness for the partial data problem. We also apply our results to partial data problems of vector fields.
Applications of Microlocal Analysis in Inverse Problems
2020
This note reviews certain classical applications of microlocal analysis in inverse problems. The text is based on lecture notes for a postgraduate level minicourse on applications of microlocal analysis in inverse problems, given in Helsinki and Shanghai in June 2019.