The Inverse Problem of Magneto-Electroencephalography is Well-Posed: it has a Unique Solution that is Stable with Respect to Perturbations

2020 ◽  
Vol 245 (2) ◽  
pp. 121-124
Author(s):  
A. S. Demidov
Keyword(s):  
Inverse Problem ◽  
Unique Solution ◽  
Well Posed

scholarly journals The Neuroelectromagnetic Inverse Problem and the Zero Dipole Localization Error

2009 ◽  
Vol 2009 ◽  
pp. 1-11 ◽  
Author(s):  
Rolando Grave de Peralta ◽  
Olaf Hauk ◽  
Sara L. Gonzalez
Keyword(s):  
Inverse Problem ◽  
Unique Solution ◽  
A Priori ◽  
Inverse Solution ◽  
Localization Error ◽  
A Priori Information ◽  
Dipole Localization ◽  
Inverse Solutions ◽  
Priori Information ◽  
Additional Constraints

A tomography of neural sources could be constructed from EEG/MEG recordings once the neuroelectromagnetic inverse problem (NIP) is solved. Unfortunately the NIP lacks a unique solution and therefore additional constraints are needed to achieve uniqueness. Researchers are then confronted with the dilemma of choosing one solution on the basis of the advantages publicized by their authors. This study aims to help researchers to better guide their choices by clarifying what is hidden behind inverse solutions oversold by their apparently optimal properties to localize single sources. Here, we introduce an inverse solution (ANA) attaining perfect localization of single sources to illustrate how spurious sources emerge and destroy the reconstruction of simultaneously active sources. Although ANA is probably the simplest and robust alternative for data generated by a single dominant source plus noise, the main contribution of this manuscript is to show that zero localization error of single sources is a trivial and largely uninformative property unable to predict the performance of an inverse solution in presence of simultaneously active sources. We recommend as the most logical strategy for solving the NIP the incorporation of sound additional a priori information about neural generators that supplements the information contained in the data.

scholarly journals Fluid velocity profile reconstruction for non-Newtonian shear dispersive flow

2001 ◽  
Vol 5 (2) ◽  
pp. 87-104 ◽  
Author(s):  
Paul R. Shorten ◽  
David J. N. Wall
Keyword(s):  
Inverse Problem ◽  
Velocity Profile ◽  
Fluid Velocity ◽  
Average Concentration ◽  
Two Dimensional ◽  
Cross Sectional ◽  
Dispersive Flow ◽  
Ill Posed ◽  
Profile Reconstruction ◽  
Well Posed

An inverse problem associated with the mass transport of a material concentration down a pipe where the flowing non-Newtonian medium has a two-dimensional velocity profile is examined. The problem of determining the two-dimensional fluid velocity profile from temporally varying cross-sectional average concentration measurements at upstream and downstream locations is considered. The special case of a known input upstream concentration with a time zero step, and a strictly decreasing velocity profile is shown to be a well-posed problem. This inverse problem is in general ill-posed and mollification is used to obtain a well conditioned problem.

scholarly journals Logarithm of a Function, a Well-Posed Inverse Problem

2014 ◽  
Vol 04 (01) ◽  
pp. 1-5 ◽  
Author(s):  
Silvia Reyes Mora ◽  
Víctor A. Cruz Barriguete ◽  
Denisse Guzmán Aguilar
Keyword(s):  
Inverse Problem ◽  
Well Posed

scholarly journals Analysis of direct and inverse problems for a fractional elastoplasticity model

Filomat ◽  
2017 ◽  
Vol 31 (3) ◽  
pp. 699-708 ◽  
Author(s):  
Salih Tatar ◽  
Süleyman Ulusoy
Keyword(s):  
Inverse Problem ◽  
Inverse Problems ◽  
Unique Solution ◽  
Continuous Dependence ◽  
Direct Problem ◽  
Direct And Inverse Problems

This study is devoted to a nonlinear time fractional inverse coeficient problem. The unknown coeffecient depends on the gradient of the solution and belongs to a set of admissible coeffecients. First we prove that the direct problem has a unique solution. Afterwards we show the continuous dependence of the solution of the corresponding direct problem on the coeffecient, the existence of a quasi-solution of the inverse problem is obtained in the appropriate class of admissible coeffecients.

3D Mean-Stream-Line Method: A New Engineering Approach to the Inverse Problem of 3D Cascade

1989 ◽  
Author(s):  
Yifang Gong ◽  
Ruixian Cai
Keyword(s):  
Inverse Problem ◽  
Potential Function ◽  
Analytical Solutions ◽  
Function Method ◽  
Computer Code ◽  
Stream Line ◽  
Engineering Approach ◽  
Line Method ◽  
The Mean ◽  
Well Posed

The Mean-Stream-Line Method proposed by Wu and Brown (1952) for 2D cascade is developed and extended to solve the 3D inverse problem of turbomachine flow. The equations suitable to the 3D case are derived and the well-posed design conditions are discussed within the scope of the annular wall constraint of turbomachines mentioned by Cai (1983). A computer code has been programmed for this method and computation results compare well with 3D incompressible potential analytical solutions. These solutions, discovered by Cai et al. (1984), are similar to turbomachine flow. A comparison of computation results between this method and the conventional 3D potential function method is also presented. However, in solving the inverse problem, the 3D Mean-Stream-Line Method is much more simple and faster than all other methods already available. In addition, some ideas to solve direct and hybrid problems are also suggested.

A well posed inverse problem for automatic pavement parameter estimation based on GPR data

2014 ◽  
Vol 65 ◽  
pp. 22-27 ◽  
Author(s):  
Diogo B. Oliveira ◽  
Douglas A.G. Vieira ◽  
Adriano C. Lisboa ◽  
Fillipe Goulart
Keyword(s):  
Inverse Problem ◽  
Parameter Estimation ◽  
Well Posed

scholarly journals ANALYSIS OF AN INVERSE PROBLEM ARISING IN PHOTOLITHOGRAPHY

2012 ◽  
Vol 22 (05) ◽  
pp. 1150026 ◽  
Author(s):  
LUCA RONDI ◽  
FADIL SANTOSA
Keyword(s):  
Inverse Problem ◽  
Variational Formulation ◽  
Diffractive Optics ◽  
Shape Design ◽  
Two Dimensional ◽  
Convergence Properties ◽  
Approximate Problem ◽  
The Relationship ◽  
Well Posed ◽  
Dimensional Domain

We consider the inverse problem of determining an optical mask that produces a desired circuit pattern in photolithography. We set the problem as a shape design problem in which the unknown is a two-dimensional domain. The relationship between the target shape and the unknown is modeled through diffractive optics. We develop a variational formulation that is well-posed and propose an approximation that can be shown to have convergence properties. The approximate problem can serve as a foundation to numerical methods, much like the Ambrosio–Tortorelli's approximation of the Mumford–Shah functional in image processing.

scholarly journals An inverse source problem for the stochastic wave equation

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Xiaoli Feng ◽  
Meixia Zhao ◽  
Peijun Li ◽  
Xu Wang
Keyword(s):  
Inverse Problem ◽  
Wave Equation ◽  
Direct Problem ◽  
Inverse Source Problem ◽  
Time Data ◽  
Final Time ◽  
Stochastic Wave Equation ◽  
Unique Mild Solution ◽  
Well Posed ◽  
Class Of Functions

<p style='text-indent:20px;'>This paper is concerned with an inverse source problem for the stochastic wave equation driven by a fractional Brownian motion. Given the random source, the direct problem is to study the solution of the stochastic wave equation. The inverse problem is to determine the statistical properties of the source from the expectation and covariance of the final-time data. For the direct problem, it is shown to be well-posed with a unique mild solution. For the inverse problem, the uniqueness is proved for a certain class of functions and the instability is characterized. Numerical experiments are presented to illustrate the reconstructions by using a truncation-based regularization method.</p>

scholarly journals On initial inverse problem for nonlinear couple heat with Kirchhoff type

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Danh Hua Quoc Nam
Keyword(s):  
Inverse Problem ◽  
Fixed Point ◽  
Main Tool ◽  
Cauchy Data ◽  
Final Model ◽  
Kirchhoff Type ◽  
Biological Phenomena ◽  
Banach’S Fixed Point Theorem ◽  
Unique Mild Solution ◽  
Well Posed

AbstractThe main objective of the paper is to study the final model for the Kirchhoff-type parabolic system. Such type problems have many applications in physical and biological phenomena. Under some smoothness of the final Cauchy data, we prove that the problem has a unique mild solution. The main tool is Banach’s fixed point theorem. We also consider the non-well-posed problem in the Hadamard sense. Finally, we apply truncation method to regularize our problem. The paper is motivated by the work of Tuan, Nam, and Nhat [Comput. Math. Appl. 77(1):15–33, 2019].

Sign in / Sign up

Export Citation Format

Share Document