scholarly journals Simultaneous Reconstruction of Coefficients and Source Parameters in Elliptic Systems Modelled with Many Boundary Value Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-15 ◽  
Author(s):  
Nilson C. Roberty
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Source Parameters ◽  
Elliptic Systems ◽  
Iterative Solution ◽  
Cauchy Data ◽  
Unknown Parameters ◽  
Simultaneous Reconstruction ◽  
Quadratic Finite Elements ◽  
Interior Domain

In this work the inverse problem for determination of unknown parameters related to both intensities and support of sources and materials coefficients in second-order elliptic equations models is posed with over specification of data on the boundary. A discrepancy function based on difference of two mixed problems formulated by splitting the Cauchy data is introduced. This function controls the measured difference between the two solutions for the same set of Cauchy data. Parameters can be determined by minimization of this function under guess values. The concept of Calderón projector gap is introduced as a tool for checking the consistency of Cauchy data. Numerical implementations based on quadratic finite elements are presented in a two dimensional square (−1, +1) × (−1, +1) model with unknown source, conductivity, and absorption supported by an also unknown characteristic square shape interior domain. Since this minimization involves the iterative solution of a huge number of direct boundary value problems, the adoption of a non-differentiable minimization algorithm is recommended and the Nelder-Mead simplex method is used to search for optimal parameters.

scholarly journals On the Identication of Coecient and Source Parameters in Elliptic Systems Modelled with Many Boundary Values Problems

2020 ◽  
pp. 73-89
Author(s):  
Nilson C. Roberty
Keyword(s):  
Mixed Boundary ◽  
Constitutive Relations ◽  
Source Parameters ◽  
Elliptic Systems ◽  
Cauchy Data ◽  
Lipschitz Boundary ◽  
Boundary Values ◽  
Unknown Parameters ◽  
Second Order Systems ◽  
Determination Of Parameters

The inverse problem for determination of parameters related to the support and/or functions describing the intensity of coefficient and sources in models based strongly elliptic second order systems is posed with Cauchy data over specification at boundary. This stablish a set of various boundary value problems associated with the same group of unknown parameters. A Lipschitz boundary dissection is used for decomposing each Cauchy data into pairs of complementary mixed boundary values problems. The concept of Calderon projector is introduced as a tool to check the consistency of the Cauchy data and to demonstrate the equivalence of these two problems. This lets you define a discrepancy function to measure the distance between the solutions of problems obtained by dissecting Lipschitz Cauchy data. This discrepancy appears as a consequence of inadequate parameters values in the constitutive relations. For Cauchy noisy data, the difference between these solutions would be small if the parameters used in the solution are correct. The methodology we propose explores concepts as Lipschitz Boundary Dissection, Complementary Mixed Problems with trial parameters and Internal Discrepancy fields. Differentiable and non-differentiable optimizations algorithms can then be used in the reconstruction of these parameters simultaneously. Numerical experiments are presented.

scholarly journals The Unique Positive Solution for Singular Hadamard Fractional Boundary Value Problems

2019 ◽  
Vol 2019 ◽  
pp. 1-6 ◽  
Author(s):  
Jinxiu Mao ◽  
Zengqin Zhao ◽  
Chenguang Wang
Keyword(s):  
Exact Solution ◽  
Convergence Rate ◽  
Boundary Value Problems ◽  
Positive Solution ◽  
Existence And Uniqueness ◽  
Boundary Value ◽  
Approximation Error ◽  
Iterative Solution ◽  
Unique Positive Solution ◽  
Fractional Boundary Value Problems

In this paper, we investigate singular Hadamard fractional boundary value problems. The existence and uniqueness of the exact iterative solution are established only by using an iterative algorithm. The iterative sequences have been proved to converge uniformly to the exact solution, and estimation of the approximation error and the convergence rate have also been derived.

scholarly journals The Sharpiro–Lopatinskij Condition for Elliptic Boundary Value Problems

2006 ◽  
Vol 9 ◽  
pp. 287-329 ◽  
Author(s):  
Katsiaryna Krupchyk ◽  
Jukka Tuomela
Keyword(s):  
Boundary Value Problems ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Elliptic Systems ◽  
Formal Theory ◽  
Elliptic Boundary Value Problems ◽  
Constructive Method ◽  
Elliptic Boundary Value ◽  
The Given ◽  
Well Posed

AbstractElliptic boundary value problems are well posed in suitable Sobolev spaces, if the boundary conditions satisfy the Shapiro–Lopatinskij condition. We propose here a criterion (which also covers over-determined elliptic systems) for checking this condition. We present a constructive method for computing the compatibility operator for the given boundary value problem operator, which is also necessary when checking the criterion. In the case of two independent variables we give a formulation of the criterion for the Shapiro–Lopatinskij condition which can be checked in a finite number of steps. Our approach is based on formal theory of PDEs, and we use constructive module theory and polynomial factorisation in our test. Actual computations were carried out with computer algebra systems Singular and MuPad.

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