ON THE CONDITIONS FOR UNIQUENESS AND EXISTENCE OF THE SOLUTION TO AN ACOUSTIC INVERSE PROBLEM I: THEORY

1993 ◽  
Vol 01 (02) ◽  
pp. 229-247 ◽  
Author(s):  
R. P. GILBERT ◽  
ZHONGYAN LIN
Keyword(s):  
Inverse Problem ◽  
Unit Disk ◽  
Numerical Experiments ◽  
Sufficiency Condition ◽  
Data Pair ◽  
Uniqueness And Existence

This paper which is Part I of a sequence deals with the problem of determining a radially dependent coefficient n (r) in the equation ∆ u − n2 (r) u = 0, in the unit disk Ω from the Dirichlet–Neumann data pair [Formula: see text]. We prove that the sufficiency condition for uniqueness established in Ref. 2 is, in some instances, also a necessity for uniqueness. We also discuss the solvability of this inverse problem. In Part II numerical experiments will be presented which illustrate the theory developed here.

scholarly journals Time- or Space-Dependent Coefficient Recovery in Parabolic Partial Differential Equation for Sensor Array in the Biological Computing

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Guanglu Zhou ◽  
Boying Wu ◽  
Wen Ji ◽  
Seungmin Rho
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Inverse Problem ◽  
Iteration Method ◽  
Numerical Experiments ◽  
Sensor Array ◽  
Variational Iteration Method ◽  
Parabolic Partial Differential Equation ◽  
Partial Differential ◽  
Variational Iteration

This study presents numerical schemes for solving a parabolic partial differential equation with a time- or space-dependent coefficient subject to an extra measurement. Through the extra measurement, the inverse problem is transformed into an equivalent nonlinear equation which is much simpler to handle. By the variational iteration method, we obtain the exact solution and the unknown coefficients. The results of numerical experiments and stable experiments imply that the variational iteration method is very suitable to solve these inverse problems.

Regularization by Denoising for simultaneous source separation

Geophysics ◽  
2021 ◽  
pp. 1-56
Author(s):  
Breno Bahia ◽  
Rongzhi Lin ◽  
Mauricio Sacchi
Keyword(s):  
Inverse Problem ◽  
Inverse Problems ◽  
Data Processing ◽  
Spectrum Analysis ◽  
Numerical Experiments ◽  
Source Separation ◽  
Singular Spectrum Analysis ◽  
Source Data ◽  
Blended Data ◽  
Simultaneous Source

Denoisers can help solve inverse problems via a recently proposed framework known as regularization by denoising (RED). The RED approach defines the regularization term of the inverse problem via explicit denoising engines. Simultaneous source separation techniques, being themselves a combination of inversion and denoising methods, provide a formidable field to explore RED. We investigate the applicability of RED to simultaneous-source data processing and introduce a deblending algorithm named REDeblending (RDB). The formulation permits developing deblending algorithms where the user can select any denoising engine that satisfies RED conditions. Two popular denoisers are tested, but the method is not limited to them: frequency-wavenumber thresholding and singular spectrum analysis. We offer numerical blended data examples to showcase the performance of RDB via numerical experiments.

Inverse problem for the Hill equation. Numerical experiments

1992 ◽  
Vol 61 (6) ◽  
pp. 2442-2445
Author(s):  
O. A. Zadvornov ◽  
L. D. �skin
Keyword(s):  
Inverse Problem ◽  
Numerical Experiments ◽  
Hill Equation ◽  
The Hill

Numerical Modeling of Two Phase Flow under Centrifugation

2012 ◽  
Vol 326-328 ◽  
pp. 221-226
Author(s):  
Jozef Kačur ◽  
Benny Malengier ◽  
Pavol Kišon
Keyword(s):  
Inverse Problem ◽  
Numerical Modeling ◽  
Numerical Experiments ◽  
Two Phase Flow ◽  
Phase Flow ◽  
Two Phase ◽  
Saturated Sample ◽  
Dae Systems ◽  
Capillary Pressure Curves ◽  
Wetting Liquid

Numerical modeling of two-phase flow under centrifugation is presented in 1D.A new method is analysed to determine capillary-pressure curves. This method is based onmodeling the interface between the zone containing only wetting liquid and the zone containingwetting and non wetting liquids. This interface appears when into a fully saturated sample withwetting liquid we inject a non-wetting liquid. By means of this interface an efficient and correctnumerical approximation is created based upon the solution of ODE and DAE systems. Bothliquids are assumed to be immiscible and incompressible. This method is a good candidate tobe used in solution of inverse problem. Some numerical experiments are presented.

scholarly journals Quasiconformal extensions for some geometric subclasses of univalent functions

1984 ◽  
Vol 7 (1) ◽  
pp. 187-195 ◽  
Author(s):  
Johnny E. Brown
Keyword(s):  
Recent Result ◽  
Unit Disk ◽  
Half Plane ◽  
Complex Plane ◽  
Univalent Functions ◽  
Sufficient Conditions ◽  
Sufficiency Condition ◽  
Extended Complex ◽  
Extended Complex Plane

LetSdenote the set of all functionsfwhich are analytic and univalent in the unit diskDnormalized so thatf(z)=z+a2z2+…. LetS∗andCbe those functionsfinSfor whichf(D)is starlike and convex, respectively. For0≤k<1, letSkdenote the subclass of functions inSwhich admit(1+k)/(1−k)-quasiconformal extensions to the extended complex plane. Sufficient conditions are given so that a functionfbelongs toSk⋂S∗orSk⋂C. Functions whose derivatives lie in a half-plane are also considered and a Noshiro-Warschawski-Wolff type sufficiency condition is given to determine which of these functions belong toSk. From the main results several other sufficient conditions are deduced which include a generalization of a recent result of Fait, Krzyz and Zygmunt.

scholarly journals Inverse Problem of Recovering the Initial Condition for a Nonlinear Equation of the Reaction–Diffusion–Advection Type by Data Given on the Position of a Reaction Front with a Time Delay

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 342
Author(s):  
Dmitry Lukyanenko ◽  
Tatyana Yeleskina ◽  
Igor Prigorniy ◽  
Temur Isaev ◽  
Andrey Borzunov ◽  
...  
Keyword(s):  
Inverse Problem ◽  
Reaction Front ◽  
Numerical Experiments ◽  
Reaction Diffusion ◽  
Singularly Perturbed ◽  
Initial Moment ◽  
Advection Equation ◽  
Initial Condition ◽  
Additional Information ◽  
The Cost

In this paper, approaches to the numerical recovering of the initial condition in the inverse problem for a nonlinear singularly perturbed reaction–diffusion–advection equation are considered. The feature of the formulation of the inverse problem is the use of additional information about the value of the solution of the equation at the known position of a reaction front, measured experimentally with a delay relative to the initial moment of time. In this case, for the numerical solution of the inverse problem, the gradient method of minimizing the cost functional is applied. In the case when only the position of the reaction front is known, the method of deep machine learning is applied. Numerical experiments demonstrated the possibility of solving such kinds of considered inverse problems.

scholarly journals Recovery of dipolar sources and stability estimates

Filomat ◽  
2019 ◽  
Vol 33 (13) ◽  
pp. 4095-4114
Author(s):  
Ridha Mdimagh
Keyword(s):  
Heat Flux ◽  
Inverse Problem ◽  
Heat Equation ◽  
Numerical Experiments ◽  
Time Dependent ◽  
Stability Estimates ◽  
Lateral Boundary ◽  
Lipschitz Stability ◽  
Spatial Locations ◽  
Stability Results

The inverse problem of identifying dipolar sources with time-dependent moments, located in a bounded domain, via the heat equation is investigated, by applying a heat flux, and from a single lateral boundary measurement of temperature. An uniqueness, and local Lipschitz stability results for this inverse problem are established which are the main contributions of this work. A non-iterative algebraic algorithm based on the reciprocity gap concept is proposed, which permits to determine the number, the spatial locations, and the time-dependent moments of the dipolar sources, Some numerical experiments are given in order to test the efficiency and the robustness of this method.

scholarly journals Uniqueness and numerical reconstruction for inverse problems dealing with interval size search

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Jone Apraiz ◽  
Jin Cheng ◽  
Anna Doubova ◽  
Enrique Fernández-Cara ◽  
Masahiro Yamamoto
Keyword(s):  
Inverse Problem ◽  
Wave Equation ◽  
Inverse Problems ◽  
Heat Equation ◽  
Numerical Experiments ◽  
Spatial Dimension ◽  
Numerical Reconstruction ◽  
Spatial Interval ◽  
Boundary Information ◽  
Size Estimates

<p style='text-indent:20px;'>We consider a heat equation and a wave equation in one spatial dimension. This article deals with the inverse problem of determining the size of the spatial interval from some extra boundary information on the solution. Under several different circumstances, we prove uniqueness, non-uniqueness and some size estimates. Moreover, we numerically solve the inverse problems and compute accurate approximations of the size. This is illustrated with several satisfactory numerical experiments.</p>

AN ACOUSTIC INVERSE PROBLEM: NUMERICAL EXPERIMENT

1995 ◽  
Vol 03 (03) ◽  
pp. 229-240 ◽  
Author(s):  
R. P. GILBERT ◽  
ZHONGYAN LIN
Keyword(s):  
Inverse Problem ◽  
Integral Equation ◽  
Numerical Experiment ◽  
Fredholm Integral Equation ◽  
Numerical Experiments ◽  
Goursat Problem ◽  
Index Of Refraction ◽  
Numerical Treatment ◽  
Moments Problem

As a sequel to Refs. 1 and 2, this paper gives a numerical treatment of the inverse problem associated with the determination of the index of refraction. We show that the problem can be solved in two steps. First we must recover a function from its moments, problem (IM), which we may reformulate as a Fredholm integral equation of the first kind, problem (IE). Second we solve an inverse Goursat problem, (IG). Numerical schemes for both steps are given along with the results of some numerical experiments.

scholarly journals Solution of the Stokes Problem as an Inverse Problem

2002 ◽  
Vol 2 (3) ◽  
pp. 213-232
Author(s):  
V. Agoshkov ◽  
C. Bardos ◽  
S. Buleev
Keyword(s):  
Optimal Control ◽  
Inverse Problem ◽  
Numerical Experiments ◽  
Original Problem ◽  
Stokes Equations ◽  
Stokes Problem

AbstractIn this study, we examine one of the approaches to the investigation and solution of the Stokes equations: an original problem is regarded as an inverse problem and then reduced to a problem of optimal control. Iteration algorithms are suggested and results of numerical experiments are presented.

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