scholarly journals A Numerical Approximation Method for the Inverse Problem of the Three-Dimensional Laplace Equation

Mathematics ◽  
2019 ◽  
Vol 7 (6) ◽  
pp. 487
Author(s):  
Shangqin He ◽  
Xiufang Feng
Keyword(s):  
Inverse Problem ◽  
Laplace Equation ◽  
Numerical Approximation ◽  
Regularization Parameter ◽  
A Priori ◽  
Three Dimensional ◽  
Neumann Boundary ◽  
A Priori Bound ◽  
Mollification Method ◽  
De La Vallée Poussin

In this article, an inverse problem with regards to the Laplace equation with non-homogeneous Neumann boundary conditions in a three-dimensional case is investigated. To deal with this problem, a regularization method (mollification method) with the bivariate de la Vallée Poussin kernel is proposed. Stable estimates are obtained under a priori bound assumptions and an appropriate choice of the regularization parameter. The error estimates indicate that the solution of the approximation continuously depends on the noisy data. Two experiments are presented, in order to validate the proposed method in terms of accuracy, convergence, stability, and efficiency.

scholarly journals SOLUTION OF THE PROBLEM OF COMPUTATIONAL STABILITY AND CONSISTENCY OF SAMPLE ESTIMATES OF THE CORRELATION MATRIX OF OBSERVATIONS BY THE METHOD OF DYNAMIC REGULARIZATION

2019 ◽  
Vol 26 ◽  
pp. 47-60
Author(s):  
V. SKACHKOV ◽  
Keyword(s):  
Inverse Problem ◽  
Regularization Method ◽  
Correlation Matrix ◽  
Regularization Parameter ◽  
A Priori ◽  
Computational Cost ◽  
Spectral Composition ◽  
Adaptive Antenna ◽  
Computational Stability ◽  
A Priori Uncertainty

The problem of forming sample estimates of the correlation matrix of observations that satisfy the criterion "computational stability – consistency" is considered. The variants in which the direct and inverse asymptotic forms of the correlation matrix of observations are approximated by various types of estimates formed from a sample of a fixed volume are investigated. The consistency of computationally stable estimates of the correlation matrix for their static regularization was analyzed. The contradiction inherent in the problem of regularization of the estimates with a fixed parameter is revealed. The dynamic regularization method as an alternative approach is proposed, which is based on the uniqueness theorem for solving the inverse problem with perturbed initial data. An optimal mean-square approximation algorithm has been developed for dynamic regularization of sample estimates of the correlation matrix of observations, using the law of monotonic decrease in the regularizing parameter with increasing sample size. An optimal dynamic regularization function was obtained for sample estimates of the correlation matrix under conditions of a priori uncertainty with respect to their spectral composition. The preference of this approach to the regularization of sample estimates of the correlation matrix under conditions of a priori uncertainty is proved, which allows to exclude the domain of computational instability from solving the inverse problem and obtain its solution in real time without involving prediction data and additional computational cost for finding the optimal value of the regularization parameter. The application of the dynamic regularization method is shown for solving the problem of detecting a signal at the output of an adaptive antenna array in a nondeterministic clutter and jamming environment. The results of a computational experiment that confirm the main conclusions are presented.

scholarly journals A Regularization Method to Solve a Cauchy Problem for the Two-Dimensional Modified Helmholtz Equation

Mathematics ◽  
2019 ◽  
Vol 7 (4) ◽  
pp. 360 ◽  
Author(s):  
Shangqin He ◽  
Xiufang Feng
Keyword(s):  
Helmholtz Equation ◽  
Regularization Method ◽  
Numerical Approximation ◽  
Regularization Parameter ◽  
Two Dimensional ◽  
Approximation Solution ◽  
Modified Helmholtz Equation ◽  
Strip Domain ◽  
Ill Posed ◽  
De La Vallée Poussin

In this paper, the ill-posed problem of the two-dimensional modified Helmholtz equation is investigated in a strip domain. For obtaining a stable numerical approximation solution, a mollification regularization method with the de la Vallée Poussin kernel is proposed. An error estimate between the exact solution and approximation solution is given under suitable choices of the regularization parameter. Two numerical experiments show that our procedure is effective and stable with respect to perturbations in the data.

scholarly journals An optimal quasi solution for the Cauchy problem for Laplace equation in the framework of inverse ECG

2019 ◽  
Vol 14 (2) ◽  
pp. 204
Author(s):  
Eduardo Hernandez-Montero ◽  
Andres Fraguela-Collar ◽  
Jacques Henry
Keyword(s):  
Inverse Problem ◽  
Cauchy Problem ◽  
Laplace Equation ◽  
Boundary Data ◽  
A Priori ◽  
Normal Derivative ◽  
Cauchy Data ◽  
Dirichlet Data ◽  
Data Completion ◽  
The Cauchy Problem

The inverse ECG problem is set as a boundary data completion for the Laplace equation: at each time the potential is measured on the torso and its normal derivative is null. One aims at reconstructing the potential on the heart. A new regularization scheme is applied to obtain an optimal regularization strategy for the boundary data completion problem. We consider the ℝn+1domain Ω. The piecewise regular boundary of Ω is defined as the union∂Ω = Γ1∪ Γ0∪ Σ, where Γ1and Γ0are disjoint, regular, andn-dimensional surfaces. Cauchy boundary data is given in Γ0, and null Dirichlet data in Σ, while no data is given in Γ1. This scheme is based on two concepts: admissible output data for an ill-posed inverse problem, and the conditionally well-posed approach of an inverse problem. An admissible data is the Cauchy data in Γ0corresponding to an harmonic function inC2(Ω) ∩H1(Ω). The methodology roughly consists of first characterizing the admissible Cauchy data, then finding the minimum distance projection in theL2-norm from the measured Cauchy data to the subset of admissible data characterized by givena prioriinformation, and finally solving the Cauchy problem with the aforementioned projection instead of the original measurement.

An inverse problem for the KdV equation with Neumann boundary measured data

2018 ◽  
Vol 26 (1) ◽  
pp. 133-151 ◽  
Author(s):  
Sakthivel Kumarasamy ◽  
Alemdar Hasanov
Keyword(s):  
Inverse Problem ◽  
A Priori Estimates ◽  
Kdv Equation ◽  
A Priori ◽  
Gradient Methods ◽  
Neumann Boundary ◽  
Sobolev Norm ◽  
Minimum Problem ◽  
Existence Of A Solution ◽  
Priori Estimates

AbstractIn this paper, we consider an inverse coefficient problem for the linearized Korteweg–de Vries (KdV) equation {u_{t}+u_{xxx}+(c(x)u)_{x}=0}, with homogeneous boundary conditions {u(0,t)=u(1,t)=u_{x}(1,t)=0}, when the Neumann data{g(t):=u_{x}(0,t)}, {t\in(0,T)}, is given as an available measured output at the boundary {x=0}. The inverse problem is formulated as a minimum problem for the regularized Tikhonov functional {\mathcal{J}_{\alpha}(c)=\frac{1}{2}\|u_{x}(0,\cdot\,;c)-g\|^{2}_{L^{2}(0,T)}+% \frac{\alpha}{2}\|c^{\prime}\|^{2}_{L^{2}(0,1)}} with Sobolev norm. Based on a priori estimates for the weak and regular weak solutions of the direct and adjoint problems, it is proved that the input-output operator is compact, which shows the ill-posedness of the inverse problem. Then Fréchet differentiability of the Tikhonov functional and Lipschitz continuity of the Fréchet gradient are proved. It is shown that the last result allows us to use an important advantage of gradient methods when the functional is from the class {C^{1,1}(\mathcal{M})}. In the final part, an existence of a solution of the minimum problem for the regularized Tikhonov functional {\mathcal{J}_{\alpha}(c)} is proved.

Fast numerical method of solving 3D coefficient inverse problem for wave equation with integral data

2018 ◽  
Vol 26 (4) ◽  
pp. 477-492 ◽  
Author(s):  
Anatoly B. Bakushinsky ◽  
Alexander S. Leonov
Keyword(s):  
Inverse Problem ◽  
Integral Equation ◽  
Wave Equation ◽  
Wave Field ◽  
A Priori ◽  
Three Dimensional ◽  
Simulated Data ◽  
Three Dimensions ◽  
Unknown Coefficient ◽  
Computational Resources

Abstract An inverse coefficient problem for time-dependent wave equation in three dimensions is under consideration. We are looking for a spatially varying coefficient of this equation knowing special time integrals of the wave field in an observation domain. The inverse problem has applications to the reconstruction of the refractive index of an inhomogeneous medium, as well as to acoustic sounding, medical imaging, etc. In the article, a new linear three-dimensional Fredholm integral equation of the first kind is introduced from which it is possible to find the unknown coefficient. We present and substantiate a numerical algorithm for solving this integral equation. The algorithm does not require significant computational resources and a long solution time. It is based on the use of fast Fourier transform under some a priori assumptions about unknown coefficient and observation region of the wave field. Typical results of solving this three-dimensional inverse problem on a personal computer for simulated data demonstrate high capabilities of the proposed algorithm.

scholarly journals A modified Tikhonov regularization method based on Hermite expansion for solving the Cauchy problem of the Laplace equation

2020 ◽  
Vol 18 (1) ◽  
pp. 1685-1697
Author(s):  
Zhenyu Zhao ◽  
Lei You ◽  
Zehong Meng
Keyword(s):  
Cauchy Problem ◽  
Laplace Equation ◽  
Tikhonov Regularization ◽  
Regularization Method ◽  
Regularization Parameter ◽  
Solution Process ◽  
Tikhonov Regularization Method ◽  
Hermite Expansion ◽  
The Cauchy Problem ◽  
Ill Posed

Abstract In this paper, a Cauchy problem for the Laplace equation is considered. We develop a modified Tikhonov regularization method based on Hermite expansion to deal with the ill posed-ness of the problem. The regularization parameter is determined by a discrepancy principle. For various smoothness conditions, the solution process of the method is uniform and the convergence rate can be obtained self-adaptively. Numerical tests are also carried out to verify the effectiveness of the method.

scholarly journals Regularization of backward time-fractional parabolic equations by Sobolev-type equations

2020 ◽  
Vol 28 (5) ◽  
pp. 659-676
Author(s):  
Dinh Nho Hào ◽  
Nguyen Van Duc ◽  
Nguyen Van Thang ◽  
Nguyen Trung Thành
Keyword(s):  
Error Estimates ◽  
Parabolic Equations ◽  
Regularization Parameter ◽  
A Priori ◽  
Sobolev Type ◽  
A Posteriori ◽  
Parameter Choice ◽  
Numerical Tests ◽  
Ill Posed ◽  
Parameter Choice Rules

AbstractThe problem of determining the initial condition from noisy final observations in time-fractional parabolic equations is considered. This problem is well known to be ill-posed, and it is regularized by backward Sobolev-type equations. Error estimates of Hölder type are obtained with a priori and a posteriori regularization parameter choice rules. The proposed regularization method results in a stable noniterative numerical scheme. The theoretical error estimates are confirmed by numerical tests for one- and two-dimensional equations.

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