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15 pages, 586 KiB

Open AccessArticle

Convergence Rate of Runge-Kutta-Type Regularization for Nonlinear Ill-Posed Problems under Logarithmic Source Condition

by Pornsarp Pornsawad, Elena Resmerita and Christine Böckmann

Mathematics 2021, 9(9), 1042; https://doi.org/10.3390/math9091042 - 04 May 2021

Cited by 1 | Viewed by 1712

We prove the logarithmic convergence rate of the families of usual and modified iterative Runge-Kutta methods for nonlinear ill-posed problems between Hilbert spaces under the logarithmic source condition, and numerically verify the obtained results. The iterative regularization is terminated by the a posteriori [...] Read more.

We prove the logarithmic convergence rate of the families of usual and modified iterative Runge-Kutta methods for nonlinear ill-posed problems between Hilbert spaces under the logarithmic source condition, and numerically verify the obtained results. The iterative regularization is terminated by the a posteriori discrepancy principle. Full article

(This article belongs to the Special Issue Numerical Analysis: Inverse Problems – Theory and Applications)

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15 pages, 8559 KiB

Open AccessArticle

Method of Constructing a Nonlinear Approximating Scheme of a Complex Signal: Application Pattern Recognition

by Oksana Mandrikova, Bogdana Mandrikova and Anastasia Rodomanskay

Mathematics 2021, 9(7), 737; https://doi.org/10.3390/math9070737 - 29 Mar 2021

Cited by 11 | Viewed by 1727

Abstract

A method for identification of structures of a complex signal and noise suppression based on nonlinear approximating schemes is proposed. When we do not know the probability distribution of a signal, the problem of identifying its structures can be solved by constructing adaptive [...] Read more.

A method for identification of structures of a complex signal and noise suppression based on nonlinear approximating schemes is proposed. When we do not know the probability distribution of a signal, the problem of identifying its structures can be solved by constructing adaptive approximating schemes in an orthonormal basis. The mapping is constructed by applying threshold functions, the parameters of which for noisy data are estimated to minimize the risk. In the absence of a priori information about the useful signal and the presence of a high noise level, the use of the optimal threshold is ineffective. The paper introduces an adaptive threshold, which is assessed on the basis of the posterior risk. Application of the method to natural data has confirmed its effectiveness. Full article

(This article belongs to the Special Issue Numerical Analysis: Inverse Problems – Theory and Applications)

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14 pages, 354 KiB

Open AccessArticle

Non-Iterative Solution Methods for Cauchy Problems for Laplace and Helmholtz Equation in Annulus Domain

by Mohsen Tadi and Miloje Radenkovic

Mathematics 2021, 9(3), 268; https://doi.org/10.3390/math9030268 - 29 Jan 2021

Cited by 3 | Viewed by 1461

Abstract

This note is concerned with two new methods for the solution of a Cauchy problem. The first method is based on homotopy-perturbation approach which leads to solving a series of well-posed boundary value problems. No regularization is needed in this method. Laplace and [...] Read more.

This note is concerned with two new methods for the solution of a Cauchy problem. The first method is based on homotopy-perturbation approach which leads to solving a series of well-posed boundary value problems. No regularization is needed in this method. Laplace and Helmholtz equations are considered in an annular region. It is also proved that the homotopy solution for the Laplace operator converges to the actual exact solution. The second method is also non-iterative. It is based on the application of the Green’s second identity which leads to a moment problem for the unknown boundary condition. Tikhonov regularization is used to obtain a stable and close approximation of the missing boundary condition. A number of examples are used to study the applicability of the methods with the presence of noise. Full article

(This article belongs to the Special Issue Numerical Analysis: Inverse Problems – Theory and Applications)

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22 pages, 1305 KiB

Open AccessArticle

Convergence Rate of the Modified Landweber Method for Solving Inverse Potential Problems

by Pornsarp Pornsawad, Parada Sungcharoen and Christine Böckmann

Mathematics 2020, 8(4), 608; https://doi.org/10.3390/math8040608 - 16 Apr 2020

Cited by 2 | Viewed by 1933

Abstract

In this paper, we present the convergence rate analysis of the modified Landweber method under logarithmic source condition for nonlinear ill-posed problems. The regularization parameter is chosen according to the discrepancy principle. The reconstructions of the shape of an unknown domain for an [...] Read more.

In this paper, we present the convergence rate analysis of the modified Landweber method under logarithmic source condition for nonlinear ill-posed problems. The regularization parameter is chosen according to the discrepancy principle. The reconstructions of the shape of an unknown domain for an inverse potential problem by using the modified Landweber method are exhibited. Full article

(This article belongs to the Special Issue Numerical Analysis: Inverse Problems – Theory and Applications)

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16 pages, 473 KiB

Open AccessArticle

The Impact of the Discrepancy Principle on the Tikhonov-Regularized Solutions with Oversmoothing Penalties

by Bernd Hofmann and Christopher Hofmann

Mathematics 2020, 8(3), 331; https://doi.org/10.3390/math8030331 - 03 Mar 2020

Cited by 4 | Viewed by 2311

Abstract

This paper deals with the Tikhonov regularization for nonlinear ill-posed operator equations in Hilbert scales with oversmoothing penalties. One focus is on the application of the discrepancy principle for choosing the regularization parameter and its consequences. Numerical case studies are performed in order [...] Read more.

This paper deals with the Tikhonov regularization for nonlinear ill-posed operator equations in Hilbert scales with oversmoothing penalties. One focus is on the application of the discrepancy principle for choosing the regularization parameter and its consequences. Numerical case studies are performed in order to complement analytical results concerning the oversmoothing situation. For example, case studies are presented for exact solutions of Hölder type smoothness with a low Hölder exponent. Moreover, the regularization parameter choice using the discrepancy principle, for which rate results are proven in the oversmoothing case in in reference (Hofmann, B.; Mathé, P. Inverse Probl. 2018, 34, 015007) is compared to Hölder type a priori choices. On the other hand, well-known analytical results on the existence and convergence of regularized solutions are summarized and partially augmented. In particular, a sketch for a novel proof to derive Hölder convergence rates in the case of oversmoothing penalties is given, extending ideas from in reference (Hofmann, B.; Plato, R. ETNA. 2020, 93). Full article

(This article belongs to the Special Issue Numerical Analysis: Inverse Problems – Theory and Applications)

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19 pages, 404 KiB

Open AccessArticle

Generalized-Fractional Tikhonov-Type Method for the Cauchy Problem of Elliptic Equation

by Hongwu Zhang and Xiaoju Zhang

Mathematics 2020, 8(1), 48; https://doi.org/10.3390/math8010048 - 01 Jan 2020

Viewed by 1736

Abstract

This article researches an ill-posed Cauchy problem of the elliptic-type equation. By placing the a-priori restriction on the exact solution we establish conditional stability. Then, based on the generalized Tikhonov and fractional Tikhonov methods, we construct a generalized-fractional Tikhonov-type regularized solution to recover [...] Read more.

This article researches an ill-posed Cauchy problem of the elliptic-type equation. By placing the a-priori restriction on the exact solution we establish conditional stability. Then, based on the generalized Tikhonov and fractional Tikhonov methods, we construct a generalized-fractional Tikhonov-type regularized solution to recover the stability of the considered problem, and some sharp-type estimates of convergence for the regularized method are derived under the a-priori and a-posteriori selection rules for the regularized parameter. Finally, we verify that the proposed method is efficient and acceptable by making the corresponding numerical experiments. Full article

(This article belongs to the Special Issue Numerical Analysis: Inverse Problems – Theory and Applications)

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21 pages, 427 KiB

Open AccessArticle

The Truncation Regularization Method for Identifying the Initial Value on Non-Homogeneous Time-Fractional Diffusion-Wave Equations

by Fan Yang, Qu Pu, Xiao-Xiao Li and Dun-Gang Li

Mathematics 2019, 7(11), 1007; https://doi.org/10.3390/math7111007 - 23 Oct 2019

Cited by 15 | Viewed by 2089

Abstract

In the essay, we consider an initial value question for a mixed initial-boundary value of time-fractional diffusion-wave equations. This matter is an ill-posed problem; the solution relies discontinuously on the measured information. The truncation regularization technique is used for restoring the initial value [...] Read more.

In the essay, we consider an initial value question for a mixed initial-boundary value of time-fractional diffusion-wave equations. This matter is an ill-posed problem; the solution relies discontinuously on the measured information. The truncation regularization technique is used for restoring the initial value functions. The convergence estimations are given in a priori regularization parameter choice regulations and a posteriori regularization parameter choice regulations. Numerical examples are given to demonstrate this is effective and practicable. Full article

(This article belongs to the Special Issue Numerical Analysis: Inverse Problems – Theory and Applications)

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10 pages, 479 KiB

Open AccessArticle

A New Method for De-Noising of Well Test Pressure Data Base on Legendre Approximation

by Fengbo Zhang, Yuandan Zheng, Zhenyu Zhao and Zhi Li

Mathematics 2019, 7(10), 989; https://doi.org/10.3390/math7100989 - 18 Oct 2019

Viewed by 1751

Abstract

In this paper, noise removing of the well test data is considered. We use the Legendre expansion to approximate well test data and a truncated strategy has been employed to reduce noise. The parameter of the truncation will be chosen by a discrepancy [...] Read more.

In this paper, noise removing of the well test data is considered. We use the Legendre expansion to approximate well test data and a truncated strategy has been employed to reduce noise. The parameter of the truncation will be chosen by a discrepancy principle and a corresponding convergence result has been obtained. The theoretical analysis shows that a well numerical approximation can be obtained by the new method. Moreover, we can directly obtain the stable numerical derivatives of the pressure data in this method. Finally, we give some numerical tests to show the effectiveness of the method. Full article

(This article belongs to the Special Issue Numerical Analysis: Inverse Problems – Theory and Applications)

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19 pages, 362 KiB

Open AccessArticle

Generalized Tikhonov Method and Convergence Estimate for the Cauchy Problem of Modified Helmholtz Equation with Nonhomogeneous Dirichlet and Neumann Datum

by Hongwu Zhang and Xiaoju Zhang

Mathematics 2019, 7(8), 667; https://doi.org/10.3390/math7080667 - 25 Jul 2019

Cited by 3 | Viewed by 2337 | Correction

Abstract

We investigate a Cauchy problem of the modified Helmholtz equation with nonhomogeneous Dirichlet and Neumann datum, this problem is ill-posed and some regularization techniques are required to stabilize numerical computation. We established the result of conditional stability under an a priori assumption for [...] Read more.

We investigate a Cauchy problem of the modified Helmholtz equation with nonhomogeneous Dirichlet and Neumann datum, this problem is ill-posed and some regularization techniques are required to stabilize numerical computation. We established the result of conditional stability under an a priori assumption for an exact solution. A generalized Tikhonov method is proposed to solve this problem, we select the regularization parameter by a priori and a posteriori rules and derive the convergence results of sharp type for this method. The corresponding numerical experiments are implemented to verify that our regularization method is practicable and satisfied. Full article

(This article belongs to the Special Issue Numerical Analysis: Inverse Problems – Theory and Applications)

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24 pages, 785 KiB

Open AccessFeature PaperArticle

Solutions of Direct and Inverse Even-Order Sturm-Liouville Problems Using Magnus Expansion

by Upeksha Perera and Christine Böckmann

Mathematics 2019, 7(6), 544; https://doi.org/10.3390/math7060544 - 14 Jun 2019

Cited by 8 | Viewed by 3119

Abstract

In this paper Lie group method in combination with Magnus expansion is utilized to develop a universal method applicable to solving a Sturm–Liouville problem (SLP) of any order with arbitrary boundary conditions. It is shown that the method has ability to solve direct [...] Read more.

In this paper Lie group method in combination with Magnus expansion is utilized to develop a universal method applicable to solving a Sturm–Liouville problem (SLP) of any order with arbitrary boundary conditions. It is shown that the method has ability to solve direct regular (and some singular) SLPs of even orders (tested for up to eight), with a mix of (including non-separable and finite singular endpoints) boundary conditions, accurately and efficiently. The present technique is successfully applied to overcome the difficulties in finding suitable sets of eigenvalues so that the inverse SLP problem can be effectively solved. The inverse SLP algorithm proposed by Barcilon (1974) is utilized in combination with the Magnus method so that a direct SLP of any (even) order and an inverse SLP of order two can be solved effectively. Full article

(This article belongs to the Special Issue Numerical Analysis: Inverse Problems – Theory and Applications)

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13 pages, 447 KiB

Open AccessArticle

A Numerical Method for Filtering the Noise in the Heat Conduction Problem

by Yao Sun, Xiaoliang Wei, Zibo Zhuang and Tian Luan

Mathematics 2019, 7(6), 502; https://doi.org/10.3390/math7060502 - 02 Jun 2019

Cited by 2 | Viewed by 2278

Abstract

In this paper, we give an effective numerical method for the heat conduction problem connected with the Laplace equation. Through the use of a single-layer potential approach to the solution, we get the boundary integral equation about the density function. In order to [...] Read more.

In this paper, we give an effective numerical method for the heat conduction problem connected with the Laplace equation. Through the use of a single-layer potential approach to the solution, we get the boundary integral equation about the density function. In order to deal with the weakly singular kernel of the integral equation, we give the projection method to deal with this part, i.e., using the Lagrange trigonometric polynomials basis to give an approximation of the density function. Although the problems under investigation are well-posed, herein the Tikhonov regularization method is not used to regularize the aforementioned direct problem with noisy data, but to filter out the noise in the corresponding perturbed data. Finally, the effectiveness of the proposed method is demonstrated using a few examples, including a boundary condition with a jump discontinuity and a boundary condition with a corner. Whilst a comparative study with the method of fundamental solutions (MFS) is also given. Full article

(This article belongs to the Special Issue Numerical Analysis: Inverse Problems – Theory and Applications)

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13 pages, 741 KiB

Open AccessArticle

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

by Shangqin He and Xiufang Feng

Mathematics 2019, 7(6), 487; https://doi.org/10.3390/math7060487 - 28 May 2019

Cited by 2 | Viewed by 2278

Abstract

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. [...] Read more.

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. Full article

(This article belongs to the Special Issue Numerical Analysis: Inverse Problems – Theory and Applications)

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19 pages, 278 KiB

Open AccessArticle

A Modified Asymptotical Regularization of Nonlinear Ill-Posed Problems

by Pornsarp Pornsawad, Nantawan Sapsakul and Christine Böckmann

Mathematics 2019, 7(5), 419; https://doi.org/10.3390/math7050419 - 10 May 2019

Cited by 5 | Viewed by 2119

Abstract

In this paper, we investigate the continuous version of modified iterative Runge–Kutta-type methods for nonlinear inverse ill-posed problems proposed in a previous work. The convergence analysis is proved under the tangential cone condition, a modified discrepancy principle, i.e., the stopping time T is [...] Read more.

In this paper, we investigate the continuous version of modified iterative Runge–Kutta-type methods for nonlinear inverse ill-posed problems proposed in a previous work. The convergence analysis is proved under the tangential cone condition, a modified discrepancy principle, i.e., the stopping time T is a solution of

∥ F (x^{δ} (T)) - y^{δ} ∥ = τ δ^{+}

for some

δ^{+} > δ

, and an appropriate source condition. We yield the optimal rate of convergence. Full article

(This article belongs to the Special Issue Numerical Analysis: Inverse Problems – Theory and Applications)

13 pages, 335 KiB

Open AccessArticle

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

by Shangqin He and Xiufang Feng

Mathematics 2019, 7(4), 360; https://doi.org/10.3390/math7040360 - 20 Apr 2019

Cited by 8 | Viewed by 2476

Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Numerical Analysis: Inverse Problems – Theory and Applications)

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2 pages, 151 KiB

Open AccessCorrection

Correction: Zhang, H.; Zhang, X. Generalized Tikhonov Method and Convergence Estimate for the Cauchy Problem of Modified Helmholtz Equation with Nonhomogeneous Dirichlet and Neumann Datum. Mathematics 2019, 7, 667

by Hongwu Zhang and Xiaoju Zhang

Mathematics 2019, 7(10), 888; https://doi.org/10.3390/math7100888 - 24 Sep 2019

Viewed by 1329

Abstract

The authors wish to make the following corrections to this paper [...] Full article

(This article belongs to the Special Issue Numerical Analysis: Inverse Problems – Theory and Applications)

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