NUMERICAL SOLUTION OF THE INVERSE COEFFICIENT PROBLEM FOR A MATHEMATICAL MODEL OF DESORPTION DYNAMICS

Geophysical Inverse Theory and Regularization Problems

Based on the parametric identification method and radial basis function networks, the meshfree approach to the solution of inverse coefficient problems of mathematical physics is proposed. For regularization of the solution, an iterative method is used where the number of iterations, determined by the norm of the residual, plays the role of the regulator. The effectiveness of the approach is confirmed by the example of inverse coefficient problems for elliptic equations.

The Newsletter is a key element in further enhancing the value of the journal to the inverse problems community. So why not be a part of this exciting forum by sending to our Bristol office material suitable for inclusion under any of the categories mentioned above. Your contributions will be very welcome. Book review Introduction to Inverse Problems in Imaging M Bertero and P Boccacci 1998 Bristol: Institute of Physics 362 pp ISBN 0-7503-0435-9 (pbk) £25.00, $49.00 This book shows that several problems in imaging are in fact linear inverse problems. In general inverse problems are ill-posed in the sense that small perturbations in the (measured) data have a significant influence on the output. Consequently, numerical methods developed in a general setting of inverse problems which cure the ill-posedness (such methods are called regularization methods) can be applied to solve problems in imaging. This is an excellent textbook on the principle of linear inverse problems, methods of their numerical solution and practical applications in imaging. It introduces basic ideas and methods for the solution of inverse and ill-posed problems while avoiding the mathematics of functional treatment of operator equations. This goal is achieved by a bottom-up presentation: the authors focus on the two problems of image deconvolution and tomography (bottom) and develop regularization theory specifically for these two problems. Most other books which are concerned with the numerical solution of inverse problems present a top-down approach. That is, a general regularization theory (top) is developed which is then applied to particular problems. The distinguished presentation makes the book very useful to students in applied mathematics, practitioners in engineering science and image processing. This book has the best prerequisites to establish a close link between the mathematical fields of imaging and inverse problems. Mainly the book focuses on two topics: image deconvolution and linear imaging systems. 1. In image deconvolution the goal is to restore a space invariant blurred image. This part of the book contains mathematical tools in image deconvolution, examples of space invariant imaging systems, a comparison of image deconvolution techniques and low-pass filtering techniques, the analysis of the ill-posedness of the deconvolution problem, regularization methods for deconvolution such as constrained least squares regularization, Tikhonov regularization, iterative regularization and statistical methods. Since the authors focus on particular problems they are able to analyse various methods for deconvolution on a very concrete level. As an example I would like to mention that the authors analyse Fourier methods for deconvolution. Thus they provide an optimal starting point to the theory of inverse problems for people working in the area of signal and image processing. 2. The second part of the book considers linear imaging systems that are not of a convolution type such as tomography problems, diffraction problems and inverse scattering problems. Efficient numerical methods for the solution of this class of problems such as singular value decomposition, Tikhonov regularization methods and Fourier-based methods are presented. O Scherzer Universität Linz

abstract: We consider inverse boundary value problems for general real principal type differential operators. The first results state that the Cauchy data set uniquely determines the scattering relation of the operator and bicharacteristic ray transforms of lower order coefficients. We also give two different boundary determination methods for general operators, and prove global uniqueness results for determining coefficients in nonlinear real principal type equations. The article presents a unified approach for treating inverse boundary problems for transport and wave equations, and highlights the role of propagation of singularities in the solution of related inverse problems.

Parents of children with genetically determined leukoencephalopathies play a major role in their children's health care. Because of the COVID-19 pandemic, many health care services were suspended, delayed or delivered remotely with telemedicine. We sought to explore the experience of parents of children with genetically determined leukoencephalopathies during the pandemic given the adapted health care services. We conducted semistructured interviews with 13 parents of 13 affected children. Three main themes were identified using thematic analysis: perceived impact of COVID-19 on health care services, benefits and challenges of telemedicine, and expectations of health care after the pandemic. Parents perceived a loss/delay in health care services while having a positive response to telemedicine. Parents wished telemedicine would remain in their care after the pandemic. This is the first study assessing the impact of COVID-19 on health care services in this population. Our results suggest that parents experience a higher level of stress owing to the shortage of services and the children's vulnerability.

A Weighted Dynamic Inverse Problem Solution is proposed for electroencephalographic current density reconstruction. The method considers physiologically based models that takes both spatial and temporal dynamics into account and a weighting stage to obtain the covariance of the measurement equation from observations. The calculated weighting matrix is included in the cost function used to solve the dynamic inverse problem. Additionally, an iterative method for inverse problem solution is proposed which considers the time varying weighting matrix in the solution. In this way, a weighted dynamic inverse problem solution is proposed including a time varying weighting matrix. The method performance is analyzed in terms of standard error, projected error, and residual error for several SNRs by using simulated signals through complex dynamical models.

This work is devoted to the development and implementation of a two-step method for solving the vector three-dimensional inverse diffraction problem on an inhomogeneous dielectric scatterer having the form of a hemisphere characterized by piecewise constant permittivity. The original boundary value problem for Maxwell’s equations is reduced to a system of integro-differential equations. An integral formulation of the vector inverse diffraction problem is proposed and the uniqueness of the solution of the first-kind integro-differential equation in special function classes is established. A two-step method for solving the vector inverse diffraction problem on the hemisphere is developed. Unlike traditional approaches, the two-step method for solving the inverse problem is non-iterative and does not require knowledge of the exact initial approximation. Consequently, there are no issues related to the convergence of the numerical method. The results of calculations of approximate solutions to the inverse problem are presented. It is shown that the two-step method is an efficient approach to solving vector problems in near-field tomography.

The solution of direct and inverse transient heat conduction problems with layered materials using exponential basis functions

Identification of time-dependent convection coefficient in a time-fractional diffusion equation

The purpose of this study is to propose an efficient numerical method for solving the inverse nonlinear problem of the movement of the compressor rotor collar in a fluid film thrust bearing. Methods. A periodic thermoelastohydrodynamic (PTEHD) mathematical model of hydrodynamic and thermal processes in a bearing is constructed under the condition of the rotor collar motion. Within the framework of the model, an inverse nonlinear problem of determining the position of the collar under a given external load is formulated. An iterative solution method is proposed, which utilizes the solution of the direct problem. To reduce computational costs, a modified Dekker–Brent method is employed in conjunction with a modified Newton’s method. Results. Numerical experiments have been conducted, demonstrating the effectiveness of the proposed approaches. The suggested methods significantly reduce the required computational resources by minimizing the number of calls to the target function in the optimization problem. A software suite has been developed that allows for the calculation of the nonlinear system of rotor motion under various physical and geometric parameters. Conclusion. An efficient set of numerical methods for solving the inverse nonlinear problem of the motion of the rotor collar in the compressor fluid film thrust bearing is proposed. The method’s effectiveness lies in substantial savings of computational resources. The method’s efficiency has been demonstrated in numerical experiments.

We consider the inverse coefficient problem of identifying the principal coefficient in the damped wave equation , subject to the boundary conditions , , from the Neumann boundary output f (t): = r(0)ux(0,t), . We propose detailed microlocal analysis of the regularity of the solution of the wave equation in each subdomain defined by the characteristics of the wave equation. This analysis allows us to derive sufficient conditions for the regularity of the weak solution of the direct problem, especially along the characteristic lines, as well as to prove necessary energy estimates including also stability estimate for auxiliary hyperbolic problem. Based on this analysis we prove the compactness and Lipschitz continuity of the Dirichlet-to-Neumann operator , corresponding to the inverse problem. The last property allows us to prove an existence of a quasi-solution of the inverse problem defined as a minimum of the Tikhonov functional and also Fréchet differentiability of this functional. For the case when and , a uniqueness theorem is derived. An explicit formula for the Fréchet gradient of the Tikhonov functional and its justification are derived by making use of the unique solution to corresponding adjoint problem. The approach proposed in this paper is expected to lead to very effective computational identification algorithms.

Summary Inverse eigenvalue problems associated with self-adjoint differential equations of order two or higher are considered. The question of what kind of data are necessary and sufficient to insure the existence of a unique solution is examined. A method of solution for well-posed inverse eigenvalue problems is then presented. The implications of this analysis to the inverse problem for the Earth's spheroidal modes of vibration is also discussed. In this talk, I propose to present a partial survey of the mathematical literature on inverse eigenvalue problems, with an eye toward geophysical applications, and to state some new results which I obtained recently. Perhaps the inverse eigenvalue problem which is best known to geophysicists is that related to the normal modes of vibration of the Earth. It can be stated thus: assuming that the Earth is spherically symmetric and given the natural frequencies ,S, and ,T, of the spheroidal and torsional modes of vibration, can we deduce the density p(r) and the speed of propagation Vr Takeuchi & Saito 1972) and very little is known about inverse eigenvalue problems of order higher than two. In order to gain an understanding of inverse eigenvalue problems one should perhaps divorce oneself from a specific problem and investigate rather a sequence of problems of increasing complexity. If you allow me to adopt this attitude, then the first task would be a rough classification of eigenvalue problems. Without pretending to be exhaustive, let me divide eigenvalue problems into two classes depending upon whether they are associated with ordinary or partial differential equations, i.e. depending upon whether they are related to one-dimensional or multi-dimensional problems. The following Sturm-Liouville problem: u+{n-q(x)}u = 0

<abstract><p>In this paper, an inverse problem of determining the space-dependent volatility from the observed market prices of options with different strikes is studied. Being different from other inverse volatility problem with classical parabolic equations, we apply the linearization method and introduce some variable substitutions to convert the original problem into an inverse source problem in a degenerate parabolic equation in a bounded area, from which an unknown volatility can be recovered and deficiencies caused by artificial truncation can be solved. Based on the optimal control framework, the problem is transformed into an optimization problem and the existence of the minimizer is established. After the necessary conditions are deduced, the uniqueness and stability of the minimizer are proved. Then, the Landweber iterative method is used to obtain a stable numerical solution of the inverse problem and some numerical experiments are also performed. The numerical results show that the algorithm which we proposed is robust and the unknown coefficient is recovered quite well.</p></abstract>

This paper proposes a new iterative integration regularization method for robust solution of ill-posed inverse problems. The proposed method is motivated from the fact that inversion of a positive definite matrix can be expressed in an integral form. Then, the development of the proposed method is mainly twofold. Firstly, two ways—including the linear iteration and the exponential ($$2^j$$) iteration—are invoked to compute the integral, of which the exponential iteration is often preferred due to its fast convergence. Secondly, after stability analysis, the proposed method is shown able to filter out the undesired effect of relatively small singular values, while preserving the desired terms of relatively large singular values, i.e., the proposed method has the guaranteed regularization effect. Numerical examples on three typical ill-posed problems are conducted with detailed comparison to some usual direct and iterative regularization methods. Final results have highlighted the proposed method: (a) due to the iterative nature, the proposed method often turns out to be more efficient than the conventional direct regularization methods including the Tikhonov regularization and the truncated singular value decomposition (TSVD), (b) the proposed method converges much faster than the Landweber method and (c) the regularization effect is guaranteed in the proposed method, while may not be in the conjugate gradient method for least squares problem (CGLS).

Coefficient inverse problems are reformulated to a unified integral differential equation. The presented method of conversion of the considered inverse problems to a unified Volterra integral-differential equation gives an opportunity to distribute the acquired results also to analogous inverse problems for non-linear parabolic equations of different types.