Inverse Cauchy Problem Research Articles

On the resolution of the non-smooth inverse Cauchy problem by the primal-dual method

On the resolution of the non-smooth inverse Cauchy problem by the primal-dual method

Inverse Cauchy problem in the framework of an RBF-based meshless technique and trigonometric basis functions

Inverse Cauchy problem in the framework of an RBF-based meshless technique and trigonometric basis functions

Numerical Method for Solving an inverse Cauchy problem for Helmholtz equation using CGLS, BICGSTAB and PCG

Numerical Method for Solving an inverse Cauchy problem for Helmholtz equation using CGLS, BICGSTAB and PCG

Resolving an Inverse Cauchy Problem of Mixed Boundary Value of Bi-harmonic using a meshless collocation method

Resolving an Inverse Cauchy Problem of Mixed Boundary Value of Bi-harmonic using a meshless collocation method

The meshless backward substitution method for inverse Cauchy problems in electroelastic piezoelectric structures

The backward substitution method is a recently proposed semi-analytical meshless collocation method. The main idea of the back substitution method is to obtain the boundary approximation from the boundary data, and this approximation does not need to satisfy the governing equation. Then, the traditional linear combination of basis functions is used to construct a correcting approximation that satisfies the homogeneous boundary conditions, the numerical solution is given by a linear combination system of boundary approximation and modified approximation, which satisfies the governing equation. In this way, the backward substitution method can be easily used to solve general problems without fundamental solutions or particular solutions. This paper makes the first attempt to extend the backward substitution method to the simulation of inverse Cauchy problems in piezoelectric structures. For the inverse Cauchy problems, the absence of data on some boundaries often leads to extreme instability of numerical solutions. A simple method is applied in this paper. For the points on the boundary without boundary information, the governing equation is enforced. Numerical results demonstrate that it will greatly increase the stability of the solution process.

On the well posedness and Steffensen’s based numerical approximation of an inverse Cauchy problem

This work is devoted to the study of an inverse Cauchy problem governed by Poisson’s equation, which is one of highly ill-posed problems in the Hadamard’s sense (Hadamard, 1953). We propose an efficient approach for solving this data completion problem, based on its reformulation into a root finding problem which is equivalent to a fixed point one involving a Steklov-Poincaré like operator (Chakib et al., 2018, Nachaoui et al., 2021). We propose first another proof of the existence of the fixed point using the Fredholm alternative. This allows us to establish also the uniqueness and the stability of the solution, in contrast to the previous work (Chakib et al., 2018, Nachaoui et al., 2021), where only the existence result was established. Then, the numerical approximation of the fixed point problem is investigated using an iterative process performed by a Steffensen’s approach (Greenbaum and Chartier, 2012) and the P1 finite element discretization. Finally, we present some numerical results showing the efficiency of the proposed approach in term of the solution accuracy and the CPU execution time.

Nonlinear Cauchy/Robin inverse problems solved by an optimal splitting-linearizing method

In the paper we develop a novel optimal splitting-linearizing method (OSLM) to iteratively solve a nonlinear inverse Cauchy problem in a simply-connected domain. The nonlinear term in the nonlinear elliptic equation is decomposed at two sides through a splitting parameter, which is then linearized around the value at the previous iteration step. The multiple-scale Pascal-polynomial method together with the OSLM is employed to solve the Cauchy problem, of which the optimal value of the splitting parameter is achieved by minimizing a theoretic merit function. Then, we solve the Cauchy/Robin inverse problem of a nonlinear elliptic equation in a doubly-connected domain for recovering the unknown Cauchy data and Robin transfer coefficient on an inner boundary. Two-parameter bases are derived to automatically satisfy the prescribed Cauchy boundary conditions on the outer boundary. When the solution is convergent after solving a sequence of linear systems, one can retrieve the Cauchy data very accurately. Simultaneously, the unknown Robin transfer coefficient is recovered from a given convective boundary condition on the inner boundary by either a division method or a linear system method. To overcome the ill-posed property of Cauchy/Robin problems, the optimal splitting parameter and a scaling factor play the role as regularization parameters. These methods assembled are new techniques to solve the Cauchy/Robin inverse problems. Although a few overspecified data are merely given on the outer boundary, the novel method is quite accurate, robust against large noise, and is convergent very fast to find the entire solution, the Cauchy data and the Robin transfer coefficient. We assess the convergence by the computed order of convergence (COC) of the proposed iterative algorithms.

A CCBM-based generalized GKB iterative regularization algorithm for inverse Cauchy problems

This paper examines inverse Cauchy problems that are governed by a kind of elliptic partial differential equation. The inverse problems involve recovering the missing data on an inaccessible boundary from the measured data on an accessible boundary, which is severely ill-posed. By using the coupled complex boundary method (CCBM), which integrates both Dirichlet and Neumann data into a single Robin boundary condition, we reformulate the underlying problem into an operator equation. Based on this new formulation, we study the solution existence issue of the reduced problem with noisy data. A Golub–Kahan bidiagonalization (GKB) process together with Givens rotation is employed for iteratively solving the proposed operator equation. The regularizing property of the developed method, called CCBM-GKB, and its convergence rate results are proved under a posteriori stopping rule. Finally, a linear finite element method is used for the numerical realization of CCBM-GKB. Various numerical experiments demonstrate that CCBM-GKB is a kind of accelerated iterative regularization method, as it is much faster than the classic Landweber method.

Polynomial Approximation of an Inverse Cauchy Problem for Modified Helmholtz Equations

In this paper, an inverse problem for the modified Helmholtz equation arising in heat conduction in the fin is considered. The goal of this paper is the determination of the temperature at the under-specified boundary (inner boundary of an annular domain) benefiting from the accessible part of the boundary with Cauchy data (boundary temperature and heat flux). This problem is solved numerically using the meshless method proposed in [2]. The stability is confirmed by applying a noise for the Cauchy data.

ОБРАТНАЯ ЗАДАЧА КОШИ ДЛЯ СТОЕЧНО-БАЛОЧНОЙ КОНСТРУКТИВНОЙ СИСТЕМЫ

The object of this study is the building frames with rigid beam-to-column assemblies. The aim of the study is to assess the impact on the accuracy of the solution of the problem of the error of the input data and the number of given coefficients of the deflection equation. The studies were carried out using analytical and experimental methods of regularization, reduction of measurements, solutions on a measuring compact, polynomial approximation, linear Lagrangian interpolation and numerical differentiation. Rigid coupling of a beam with a rack is modeled analytically and by a full-scale experiment. For a quantitative assessment of the effectiveness of solving the problem, the values of the target parameter and the optimization criterion are determined through the minimum of the Lebesgue function. It is proposed to use the obtained results of solving the inverse Cauchy problem in experimental and theoretical studies of rack-and-beam structures.

A Haar wavelets-based direct reconstruction method for the Cauchy problem of the Poisson equation

In this paper, we develop a Haar wavelet-based reconstruction method to recover missing data on an inaccessible part of the boundary from measured data on another accessible part. The technique is developed to solve inverse Cauchy problems governed by the Poisson equation which is severely ill-posed. The new method is mathematically simple to implement and can be easily applied to Cauchy problems governed by other partial differential equations appearing in various fields of natural science, engineering, and economics. To take into account the ill-conditioning of the obtained linear system due to the ill-posedness of the Cauchy problem, a preconditioning strategy combined with a regularization has been developed. Comparing the numerical results produced by a meshless method based on the polynomial expansion with those produced by the proposed technique illustrates the superiority of the latter. Other numerical results show its effectiveness.

Local knot method for solving inverse Cauchy problems of Helmholtz equations on complicated two‐ and three‐dimensional domains

AbstractThis article presents a local knot method (LKM) to solve inverse Cauchy problems of Helmholtz equations in arbitrary 2D and 3D domains. The Moore–Penrose pseudoinverse using the truncated singular value decomposition is employed in the local approximation of supporting domain instead of the moving least squares method. The developed approach is a semi‐analytical and local radial basis function collocation method using the non‐singular general solution as the basis function. Like the traditional boundary knot method, the LKM is simple, accurate and easy‐to‐program in solving inverse Cauchy problems associated with Helmholtz equations. Unlike the boundary knot method, the new scheme can directly reconstruct the unknowns both inside the physical domain and along its boundary by solving a sparse linear system, and can achieve a satisfactory solution. Numerical experiments, involving the complicated geometry and the high noise level, confirm the accuracy and reliability of the proposed method for solving inverse Cauchy problems of Helmholtz equations.

MULTIFACTORIAL APPROACH TO THE DETERMINATION OF HEAT TRANSFER IN MULTILAYER SURFACES

Development of the territories with specific climatic conditions requires the selection of effective materials. Multilayer composite materials are suitable for these purposes in road construction in the Arctic. The article presents a design scheme for a multilayer road surface, which is used to determine the collection of traffic loads. The method of solving the inverse Cauchy problem and the empirical formula to determine the value of the temperature arising from deformation at the contact point of the surface of a multilayer road surface and transport are determined. The experimental studies conducted were aimed to determine the heat transfer through a three-layer road surface made of structural foam concrete of the D1200 brand. The processing of experimental data on temperature values made it possible to derive the formula to determine the change in the layers’ temperature depending on their number.

Localized Method of Fundamental Solutions for Two-Dimensional Inhomogeneous Inverse Cauchy Problems

Due to the fundamental solutions are employed as basis functions, the localized method of fundamental solution can obtain more accurate numerical results than other localized methods in the homogeneous problems. Since the inverse Cauchy problem is ill posed, a small disturbance will lead to great errors in the numerical simulations. More accurate numerical methods are needed in the inverse Cauchy problem. In this work, the LMFS is firstly proposed to analyze the inhomogeneous inverse Cauchy problem. The recursive composite multiple reciprocity method (RC-MRM) is adopted to change original inhomogeneous problem into a higher-order homogeneous problem. Then, the high-order homogeneous problem can be solved directly by the LMFS. Several numerical experiments are carried out to demonstrate the efficiency of the LMFS for the inhomogeneous inverse Cauchy problems.

A meshless collocation method for solving the inverse Cauchy problem associated with the variable-order fractional heat conduction model under functionally graded materials

A localized meshless collocation method, namely the generalized finite difference method (GFDM), is introduced to cope with the inverse Cauchy problem associated with the fractional heat conduction model under functionally graded materials (FGMs). The variable-order time-fractional heat conduction equation under the Caputo definition is employed to describe anomalous heat conduction problems. In the present numerical framework, temporal-discretization is implemented by using the standard implicit finite difference method with the spatial-discretization through the GFDM. On the basis of moving least squares and Taylor series expansion, the GFDM is capable of avoiding the ill-posedness in inverse Cauchy problems and solving the time fractional heat conduction equations (TFHCE) under FGMs. To give evidence of the efficiency and accuracy of the proposed approach for solving inverse heat conduction of FGMs, three numerical experiments are considered in the results and discussions section.

Localized Boundary Knot Method for Solving Two-Dimensional Inverse Cauchy Problems

In this paper, a localized boundary knot method is adopted to solve two-dimensional inverse Cauchy problems, which are controlled by a second-order linear differential equation. The localized boundary knot method is a numerical method based on the local concept of the localization method of the fundamental solution. The approach is formed by combining the classical boundary knot method with the localization method. It has the potential to solve many complex engineering problems. Generally, in an inverse Cauchy problem, there are no boundary conditions in specific boundaries. Additionally, in order to be close to the actual engineering situation, a certain level of noise is added to the known boundary conditions to simulate the measurement error. The localized boundary knot method can be used to solve two-dimensional Cauchy problems more stably and is truly free from mesh and numerical quadrature. In this paper, the stability of the method is verified by using multi-connected domain and simply connected domain examples in Laplace equations.

Information and Measurement System for Monitoring Beams in Building Structures

Purpose of research. The development of a method and algorithm for reducing measurements of beam identification parameters in an information and measurement system for monitoring building structures with measurement of deflections and recovery of actual values of beam initial parameters and external load when solving the inverse Cauchy problem.Methods. The solution of the problem is carried out through formulating the transverse bending of the beam according to the Euler – Bernoulli theory using the method of regularization and reduction of measurements by solving the inverse Cauchy problem by means of linear Lagrangian approximation in the procedure of numerical differentiation of the beam deflection function. A methodology is formulated for identifying insignificant beam identification parameters by comparing the deflection of the beam caused by the parameter under study with the sensitivity threshold of measuring instruments. In this case, the modification of the state space of identification parameters with a decrease in its dimension is simulated.Results. The working capability of the formulated experimental calculation method is confirmed by numerical experiment with a load on the beam in the form of a bending moment, concentrated and (or) constant distributed load. It has been established that when identifying insignificant initial parameters and loads acting on the beam, the reduction of measurements increases the accuracy of restoring the beam identification parameters.Conclusion. The developed methodology can be used to improve the accuracy of inspection methods of construction facilities at the stage of experimental and theoretical research.

Chebyshev Alternance when Approximating Initial Conditions of the Inverse Cauchy Problem

Purpose of research. The work is devoted to a range of questions related to Cauchy problem on the segment of real axis with the application of the inverse Cauchy problem, in which real constants are initial conditions which are optimally restored according to experimental or tabular values of the solution of the differential equation. The object of the study is an information-measuring system, in which approximate values of initial conditions are calculated from discrete function values of Cauchy problem solving.Methods. The following problems are solved for this purpose: parameters of measuring section placement on the investigated object and approximation grid on measuring section are developed. Characteristics of recovery accuracy of initial conditions of the task are formulated.Results. An experimental-calculated method of determining initial conditions in the inverse Cauchy problem is proposed. It is based on the concept of objective function of regularization of the problem. Task regularization parameter in the form of minimum value by Lebesgue function is proposed.Conclusion. The reaction of uniformly approximating method of the initial conditions of the inverse Cauchy problem to the deviation of the approximation grid coordinates nodes from the coordinates of Chebyshev alternance was described. Graphs of method reaction to deviation of grid pitch from optimal pitch are given.

A finite-difference and Haar wavelets hybrid collocation technique for non-linear inverse Cauchy problems

In this research work, a finite-difference and Haar wavelet hybrid collocation scheme is introduced for the ill-posed non-linear inverse Cauchy problem with a source depending on space variable along with an unknown solution and unknown right side boundary. The first-order finite-difference approach is adopted to approximate the part and two different Haar series are managed to approximate part and source term respectively. A simple linearization procedure is used to convert the non-linear problem into a linear form. In contradiction to various numerical schemes, the current introduced method generates a well-conditioned system of algebraic equations, therefore it is not required to apply a regularization approach. The results of the proposed method are stable and converge to the exact solution. Some numerical tests are also performed to confirm the accuracy, well-conditioning of the algebraic equations and easy applicability of the scheme on linear and non-linear cases.

ОБРАТНАЯ ЗАДАЧА КОШИ ДЛЯ БАЛОК В СТРОИТЕЛЬНЫХ КОНСТРУКЦИЯХ

A quality index of the coefficient grid inverse Cauchy problem for beams in building structures is proposed. The indicator is based on the theory of regularization of inverse problems. An articulated support of a beam on a column is modeled analytically and by a full-scale experiment. Models of measurement and calculation are investigated for a uniform continuous error rate of deflection measurement and calculation of beam identification parameters. Models differ in various combinations of types of external load. A measure of the influence of the error of the measuring instrument and the distribution of approximation grid nodes on the error in determining the coefficients of the beam deflection equation with a fixed first coefficient is proposed. The measure of influence is described by the dimensionless absolute condition number of the problem. The values of the dimensionless absolute condition number and the quality index of the problem are analyzed depending on the distribution of approximation grid nodes, the error of the measuring instrument, and the type of measurement and calculation model. It is proposed to use the obtained analytical dependencies for the analysis of building structures at the stage of experimental and theoretical studies.