A modification of the semi-analytic inversion method: determination of the yield stress and a comparison with the parametrization algorithm

The problem related to controlled potential experiments in electrochemistry is studied. Ion transport is regarded as the superposition of diffusion and migration. Modelling of the experiment leads to a problem for a nonlinear parabolic equation with additional condition. Driven by the needs of theoretical analysis, from the point of view a inverse coefficient problem, we analyze the monotonicity of input-output mappings in inverse coefficient and source problems for this parabolic equation. Additionally, we extend the nonlinear parabolic equation to a more general case. Under some proper conditions, we investigate the existence of quasisolution of the generalized nonlinear parabolic equation.

Two types of inverse source problems of identifying asynchronously distributed spatial loads governed by the Euler–Bernoulli beam equation , with hinged–clamped ends ( ), are studied. Here are linearly independent functions, describing an asynchronous temporal loading, and are the spatial load distributions. In the first identification problem the values , of the deflection , are assumed to be known, as measured output data, in a neighbourhood of the finite set of points , corresponding to the internal points of a continuous beam, for all . In the second identification problem the values , of the slope , are assumed to be known, as measured output data in a neighbourhood of the same set of points P for all . These inverse source problems will be defined subsequently as the problems ISP1 and ISP2. The general purpose of this study is to develop mathematical concepts and tools that are capable of providing effective numerical algorithms for the numerical solution of the considered class of inverse problems. Note that both measured output data and contain random noise. In the first part of the study we prove that each measured output data and can uniquely determine the unknown functions . In the second part of the study we will introduce the input–output operators , and , , corresponding to the problems ISP1 and ISP2, and then reformulate these problems as the operator equations: and , where and . Since both measured output data contain random noise, we use the most prominent regularisation method, Tikhonov regularisation, introducing the regularised cost functionals and . Using a priori estimates for the weak solution of the direct problem and the Tikhonov regularisation method combined with the adjoint problem approach, we prove that the Fréchet gradients and of both cost functionals can explicitly be derived via the corresponding weak solutions of adjoint problems and the known temporal loads . Moreover, we show that these gradients are Lipschitz continuous, which allows the use of gradient type iteration convergent algorithms. Two applications of the proposed theory are presented. It is shown that solvability results for inverse source problems related to the synchronous loading case, with a single interior measured data, are special cases of the obtained results for asynchronously distributed spatial load cases.

This article deals with the mathematical analysis of the inverse coefficient problem of identifying the unknown coefficient k(x) in the linear time fractional parabolic equation D t α u(x,t)= ( k ( x ) u x ) x , 0<α≤1, with mixed boundary conditions u(0,t)= ψ 0 (t), u x (1,t)= ψ 1 (t). By defining the input-output mappings Φ[⋅]:K→ C 1 [0,T] and Ψ[⋅]:K→C[0,T], the inverse problem is reduced to the problem of their invertibility. Hence the main purpose of this study is to investigate the distinguishability of the input-output mappings Φ[⋅] and Ψ[⋅]. This work shows that the input-output mappings Φ[⋅] and Ψ[⋅] have the distinguishability property. Moreover, the value k(0) of the unknown diffusion coefficient k(x) at x=0 can be determined explicitly by making use of measured output data (boundary observation) k(0) u x (0,t)=f(t), which brings greater restriction on the set of admissible coefficients. It is also shown that the measured output data f(t) and h(t) can be determined analytically by a series representation, which implies that the input-output mappings Φ[⋅]:K→ C 1 [0,T] and Ψ[⋅]:K→C[0,T] can be described explicitly.

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.

Two main aims of this paper are to develop a numerical method to solve an inverse source problem for parabolic equations and apply it to solve a nonlinear coefficient inverse problem. The inverse source problem in this paper is the problem to reconstruct a source term from external observations. Our method to solve this inverse source problem consists of two stages. We first establish an equation of the derivative of the solution to the parabolic equation with respect to the time variable. Then, in the second stage, we solve this equation by the quasi-reversibility method. The inverse source problem considered in this paper is the linearization of a nonlinear coefficient inverse problem. Hence, iteratively solving the inverse source problem provides the numerical solution to that coefficient inverse problem. Numerical results for the inverse source problem under consideration and the corresponding nonlinear coefficient inverse problem are presented.

An effective technique for inverse coefficient problem solving of the shear modulus reconstruction for both inhomogeneous elastic and viscoelastic cylindrical waveguide is proposed. The identification is carried out using the displacement field data measured on the finite part of the outer boundary of the cylindrical waveguide in the torsional vibrations mode. An iterative scheme for solving the inverse problem is developed. At each iteration step, the correction to the shear modulus distribution function is calculated using the Fredholm integral equation of the first kind with a smooth kernel. Numerical results are obtained for both direct and inverse problems for different laws of the shear modulus variation.

See the attached PDF file In this paper, the Coefficient Inverse Problems (I.P.), the Geometrical I.P. and Microstructure (I.P.) in elasticity, plasticity, hardening, fatigue, fracture and microstructure are formulated and solved. The retrospective I.P. and the boundary I.P. are commented. It was prepared equations for evaluation of the velocities for propagation of longitudinal and transversal ultrasonic wave. The relationships between following mechanical properties: modulus of elasticity, Henki’s coefficient, parameters in hardening law, parameters in Wohler ‘s curve, fatigue limit, coefficient in fracture mechanics and acoustics characteristics of materials are obtained. The geometrical I.P. for reflected area and crack depth are solved by means reflected and diffraction methods respectively.The microstructure I.P. is solved by means mathematical - ultrasonic dualism. KEY WORDS: Inverse Problems, Solid Mechanics, Ultrasonic.

The paper presents the coefficient inverse problem of heat and mass transfer. It is known that the process of moisture and heat transfer in soil is described by a system of partial differential equations. The inverse coefficient problem is solved for finding the thermophysical characteristics of the heat and mass transfer equations. The initial-boundary conditions for the heat and mass transfer system are set. To solve the problem, the measured values of temperature and moisture are additionally set at the accessible boundary of the considered area. To solve the inverse problem, the Dufort- Frankel scheme and Matlab Optimization Toolbox are used. The Jacobian is used to reduce the number of iterations. The results of numerical calculations with and without Jacobian are presented.

An inverse coefficient problem is considered for a stationary nonlinear convection- diffusion-reaction equation, in which reaction coefficient has a rather common dependence on substance concentration and on spacial variable. The solvability of the considered nonlinear boundary value problem is proved in a general case. The existence of solutions of the inverse problem is proved for the reaction coefficients, which are defined by the product of two functions. The first function depends on a spatial variable, the second one depends nonlinearly on the solution of the boundary value problem. The mentioned inverse problem consists in reconstructing the first function with the help of additional information provided by the solution of the boundary value problem.

: Coefficient Inverse Problems (CIPs) for Partial Differential Equations (PDEs) represent a very important tool for such needs of the Army as imaging of unknown targets hidden in cluttered heterogeneous backgrounds. The goal of this project is the development of globally convergent numerical methods for a wide class of CIPs. These methods are tested on mathematical models of the interest to the Army such as imaging of antipersonnel land mines and targets on battlefields covered by smogs and flames. In our definition global entails: (1) a rigorous convergence analysis that does not depend on the quality of the initial guess, and (2) numerical simulations that confirm the advertised convergence property. A conventional way to solve a CIP is via the minimization of a least squares objective functional. This functional characterizes misfit between the data and the solution of that PDE with a for the unknown coefficient. However, it is well known to researchers working on computations of inverse problems that the phenomenon of multiple local minima of these functionals represents the major obstacle for the development of reliable numerical methods for multidimensional CIPs. This phenomenon in turn is caused by the above mentioned non-linearity and ill-posedness. Because of local minima, one should somehow guess in advance about a good approximation for the solution. Without the availability of a first good guess, however, there is no guarantee that the calculated coefficient is indeed close to the correct one. In our terminology these are locally convergent numerical methods. In other words, their convergence to the correct solution can be guaranteed only if the starting point is located in a small neighborhood of this solution. Because of local minima, conventional numerical methods for multidimensional CIPs are locally convergent ones. However, in many important applications the first good guess is unavailable.

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

An inverse coefficient problem for a mathematical model of desorption dynamics is considered. The inverse problem is reduced to a nonlinear operator equation for an unknown coefficient. The operator equation is used to construct an iterative numerical method for solving the inverse problem. To prove the convergence of the iterative method, the contraction mappings principle is used. Examples of the application of the iterative method for the numerical solution of the inverse problem are given.

Channel modeling is a challenging vital step towards the development of diffusion-based molecular communication networks (DMCNs). Analytical approaches for diffusion channel modeling are limited to simple and specific geometries and boundary conditions. Also, simulation- and experiment-driven methods are very time-consuming and computationally complex. In this paper, the channel model for DMCN employing the fundamental concentration Green's function (CGF) is characterized. A general homogeneous boundary condition framework is considered that includes any linear reaction systems at the boundaries in the environment. To obtain the CGF for a general DMCN including multiple transmitters, receivers, and other objects with arbitrary geometries and boundary conditions, a semi-analytical method (SAM) is proposed. The CGF linear integral equation (CLIE) is analytically derived. By employing the numerical method of moments, the problem of CGF derivation from CLIE is transformed into an inverse matrix problem. Moreover, a sequential SAM is proposed that converts the inversion problem of a large matrix into multiple smaller matrices reducing the computational complexity. Particle-based simulator confirms the results obtained from the proposed SAM. The convergence and run time for the proposed method are examined. Further, the error probability of a simple diffusion-based molecular communication system is analyzed and examined using the proposed method.

Temperature dependent thermal conductivity determination and source identification for nonlinear heat conduction by means of the Trefftz and homotopy perturbation methods

Multi-parameters identification problem for a degenerate parabolic equation