Identification of a multi-dimensional space-dependent heat source from boundary data

We investigate the linear but ill-posed inverse problem of determining a multi-dimensional space-dependent heat source in the parabolic heat equation from Cauchy boundary data. This model is important in practical applications where the distribution of internal sources is to be monitored and controlled with care and accuracy from non-invasive and non-intrusive boundary measurements only. The mathematical formulation ensures that a solution of the inverse problem is unique but the existence and stability are still issues to be dealt with. Even if a solution exists it is not stable with respect to small noise in the measured boundary data hence the inverse problem is still ill-posed. The Landweber method is developed in order to restore stability through iterative regularization. Furthermore, the conjugate gradient method is also developed in order to speed up the convergence. An alternating direction explicit finite-difference method is employed for discretising the well-posed problems resulting from these iterative procedures. Numerical results in two-dimensions are illustrated and discussed

Simultaneous determination of two unknown thermal coefficients through a mushy zone model with an overspecified convective boundary condition

The simultaneous determination of two unknown thermal coefficients for a semi-infinite material under a phase-change process with a mushy zone according to the Solomon-Wilson-Alexiades model is considered. The material is assumed to be initially liquid at its melting temperature and it is considered that the solidification process begins due to a heat flux imposed at the fixed face. The associated free boundary value problem is overspecified with a convective boundary condition with the aim of the simultaneous determination of the temperature of the solid region, one of the two free boundaries of the mushy zone and two thermal coefficients among the latent heat by unit mass, the thermal conductivity, the mass density, the specific heat and the two coefficients that characterize the mushy zone. The another free boundary of the mushy zone, the bulk temperature and the heat flux and heat transfer coefficients at the fixed face are assumed to be known. According to the choice of the unknown thermal coefficients, fifteen phase-change problems arise. The study of all of them is presented and explicit formulae for the unknowns are given, beside necessary and sufficient conditions on data in order to obtain them. Formulae for the unknown thermal coefficients, with their corresponding restrictions on data, are summarized in a table.Comment: 27 pages, 1 Table, 1 Appendi

Convergence analysis of a Crank-Nicolson Galerkin method for an inverse source problem for parabolic equations with boundary observations

This work is devoted to an inverse problem of identifying a source term depending on both spatial and time variables in a parabolic equation from single Cauchy data on a part of the boundary. A Crank-Nicolson Galerkin method is applied to the least squares functional with an quadratic stabilizing penalty term. The convergence of finite dimensional regularized approximations to the sought source as measurement noise levels and mesh sizes approach to zero with an appropriate regularization parameter is proved. Moreover, under a suitable source condition, an error bound and corresponding convergence rates are proved. Finally, several numerical experiments are presented to illustrate the theoretical findings.Comment: Inverse source problem, Tikhonov regularization, Crank-Nicolson Galerkin method, Source condition, Convergence rates, Ill-posedness, Parabolic proble

Inverse space-dependent force problems for the wave equation

The determination of the displacement and the space-dependent force acting on a vibrating structure from measured final or time-average displacement observation is thoroughly investigated. Several aspects related to the existence and uniqueness of a solution of the linear but ill-posed inverse force problems are highlighted. After that, in order to capture the solution a variational formulation is proposed and the gradient of the least-squares functional that is minimized is rigorously and explicitly derived. Numerical results obtained using the Landweber method and the conjugate gradient method are presented and discussed illustrating the convergence of the iterative procedures for exact input data. Furthermore, for noisy data the semi-convergence phenomenon appears, as expected, and stability is restored by stopping the iterations according to the discrepancy principle criterion once the residual becomes close to the amount of noise. The present investigation will be significant to researchers concerned with wave propagation and control of vibrating structures

Leak identification in saturated unsteady flow via a Cauchy problem

This work is an initial study of a numerical method for identifying multiple leak zones in saturated unsteady flow. Using the conventional saturated groundwater flow equation, the leak identification problem is modelled as a Cauchy problem for the heat equation and the aim is to find the regions on the boundary of the solution domain where the solution vanishes, since leak zones correspond to null pressure values. This problem is ill-posed and to reconstruct the solution in a stable way, we therefore modify and employ an iterative regularizing method proposed in [1] and [2]. In this method, mixed well-posed problems obtained by changing the boundary conditions are solved for the heat operator as well as for its adjoint, to get a sequence of approximations to the original Cauchy problem. The mixed problems are solved using a Finite element method (FEM), and the numerical results indicate that the leak zones can be identified with the proposed method

Identification of conductivity in inhomogeneous orthotropic media

Purpose - The purpose of this paper is to solve numerically the identification of the thermal conductivity of an inhomogeneous and possibly anisotropic medium from interior/internal temperature measurements. Design/methodology/approach - The formulated coefficient identification problem is inverse and ill-posed and therefore, in order to obtain a stable solution, a nonlinear regularized least-squares approach is employed. For the numerical discretisation of the orthotropic heat equation, the finite-difference method is applied, whilst the nonlinear minimization is performed using the MATLAB toolbox routine lsqnonlin. Findings - Numerical results show the accuracy and stability of solution even in the presence of noise (modelling inexact measurements) in the input temperature data. Research limitations/implications - The mathematical formulation uses temporal tem- perature measurements taken at many points inside the sample and this may be too much information that is provided to identify a spacewise dependent only conductivity tensor. Practical implications - Since noisy data are inverted, the study models real situations in which practical temperature measurements recorded using thermocouples are inherently contaminated with random noise. Social implications - The identification of the conductivity of inhomogeneous and orthotropic media will be of great interest to the inverse problems community with applications in geophysics, groundwater flow and heat transfer. Originality/value - The current investigation advances the field of coefficient identification problems by generalising the conductivity to be orthotropic in addition of being heterogeneous. The originality lies in performing, for the first time, numerical simulations of inver- sion to find the anisotropic and inhomogeneous thermal conductivity form noisy temperature measurements. Further value and physical significance is brought in by determining the degree of cure in a resin transfer molding process, in addition to obtaining the inhomogeneous thermal conductivity of the tested material

Determination of a time-dependent diffusivity in a nonlinear parabolic problem

In this article, an inverse nonlinear convection-diffusion problem is considered for the identification of an unknown solely time-dependent diffusion coefficient in a subregion of a bounded domain in . The missing data are compensated by boundary observations on a part of the surface of the subdomain: the total flux through that surface or the values of the solution at that surface are measured. Two solution methods are discussed. In both cases, the solvability of the problem is proved using coefficient to data mappings. More specific, a nonlinear numerical algorithm based on Rothe's method is designed and the convergence of approximations towards the weak solution in suitable function spaces is shown. In the proofs, also the monotonicity methods and the Minty-Browder argument are employed. The results of numerical experiments are discussed

Optimal Control Method of Parabolic Partial Differential Equations and Its Application to Heat Transfer Model in Continuous Cast Secondary Cooling Zone

Our work is devoted to a class of optimal control problems of parabolic partial differential equations. Because of the partial differential equations constraints, it is rather difficult to solve the optimization problem. The gradient of the cost function can be found by the adjoint problem approach. Based on the adjoint problem approach, the gradient of cost function is proved to be Lipschitz continuous. An improved conjugate method is applied to solve this optimization problem and this algorithm is proved to be convergent. This method is applied to set-point values in continuous cast secondary cooling zone. Based on the real data in a plant, the simulation experiments show that the method can ensure the steel billet quality. From these experiment results, it is concluded that the improved conjugate gradient algorithm is convergent and the method is effective in optimal control problem of partial differential equations

Inverse force problems for the wave equation

Inverse problems have become more and more important in various fields of science and technology, and have certainly been one of the fastest growing areas in applied mathematics over the last three decades. However, as inverse problems typically lead to mathematical models which are ill-posed, their solutions are unstable under data perturbations and classical numerical techniques fail to provide accurate and stable solutions. The work in thesis focuses on inverse force problems for the wave equation which consists of determining an unknown space/time-dependent force function acting on a vibrating structure from Cauchy boundary, final time displacement or integral data. The novel contribution of this thesis involves the development of efficient numerical algorithms for these inverse but ill-posed problems. We have used the boundary element method (BEM) to discretise the wave equation with a constant wave speed, and the finite difference method (FDM) for non-constant wave speed and/or inhomogeneous wave propagating medium. Imposing the available boundary and additional conditions, upon discretisation the inverse and ill-posed problem is recast into one of solving an ill- conditioned system of equations. The accuracy and convergence of the numerical results are investigated for various test force functions. The stability of the numerical solutions is investigated by introducing random noise into the input data which yields unstable results if no regularisation is used. The Tikhonov regularization method is employed in order to reduce the influence of the measurement errors on the numerical results. The choice of the regularization parameter is based on trial and error or on the L-curve criterion. Iterative regularizing methods such as the Landweber and conjugate gradient methods are also employed in one chapter. The inverse numerical solutions are compared with their known analytical solutions, where available, and with the corresponding direct numerical solutions otherwise