Development of a Numerical Method for Solving the Inverse Cauchy Problem for the Heat Equation

In this work, the initial temperature has been investigated in the Cauchy inverse problem for linear heat conduction equation that it depends on the given temperature at specification time. In this problem, the initial temperature distribution is unknown, but instead, there is a known temperature at the time, t = T > 0. The heat conduction problem can be formulated as Fredholm integral first kind equation. It is well known that this problem is an ill-posed problem and direct solution to this problem is unacceptable. An algorithm has been used to define a finite-dimensional operator for this problem also used the generalized discrepancy method to reduce the conditional extremum variation problem to unconditional extremum variation problem for the integral equation. The discretization of the integral equation has made it possible to reduce this problem to a system of linear algebraic equations. Then, Tikhonov’s regularization inversion method has been used to find an approximation solution. Finally, the numerical computation example has been presented to verify the accuracy of the estimated solution.In this work, the initial temperature has been investigated in the Cauchy inverse problem for linear heat conduction equation that it depends on the given temperature at specification time. In this problem, the initial temperature distribution is unknown, but instead, there is a known temperature at the time, t = T > 0. The heat conduction problem can be formulated as Fredholm integral first kind equation. It is well known that this problem is an ill-posed problem and direct solution to this problem is unacceptable. An algorithm has been used to define a finite-dimensional operator for this problem also used the generalized discrepancy method to reduce the conditional extremum variation problem to unconditional extremum variation problem for the integral equation. The discretization of the integral equation has made it possible to reduce this problem to a system of linear algebraic equations. Then, Tikhonov’s regularization inversion method has been used to find an approximation solution. Finally, the numerical computation example has been presented to verify the accuracy of the estimated solution

Multigrid Method for Solving Inverse Problems for Heat Equation

In this paper, the inverse problems for the boundary value and initial value in a heat equation are posed and solved. It is well known that those problems are ill posed. The problems are reformulated as integral equations of the first kind by using the separation-of-variables method. The discretization of the integral equation allowed us to reduce the integral equation to a system of linear algebraic equations or a linear operator equation of the first kind on Hilbert spaces. The Landweber-type iterative method was used in order to find an approximation solution. The V-cycle multigrid method is used to obtain more frequent and fast convergence for iteration. The numerical computation examples are presented to verify the accuracy and fast computing of the approximation solution

Updating the Landweber Iteration Method for Solving Inverse Problems

The Landweber iteration method is one of the most popular methods for the solution of linear discrete ill-posed problems. The diversity of physical problems and the diversity of operators that result from them leads us to think about updating the main methods and algorithms to achieve the best results. We considered in this work the linear operator equation and the use of a new version of the Landweber iterative method as an iterative solver. The main goal of updating the Landweber iteration method is to make the iteration process fast and more accurate. We used a polar decomposition to achieve a symmetric positive definite operator instead of an identity operator in the classical Landweber method. We carried out the convergence and other necessary analyses to prove the usability of the new iteration method. The residual method was used as an analysis method to rate the convergence of the iteration. The modified iterative method was compared to the classical Landweber method. A numerical experiment illustrates the effectiveness of this method by applying it to solve the inverse boundary value problem of the heat equation (IBVP)

Updating the Landweber Iteration Method for Solving Inverse Problems

The Landweber iteration method is one of the most popular methods for the solution of linear discrete ill-posed problems. The diversity of physical problems and the diversity of operators that result from them leads us to think about updating the main methods and algorithms to achieve the best results. We considered in this work the linear operator equation and the use of a new version of the Landweber iterative method as an iterative solver. The main goal of updating the Landweber iteration method is to make the iteration process fast and more accurate. We used a polar decomposition to achieve a symmetric positive definite operator instead of an identity operator in the classical Landweber method. We carried out the convergence and other necessary analyses to prove the usability of the new iteration method. The residual method was used as an analysis method to rate the convergence of the iteration. The modified iterative method was compared to the classical Landweber method. A numerical experiment illustrates the effectiveness of this method by applying it to solve the inverse boundary value problem of the heat equation (IBVP)