Application of the Ritz–Galerkin method for recovering the spacewise-coefficients in the wave equation

An inverse problem for the determination of the unknown spacewise-dependent coefficients in a hyperbolic equation through additional boundary measurements is considered. For the sake of simplicity, the problem has been considered in one dimension, however the method is applicable for problems with regular and bounded domains in higher dimensions. The technique provides space–time approximations for the wave equation by expanding the required approximate solutions using the Bernstein multi-scaling functions. The key feature of the approach is applying the Ritz–Galerkin method along with utilizing the satisfier function which fulfills all the initial and boundary conditions as well. As the consequences, only a low number of basis is required to find satisfactory results which lead to reliably less computations in comparison to other published methods. Despite the assumptions of sufficiently smooth initial and boundary conditions to guarantee a unique solution, this solution is unstable. This fact is confirmed by the numerical findings that show the errors in the approximations are relatively large, i.e. they are not of the same order, compared to the noise levels greater than 3%. Hence the problem is ill-posed. To alleviate the difficulties arising from solving the ill-posed problems, a type of particular regularization technique which consists in repeatedly linearizing the operator equation F(x)=yF(x)=y is utilized. Numerical results are presented for the typical benchmark test examples, which have the input measured data perturbed by increasing amounts of noise. The numerical investigations are included, showing that accurate and stable results can be obtained efficiently with small computational cost.