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Abstract: In this paper we present an iterative method, inspired by the inverse iteration with shift technique of finite linear algebra, designed to find the eigenvalues and eigenfunctions of the Laplacian with homogeneous Dirichlet boundary condition for arbitrary bounded domains . This method, which has a direct functional analysis approach, does not approximate the eigenvalues of the Laplacian as those of a finite linear operator. It is based on the uniform convergence away from nodal surfaces and can produce a simple and fast algorithm for computing the eigenvalues with minimal computational requirements, instead of using the ubiquitous Rayleigh quotient of finite linear algebra. Also, an alternative expression for the Rayleigh quotient in the associated infinite dimensional Sobolev space which avoids the integration of gradients is introduced and shown to be more efficient. The method can also be used in order to produce the spectral decomposition of any given function . [Copyright &y& Elsevier]

Abstract: This paper is devoted to the class of inverse problems for coefficient identification with an adjoint problem approach related to nonlinear parabolic partial differential equations. The unknown coefficient depends on the gradient of the solution and belongs to a set of admissible coefficients. First, we present a coefficient-to-data mapping. Then, based on maximum principle and the adjoint problem of the direct problem, the integral identities will be obtained. Using these identities, we can show that the coefficient-to-data mapping is continuous and strictly monotone. Furthermore, an approximate solution to the inverse problem is constructed, and the error is analyzed. Finally, the applicability of the method is demonstrated in numerical examples. [Copyright &y& Elsevier]