An inverse polynomial method for the identification of the leading coefficient in the Sturm-Liouville operator from boundary measurements
An inverse polynomial method of determining the unknown leading coefficient k=k(x) of the linear Sturm-Liouville operator Au=-(k(x)u^'(x))^'+q(x)u(x), x@?(0,1), is presented. As an additional condition only two measured data at the boundary (x=0,x=1) are used. In absence of a singular point (u^'(x) 0,u^''(x) 0,@?x@?[0,1]) the inverse problem is classified as a well-conditioned . If there exists at least one singular point, then the inverse problem is classified as moderately ill-conditioned (u^'(x"0)=0,x"0@?(0,1);u^'(x) 0,@?x x"0;u^''(x) 0,@?x@?[0,1]) and severely ill-conditioned (u^'(x"0)=u^''(x"0)=0,x"0@?(0,1);u^'(x) 0,u^''(x) 0,@?x x"0). For each of the cases direct problem solution is approximated by corresponding polynomials and the inverse problem is reformulated as a Cauchy problem for to the first order differential equation with respect the unknown function k=k(x). An approximate analytical solution of the each Cauchy problems are derived in explicit form. Numerical simulations all the above cases are given for noise free and noisy data. An accuracy of the presented approach is demonstrated on numerical test solutions.
Error Analysis of an Equation Error Method for the Identification of the Diffusion Coefficient in a Quasi-linear Parabolic Differential Equation
We consider the inverse problem of reconstructing the diffusion coefficient in a quasi-linear parabolic differential equation in divergence form from measurements of the solution at a finite number of points in the interior of the domain. An equation error method is developed which transforms the inverse problem into a system of linear operator equations for the diffusion coefficient and which can be solved by the conjugate gradient method in a very efficient and stable manner. A detailed error analysis relates the required number of measurements with their accuracy. Numerical results illustrate the performance of the method.
A brief introduction to numerical analysis
Lecture 1: metric space some useful definitions nested balls normed space popular vector norms matrix norms equivalent norms operator norms. Lecture 2: scalar product length of a vector isometric matricies preservation of length and unitary matricies Schur theorum normal matricies positive definite matricies the singular value decomposition unitarily invariant norms a short way to the SVD approximations of a lower rank smoothness and ranks. Lecture 3: perturbation theory condition of a matrix convergent matricies and series the simplest iteration method inverses and series condition of a linear system consistency of matrix and right-hand side eigenvalue perturbations continuity of the polynomial roots. Lecture 4: diagonal dominance Gerschgorin disks small perturbations of eigen values and vectors condition of a simple eigenvalue analitic perturbations. Lecture 5: spectral distances "symmetric" theorums Hoffman-Wielandt theorum permutation vector of a matrix "unnormal" extension eigenvalues of Hermitian matrices interlacing properties what are clusters? singular value clusters eigenvalue clusters. Lecture 6: floating-point numbers computer arithmetic axioms round-off errors for the scalar product forward and backward analysis some philosophy an example of "bad" operation one more example ideal and machine tests up or down solving the triangular systems. Lecture 7: direct methods for linear systems theory of the LU decomposition round-off errors for the LU decomposition growth of matrix entries and pivoting complete pivoting the Cholesky method triangular decompositions and linear systems solution how to refine the solution. Lecture 8: the QR decomposition of a square matrix the QR decomposition of a rectangular matrix householder matrices elimination of elements by reflections Givens matricies elimination of elements by rotations computer realizations of reflections and rotations orthgonalization method loss of orthogonality modified Gram-Schmidt algorithm bidiagonalization unitary similarity reduction to the Hessenberg form. Lecture 9: the eigenvalue problem the power method subspace iterations distances between subspaces subspaces and orthoprojectors distances and orthoprojectors subspaces of equal dimension the CS decomposition convergence of subspace iterations for the block diagonal matrix convergance of subspace iterations in the general case. Lecture 10: the QR algorithm generalised QR algorithm basic formulas the QR iteration lemma convergance of the QR iterations pessimistric and optimistic Bruhat decomposition what if the inverse matrix is not strongly regular the QR iterations and the subspace iterations. Lecture 11: quadratic convergence cubic convergence what makes the QR algorithm efficient implicit QR iterations arrangement of computations how to find the singular value decomposition. Lecture 12: function approximation (Part contents)