Spectral methods for linear inverse problems with unbounded operators

Inverse spectral problems for Sturm-Liouville operators with transmission conditions

Abstract: This paper deals with the boundary value problem involving the differential equation                      -y''+q(x)y=lambda y                                 subject to the standard boundary conditions along with the following discontinuity conditions at a point              y(a+0)=a1y(a-0),    y'(a+0)=a2y'(a-0)+a3y(a-0).  We develop the Hochestadt-Lieberman’s result for Sturm-Lio...

Domain Decomposition Methods for Linear Inverse Problems with Sparsity Constraints

Quantities of interest appearing in concrete applications often possess sparse expansions with respect to a preassigned frame. Recently, there were introduced sparsity measures which are typically constructed on the basis of weighted l1 norms of frame coefficients. One can model the reconstruction of a sparse vector from noisy linear measurements as the minimization of the functional defined by...

Ill-posed problems with unbounded operators

Let A be a linear, closed, densely defined unbounded operator in a Hilbert space. Assume that A is not boundedly invertible. If Eq. (1) Au = f is solvable, and ‖fδ − f ‖ δ, then the following results are provided: Problem Fδ(u) := ‖Au− fδ‖2 + α‖u‖2 has a unique global minimizer uα,δ for any fδ , uα,δ = A∗(AA∗ + αI)−1fδ . There is a function α = α(δ), limδ→0 α(δ)= 0 such that limδ→0 ‖uα(δ),δ − y...

Stochastic spectral methods for efficient Bayesian solution of inverse problems

We present a reformulation of the Bayesian approach to inverse problems, that seeks to accelerate Bayesian inference by using polynomial chaos (PC) expansions to represent random variables. Evaluation of integrals over the unknown parameter space is recast, more efficiently, as Monte Carlo sampling of the random variables underlying the PC expansion. We evaluate the utility of this technique on...

Uniqueness Theorems for Multidimensional inverse Problems with Unbounded Coefficients