Transmission eigenvalues have important applications in inverse scattering theory. They can be used to obtain useful information of the physical properties, such as the index of refraction, of the scattering target. Despite considerable effort devoted to the existence and estimation for the transmission eigenvalues, the numerical treatment is limited. Since the problem is non-standard, classica...

This chapter contains an overview of some of the important techniques used to solve linear systems of equations Ax = b (1) by iterative methods. We consider methods based on two general ideas, splittings of the coeecient matrix, leading to stationary iterative methods, and Krylov subspace methods. These two ideas can also be combined to produce preconditioned iterative methods. In addition, we ...

Preconditioned iterative methods are described for the solution of an elliptic partial differential equation over an unit square region with Robbins boundary conditions. Transform based preconditioners are constructed and analyzed. The motivation is to exploit the fast inversion of transform based systems via the fast transform. We prove that transform based preconditioners can be chosen so tha...

In many applications we have to solve a linear system Ax = b with A ∈ Rn×n and b ∈ Rn given. If n is large the solution of the linear system takes a lot of operations, and standard Gaussian elimination may take too long. But in many cases most entries of the matrix A are zero and A is a so-called sparse matrix. This means each equation only couples very few of the n unknowns x1, . . . , xn. A t...

For many elliptic PDE problems, finite-difference and finite-element methods are the techniques of choice. In a finite-difference approach, a solution uk on a set of discrete gridpoints 1, . . . , k is searched for. The discretized partial differential equation and boundary conditions create linear relationships between the different values of uk. In the finite-element method, the solution is e...

This tutorial paper discusses the use of iterative restoration algorithms for the removal of linear blurs from photographic Images which may also be assumed to be degraded by pointwise nonlineariries such as film saturation and additive noise. lterative algorithms are particularly attractive for this application because they allow for the incorporation of various types of prior knowledge about ...

In these lectures, I will describe a simple iterative method that supplies new proofs of integrality of linear characterizations of various basic problems in combinatorial optimization, and also allows adaptations to design approximation algorithms for NP-hard variants of these problems involving extra “degree-like” budget constraints. It is inspired by Jain’s iterative rounding method for desi...

SUMMARY We present several preconditioned conjugate gradient methods for the iterative solution of the linear, symmetric systems of equations arising from the nite element method in displacement form, both for the h-version and the p-version. The preconditioners are based on a decomposition of the solution space into overlapping subspaces and solving separately on each subspace. A judicious cho...

Iterative methods based on small functions are used both to show local surjectivity of certain operators and a fixed point property of mappings on scales of complete metric spaces.

Although image restoration methods based on spectral filtering techniques are very efficient, they can be applied only to problems with fairly simple spatially invariant blurring operators. Iterative methods, however, are much more flexible; they can be very efficient for spatially invariant as well as spatially variant blurs, they can incorporate a variety of regularization techniques and boun...