In this paper, we are concerned with constructing interpolating scaling functions. The presented construction can be interpreted as a natural generalization of a well-known univariate approach and applies to scaling matrices A satisfying |det A| = 2. The resulting scaling functions automatically satisfy certain Strang-Fix-conditions.

We propose an efficient numerical algorithm for relative error model reduction based on balanced stochastic truncation. The method uses full-rank factors of the Gramians to be balanced versus each other and exploits the fact that for large-scale systems these Gramians are often of low numerical rank. We use the easy-to-parallelize sign function method as the major computational tool in determining these full-rank factors and demonstrate the numerical performance of the suggested implementation of balanced stochastic truncation model reduction.

Abstract not found

The quadratic matrix equation AX² +BX +C = 0 in n x n matrices arises in applications and is of intrinsic interest as one of the simplest nonlinear matrix equations. We give a complete characterization of solutions in terms of the generalized Schur decomposition and describe and compare various numerical solution techniques. In particular, we give a thorough treatment of functional iteration methods based on Bernoulli's method. Other methods considered include Newton's method with exact line searches, symbolic solution and continued fractions. We show that functional iteration applied to the quadratic matrix equation can provide an efficient way to solve the associated quadratic eigenvalue problem ( 2 A + B + C)x = 0.

We consider Morozov's discrepancy principle for Tikhonov{regularization of nonlinear operator equations. It is shown that minor restrictions to the operator F already guarantee the existence of a regularization parameter such that ky F (x )k c 1 holds. Moreover, some additional smoothness assumptions on the solution of F (x) = y ensure an optimal convergence rate. Finally we investigate some practically relevant examples, e.g. medical imaging (Single Photon Emission Computed Tomography). It is illustrated that the introduced conditions on F will be met in general by a large class of nonlinear operators.

We report on a new iterative approach for nding a global minimizer of the Tikhonov-Phillips functional with a special class of nonlinear operators F. Assuming that the operator itself can be decomposed into (or approximated by) a sum of a linear and a bilinear operator, we introduce a two-step iteration scheme based on an outer iteration over the regularization parameter and an inner iteration with a steepest descent method. Finally we present numerical results for the reconstruction of the emission function in single photon emission computed tomography (SPECT).

with exact line searches

We report on an iterative approach to reconstruct both the activity f(x) and the attenuation (x) directly from the emission sinogram data. The proposed algorithm is based on the iterative methods for solving linear operator equations. Whenever an operator F is the sum of a linear and a bilinear operator, a modified iteration sequence can be dened. Using a Taylor series about a fixed approximate distribution 0, the attenuated Radon transform can be well approximated as the sum of a linear operator in f and a bilinear operator in f and µ. The algorithm alternates between updates of f and updates of µ. In our test computations, the proposed algorithms achieve good reconstruction results both for generated and real data.

This paper is concerned with the construction of generalized Banach frames on homogeneous spaces. The major tool is a unitary group representation which is square integrable modulo a certain subgroup. By means of this representation, generalized coorbit spaces can be dened. Moreover, we can construct a specic reproducing kernel which, after a judicious discretization, gives rise to Banach frames for these coorbit spaces. We also discuss nonlinear approximation schemes based on our new Banach frames. As a classical example, we apply our construction to the problem of analyzing and approximating functions on the spheres.

This paper describes the design of a software library for solving the basic computational problems that arise in analysis and synthesis of linear control systems. The library is intended for use in high performance computing environments based on parallel distributed memory architectures. The portability of the library is ensured by using the BLACS, PBLAS, and ScaLAPACK as the basic layer of communication and computational routines. Preliminary numerical results demonstrate the performance of the developed codes on parallel computers. The suggested library can serve as a basic layer for PSLICOT, a parallel extension of the Subroutine Library in Control Theory (SLICOT). 1 Introduction In recent years, many new and reliable numerical methods have been developed for analysis and synthesis of moderate size linear time-invariant (LTI) systems. In generalized state-space form, such systems are described by the following models. Continuous-time LTI system: E x(t) = Ax(t) +Bu(t); t ? 0; ...