Multiparameter Inversion: Cramer's Rule for Pseudodifferential Operators

Linearized multiparameter inversion is a model-driven variant of amplitude-versus-offset analysis, which seeks to separately account for the influences of several model parameters on the seismic response. Previous approaches to this class of problems have included geometric optics-based (Kirchhoff, GRT) inversion and iterative methods suitable for large linear systems. In this paper, we suggest an approach based on the mathematical nature of the normal operator of linearized inversion—it is a scaling operator in phase space—and on a very old idea from linear algebra, namely, Cramer's rule for computing the inverse of a matrix. The approximate solution of the linearized multiparameter problem so produced involves no ray theory computations. It may be sufficiently accurate for some purposes; for others, it can serve as a preconditioner to enhance the convergence of standard iterative methods.

Minimization of Heat Sink Mass Using CFD and Mathematical Optimization

This paper describes the use of CFD and mathematical optimization to minimise heat sink mass given a maximum allowable heat sink temperature, a constant cooling fan power and heat source. Heat sink designers have to consider a number of conflicting parameters. Heat transfer is influenced by, amongst others, heat sink properties (such as surface area), airflow bypass and the location of heat sources, whilst size and/or mass of the heat sink needs to be minimized. This multiparameter problem lends itself naturally to optimization techniques. In this study a commercial CFD code, STAR-CD, is linked to the DYNAMIC-Q method of Snyman. Five design variables are considered for three heat source cases. Optimal designs are obtained within six design iterations. The paper illustrates how mathematical optimization can be used to design compact heat sinks for different types of electronic enclosures.

Simple eigenvalues and bifurcation for a multiparameter problem

We are concerned with the problem of bifurcation of solutions of a non-linear multiparameter problem at a simple eigenvalue of the linearised problem.Let X and Y be real Banach spaces, and let A, Bi, i = 1, …, n∈B(X, Y). Let : Rn × X → Y be a non-linear mapping. We consider the equationwhereand λ=(λ1, λ2,…,λn) ∈ Rn is an n-tuple of spectral parameters.

Multiparameter problems and joint spectra

SynopsisIn this paper, we demonstrate an intimate connection between the spectrum of a multiparameter problem and the joint spectrum of an associated set of commuting operators, and show that the spectrum of a multiparameter problem involving bounded operators is non-empty. Multiparameter systems involving compact and self-adjoint operators are considered, and some simplification of results in the literature are noted.