Marching schemes for inverse acoustic scattering problems

Summary For the numerical solution of inverse Helmholtz problems the boundary value problem for a Helmholtz equation with spatially variable wave number has to be solved repeatedly. For large wave numbers this is a challenge. In the paper we reformulate the inverse problem as an initial value problem, and describe a marching scheme for the numerical computation that needs only n² log n operations on an n × n grid. We derive stability and error estimates for the marching scheme. We show that the marching solution is close to the low-pass filtered true solution. We present numerical examples that demonstrate the efficacy of the marching scheme.     

Buchbesprechungen

Ohne Zusammenfassung

---

Book reviews

---

Notices bibliographiques

     

Low-temperature properties of Samarium monopnictides

SmP, SmAs, SmSb and SmBi order antiferromagnetically at Néel temperatures of 1.6, 1.8, 2.1 and 9 K, respectively. The crystal-field splittings _5/2 between the ₇ doublet and the ₈ quartet of theJ=5/2 ground state of the Sm³⁺ ion have been derived from caloric and magnetic measurements in the paramagnetic temperature range.     

Aus den Berichten der Commission für oceanographische Forschungen

Ohne Zusammenfassung     

über Bronzen aus Ephesus

Ohne Zusammenfassung     

Numerische Behandlung singulärer Sturm-Liouville-Probleme

Summary Generalizing the method of Wendroff [9] and using an estimate for the square integral of a normed eigenfunction outside a compact set, bounds are obtained for the eigenvalues of singular Sturm-Liouville problems from a finite difference method. The number of mesh points necessary to obtain. the accuracy behaves like ^–&frac; ln if tends to zero. Some numerical examples are given.     

Das Differenzenverfahren für singuläre Rand-Eigenwertaufgaben gewöhnlicher Differentialgleichungen

Summary The boundary value problem for a class of singular second order differential operators is defined. Using the standard three point discretisation for the differential equation and taking care of the limits involved in the boundary conditions in a natural way, finite difference approximations to the boundary value problems are defined and their convergence properties are investigated. The rate of convergence is given in terms of the data. It turns out that for problems of the first kind extrapolation is possible up to an arbitrary order after a suitable change of the independent variable, whereas for problems of the second kind neither theoretical nor numerical results indicate the possibility of extrapolation. Corresponding results hold for the eigenvalue problems. Some numerical examples show that the convergence rates given in the paper are best possible and demonstrate the effect of extrapolation.