An Adaptive Wavelet Method for Solving High-Dimensional Elliptic PDEs

Adaptive tensor product wavelet methods are applied for solving Poisson’s equation, as well as anisotropic generalizations, in high space dimensions. It will be demonstrated that the resulting approximations converge in energy norm with the same rate as the best approximations from the span of the best N tensor product wavelets, where moreover the constant factor that we may lose is independent of the space dimension n. The cost of producing these approximations will be proportional to their length with a constant factor that may grow with n, but only linearly.     

Ueber den Stickstoffgehalt des Nitroglycerins im Dynamit

Ohne Zusammenfassung     

Stereoselective 6π-electron electrocyclic ring closures of 2-halo-amidotrienes via a remote 1,6-asymmetric induction

A diastereoselective 6π-electrocyclic ring closure employing halogen-substituted 3-amidotrienes via a 1,6-remote asymmetric induction is described. This new asymmetric manifold for pericyclic ring closure further underscores the significance of the allenamide chemistry.     

High-Dimensional Adaptive Sparse Polynomial Interpolation and Applications to Parametric PDEs

We consider the problem of Lagrange polynomial interpolation in high or countably infinite dimension, motivated by the fast computation of solutions to partial differential equations (PDEs) depending on a possibly large number of parameters which result from the application of generalised polynomial chaos discretisations to random and stochastic PDEs. In such applications there is a substantial advantage in considering polynomial spaces that are sparse and anisotropic with respect to the different parametric variables. In an adaptive context, the polynomial space is enriched at different stages of the computation. In this paper, we study an interpolation technique in which the sample set is incremented as the polynomial dimension increases, leading therefore to a minimal amount of PDE solving. This construction is based on the standard principle of tensorisation of a one-dimensional interpolation scheme and sparsification. We derive bounds on the Lebesgue constants for this interpolation process in terms of their univariate counterpart. For a class of model elliptic parametric PDE’s, we have shown in Chkifa et al. (Modél. Math. Anal. Numér. 47(1):253–280, 2013) that certain polynomial approximations based on Taylor expansions converge in terms of the polynomial dimension with an algebraic rate that is robust with respect to the parametric dimension. We show that this rate is preserved when using our interpolation algorithm. We also propose a greedy algorithm for the adaptive selection of the polynomial spaces based on our interpolation scheme, and illustrate its performance both on scalar valued functions and on parametric elliptic PDE’s.     

Studies of the out-of-plane emission of pions in symmetric heavy-ion collisions

Triple differential cross-sections of midrapidity pions from ²⁰⁹Bi+²⁰⁹Bi collisions were measured with the Kaon Spectrometer at SIS at incident energies of 400, 700 and 1000 AMeV. The azimuthal emission pattern of the pions has been investigated in dependence of beam energy and impact parameter. An enhanced emission of pions perpendicular to the reaction plane is observed. The strength of the anisotropy increases with beam energy and pion transverse momentum. In contrast to the nucleons the anisotropy varies only little with the impact parameter. No difference in the behaviour of positive and negative pions is observed.     

Quadrature for hp-Galerkin BEM in {\hbox{\sf l\kern-.13em R}}^3

Summary. The Galerkin discretization of a Fredholm integral equation of the second kind on a closed, piecewise analytic surface is analyzed. High order, -boundary elements on grids which are geometrically graded toward the edges and vertices of the surface give exponential convergence, similar to what is known in the -Finite Element Method. A quadrature strategy is developed which gives rise to a fully discrete scheme preserving the exponential convergence of the -Boundary Element Method. The total work necessary for the consistent quadratures is shown to grow algebraically with the number of degrees of freedom. Numerical results on a curved polyhedron show exponential convergence with respect to the number of degrees of freedom as well as with respect to the CPU-time. Received April 22, 1996