Adjoint-based optimal control using meshfree discretizations

AbstractThe paper at hand presents a combination of optimal control approaches for PDEs with meshless discretizations. Applying a classical Lagrangian type particle method to optimization problems with hyperbolic constraints, several adjoint-based strategies differing in the sequential order of optimization and discretization of the Lagrangian or Eulerian problem formulation are proposed and compared. The numerical results confirm the theoretically predicted independence principle of the optimization approaches and show the expected convergence behavior. Moreover, they exemplify the superiority of meshless methods over the conventional mesh-based approaches for the numerical handling and optimization of problems with time-dependent geometries and freely moving boundaries

Schnelle Löser für partielle Differentialgleichungen

The workshop Schnelle Löser für partielle Differentialgleichungen, organised by Randolph E. Bank (La Jolla), Wolfgang Hackbusch(Leipzig), Gabriel Wittum (Heidelberg) was held May 22nd - May 28th, 2005. This meeting was well attended by 47 participants with broad geographic representation from 9 countries and 3 continents. This workshop was a nice blend of researchers with various backgrounds

Multiscale Partition of Unity

We introduce a new Partition of Unity Method for the numerical homogenization of elliptic partial differential equations with arbitrarily rough coefficients. We do not restrict to a particular ansatz space or the existence of a finite element mesh. The method modifies a given partition of unity such that optimal convergence is achieved independent of oscillation or discontinuities of the diffusion coefficient. The modification is based on an orthogonal decomposition of the solution space while preserving the partition of unity property. This precomputation involves the solution of independent problems on local subdomains of selectable size. We deduce quantitative error estimates for the method that account for the chosen amount of localization. Numerical experiments illustrate the high approximation properties even for 'cheap' parameter choices.Comment: Proceedings for Seventh International Workshop on Meshfree Methods for Partial Differential Equations, 18 pages, 3 figure

New Discretization Methods for the Numerical Approximation of PDEs

The construction and mathematical analysis of numerical methods for PDEs is a fundamental area of modern applied mathematics. Among the various techniques that have been proposed in the past, some – in particular, finite element methods, – have been exceptionally successful in a range of applications. There are however a number of important challenges that remain, including the optimal adaptive finite element approximation of solutions to transport-dominated diffusion problems, the efficient numerical approximation of parametrized families of PDEs, and the efficient numerical approximation of high-dimensional partial differential equations (that arise from stochastic analysis and statistical physics, for example, in the form of a backward Kolmogorov equation, which, unlike its formal adjoint, the forward Kolmogorov equation, is not in divergence form, and therefore not directly amenable to finite element approximation, even when the spatial dimension is low). In recent years several original and conceptionally new ideas have emerged in order to tackle these open problems. The goal of this workshop was to discuss and compare a number of novel approaches, to study their potential and applicability, and to formulate the strategic goals and directions of research in this field for the next five years

Parabolic Partial Differential Equations with Border Conditions of Dirichlet as Inverse Moments Problem

We considerer parabolic partial differential equations. We will see that an approximate solution can be found using the techniques of generalized inverse moments problem and also bounds for the error of estimated solution. First we transform the parabolic partial differential equation to the integral equation. Using the inverse moments problem techniques we obtain an approximate solution. Then we find a numerical approximation of when solving the integral equation, because solving the previous integral equation is equivalent to solving the equationGrupo de Aplicaciones Matemáticas y Estadísticas de la Facultad de Ingeniería (GAMEFI)Facultad de Ingeniería (FI

Meshless Euler solver using radial basis functions for solving inviscid compressible flows

The goal of this research work is to develop a meshless Euler solver using radial basis functions (RBFs). Meshless methods attempt to address the problems in computational methods arising due to their mesh dependence. The present meshless method uses the differential quadrature (DQ) technique, which states that the derivatives of a function at a point can be approximated by a linear combination of the function values at a set of scattered points or nodes in its neighborhood. The derivative evaluation is dependent only on the nodal distribution and independent of the function. RBFs are used as basis functions for the DQ technique.;The local radial basis function-differential quadrature (RBF-DQ) method is used to develop a meshless Euler solver for inviscid compressible flows. An Euler solver should take into account the direction of wave propagation associated with the hyperbolic PDEs. Hence second order a Rusanov solver is employed to evaluate fluxes at the mid-points, analogous to flux evaluation at cell interface in the finite volume method. The DQ technique is then applied to these upwind fluxes to approximate the flux gradients. Thus the conservative form of the Euler equation in differential form is discretized using RBF-DQ technique. The solver is applied to and validated by various steady state compressible flows. The present meshless Euler solver using RBFs captures the flow physics both qualitatively and quantitatively