Countries

parametric approach, additive preconditioners When one disposes of, or is able to construct a hierarchy of either physical, or purely numerical models for the same physical situation, it is natural to attempt to devise a numerical method in which the fine model is in a way to precondition the coarse. In doing this, one expects to gain efficiency, and optimally, to achieve a numerical method whose convergence characteristics are independent of certain numerical factors such as local grid size, degree of representation, etc. A prototype for such hierarchical methods is provided by the multigrid method for solving a set of discretized partial differential equations (PDE), typically a boundary-value problem of elliptic type. There, the hierarchy is associated with a sequence of grids of various degrees of refinement and the related discretizations of the PDE problem. In the lecture, we begin by reviewing classical results about multigrid to demonstrate the effectiveness of such multilevel hierarchical methods, which for the model problem achieve the optimal linear convergence: equivalently, the cost for solving a discrete problem with N degrees of freedom is (only) proportional to N. Here, N is typically proportional to the number of gridpoints in a fine discretization. This leads us naturally to raise the following question: how can hierarchical concepts be used to achieve higher, if not optimal efficiency in a numerical shape optimization related to a distributed (PDE) problem, such as, aerodynamic shape optimization of a body (wing, or configuration) immersed in a compressible flow governed by the Euler equations. Severa