A parallel fast direct solver based on the Divide & Conquer method for linear systems with separable block tridiagonal matrices is considered. Such systems appear, for example, when discretizing the Poisson equation in a rectangular domain using the ve{point nite di erence scheme or the piecewise linear nite elements on a triangulated rectangular mesh. The Divide & Conquer method has the arithmetical complexity O(N log N), and it is closely related to the cyclic reduction, but instead of using the matrix polynomial factorization the so{called partial solution technique is employed. The method is presented and analyzed in a general base q framework and based on this analysis, the base four variant is chosen for parallel implementation using the MPI standard. The generalization of the method to the case of arbitrary block dimension is described. The numerical experiments show the sequential e ciency and numerical stability of the considered method compared to the well{known BLKTRI{implementation of the generalized cyclic reduction. The good scalability properties of the parallel Divide & Conquer method are demonstrated in a distributed memory Cray T3E computer.

This article proposes an Enhanced Memetic Differential Evolution (EMDE) for designing digital filters which aim at detecting defects of the paper produced during an industrial process. Defect detection is handled by means of two Gabor filters and their design is performed by the EMDE. The EMDE is a novel adaptive evolutionary algorithm which combines the powerful explorative features of Differential Evolution with the exploitative features of three local search algorithms employing different pivot rules and neighborhood generating functions. These local search algorithms are the Hooke Jeeves Algorithm, a Stochastic Local Search, and Simulated Annealing. The local search algorithms are adaptively coordinated by means of a control parameter that measures fitness distribution among individuals of the population and a novel probabilistic scheme. Numerical results confirm that Differential Evolution is an efficient evolutionary framework for the image processing problem under investigation and show that the EMDE performs well. As a matter of fact, the application of the EMDE leads to a design of an efficiently tailored filter. A comparison with various popular metaheuristics proves the effectiveness of the EMDE in terms of convergence speed, stagnation prevention, and capability in detecting solutions having high performance.

SUMMARYWe consider the e cient numerical solution of the Helmholtz equation in a rectangular domain with a perfectly matched layer (PML) or an absorbing boundary condition (ABC). Standard bilinear (trilinear) ÿnite-element discretization on an orthogonal mesh leads to a separable system of linear equations for which we describe a cyclic reduction-type fast direct solver. We present numerical studies to estimate the re ection of waves caused by an absorbing boundary and a PML, and we optimize certain parameters of the layer to minimize the re ection.

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The application of the ctitious domain method to the solution of the threedimensional Helmholtz equation with non-re ecting boundary conditions is considered. The nite element discretization leads to a large-scale linear system, which is solved with preconditioned GMRES iterations. The preconditioner is constructed by use of the algebraic ctitious domain approach, which makes it possible to realize the iterative method in a low-dimensional subspace and thereby use partial solution method to solve the linear systems with the preconditioner. An e cient parallel implementation of the subspace iterations and the partial solution method is studied. Results of numerical experiments demonstrate good scalability properties and ability to solve large scale problems on a Cray T3E.

We consider a controllability technique for the numerical solution of the Helmholtz equation. The original time-harmonic equation is represented as an exact controllability problem for the time-dependent wave equation. This problem is then formulated as a least-squares optimization problem, which is solved by the conjugate gradient method. Such an approach was first suggested and developed in the 1990s by French researchers and we introduce some improvements to its practical realization.We use higher-order spectral elements for spatial discretization, which leads to high accuracy and lumped mass matrices. Higher-order approximation reduces the pollution effect associated with finite element approximation of time-harmonic wave equations, and mass lumping makes explicit time-stepping schemes for the wave equation very efficent. We also derive a new way to compute the gradient of the least-squares functional and use algebraic multigrid method for preconditioning the conjugate gradient algorithm.Numerical results demonstrate the significant improvements in efficiency due to the higher-order spectral elements. For a given accuracy, spectral element method requires fewer computational operations than conventional finite element method. In addition, by using higher-order polynomial basis the influence of the pollution effect is reduced.

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