a domain decomposition method for the exterior helmholtz problem a new domain decomposition method is presented for the exterior helmholtz problem the nonlocal dirichlet to neumann dtn map is used as a nonreflecting condition on the outer computational boundary the computational domain is divided into nonoverlapping subdomains with sommerfeld type conditions on the adjacent subdomain boundaries to ensure uniqueness an iterative scheme is developed where independent subdomain boundary value problems are obtained by applying the dtn operator to values from the previous iteration the independent problems are then discretized with finite elements and can be solved concurrently numerical results are presented for a two dimensional model problem and both the solution accuracy and convergence rate are investigated

a generalization of the morita mumford classes to extended mapping class groups for surfaces let g 1 be an oriented compact surface of genus g with 1 boundary component and g 1 the mapping class group of g 1 we define a bigraded series of cohomology classes mi j h2i j 2 g 1 jh1 g 1 2i j 2 1 i j 0 when j 0 the class mi 1 0 is the i th morita mumford class mo mu it is proved that hr g 1 sh1 g 1 is generated by mi j s for the case r s 2 and the case g 5 and r s 1 3 especially the johnson homomorphism extended to the whole mapping class group by morita mo3 has an implicit representation by the classes m0 3 and m0 2m1 1 over

a mixed type finite element approximation for radiation problems using fictitious domain method in the finite element approximation of the exterior helmholtz problem we propose an approximation method to implement the dtn mapping formulated as a pseudo differential operator on a computational artificial boundary the method is then combined with the fictitious domain method our method directly gives an approximation matrix for the sesqui linear form for the dtn mapping the eigenvalues of the approximation matrix are simplified to a closed form and can be computed efficiently by using a continued fraction formula solution outside the computational domain and the far field solution can also be computed efficiently by expressing them as operations of pseudo differential operators an inner artificial dtn boundary condition is also implemented by our method we prove the convergence of the solution of our method and compare the performance with the standard finite element approximation based on the fourier series expansion of the dtn operator the efficiency of our method is demonstrated through numerical examples

a priori error estimates of the dtn fem for the transmission problem in acoustics abstract this paper is concerned with a variational approach solving the two dimensional acoustic transmission problems the original problem is reduced to an equivalent nonlocal boundary value problem by introducing an exact dirichlet to neumann dtn mapping in terms of fourier expansion series on an artificial boundary uniqueness and existence of solutions in appropriate sobolev spaces are established for the corresponding variational problem and its modification due to the truncation of dtn mapping a priori error estimates containing the effects of both element meshsize and truncation order of series for the finite element approximation are derived numerical experiments are also presented to illustrate the efficiency and accuracy of the numerical scheme

a relaxation procedure for domain decomposition methods using finite elements we present the convergence analysis of a new domain decomposition technique for finite element approximations this technique was introduced in 11 and is based on an iterative procedure among subdomains in which transmission conditions at interfaces are taken into account partly in one subdomain and partly in its adjacent no global preconditioner is needed in the practice but simply single domain finite element solvers are required an optimal strategy for an automatic selection of a relaxation parameter to be used at interface subdomains is indicated applications are given to both elliptic equations and incompressible stokes equations

analysis of the kleiser schumann method the kleiser schumann algorithm for the approximation of the stokes problem by fourier legendre polynomials is analized stability when the degree of the polynomials increases is established whereas error estimates in sobolev spaces are proven

a triangulation algorithm for fast elliptic solvers based on domain imbedding the following triangulation problem is considered let r be a rectangle and omega an open domain whose closure is contained in r consider a rectangular grid covering r perturb this grid by shifting points close to the boundary partial omega onto partial omega this results in a quadrilateral almost rectangular grid divide each cell of this grid into two triangles along one of its diagonals this results in a triangulation of r a subset of this triangulation is a triangulation of an approximation omega h to omega the distance between partial omega h and partial omega is required to be o h 2 where h denotes the maximum meshwidth of the rectangular grid all triangles should be nondegenerate triangulations of this kind are needed for finite element domain imbedding methods for elliptic boundary value problems a particularly simple example of such a method is reviewed the convergence theory for this method motivates our definition of nondegeneracy of triangles f

sherman morrison woodbury formula based algorithms for the surface smoothing problem abstract surface smoothing applied to range elevation data acquired using a variety of sources has been a very active area of research in computational vision over the past decade generalized splines have emerged as the single most popular approximation tool to this end in this paper we present a new and fast algorithm for solving the surface smoothing problem using a membrane a thin plate or a thin plate membrane spline for data containing discontinuities our approach involves imbedding the surface smoothing problem specified on an irregular domain in the sense of discontinuities and boundaries in a rectangular region using the capacitance matrix method based on the sherman morrison woodbury formula of matrix analysis this formula is used in converting the problem of solving the original linear system resulting from a finite element discretization of the variational formulation of the surface smoothing problem to solving a lyapunov matrix equation or a cascade of two lyapunov matrix equations the reduced problem can then be solved very efficiently using the adi method and the bi conjugate gradient technique our solution requires the generation of a dense capacitance matrix for which we propose a practical and efficient method we demonstrate the efficiency of our algorithm via experiments on sparse data surface smoothing making performance comparisons with the conjugate gradient and hierarchical basis preconditioned conjugate gradient algorithms

relations among tautological classes revisited abstract we give a simple generalisation of a theorem of morita 1989 10 11 which leads to a great number of relations among tautological classes on moduli spaces of riemann surfaces

fictitious domain methods for the numerical solution of two dimensional scattering problems fictitious domain methods for the numerical solution of two dimensional scattering problems are considered the original exterior boundary value problem is approximated by truncating the unbounded domain and by imposing a nonreflecting boundary condition on the artificial boundary first order second order and exact nonreflecting boundary conditions are tested on rectangular and circular boundaries the finite element discretizations of the corresponding approximate boundary value problems are performed using locally fitted meshes and the discrete equations are solved with fictitious domain methods a special finite element method using nonmatching meshes is considered this method uses the macro hybrid formulation based on domain decomposition to couple polar and cartesian coordinate systems a special preconditioner based on fictitious domains is introduced for the arising algebraic saddle point system such that the subspace of constraints becomes invariant with respect to the preconditioned iterative procedure the performance of the new method is compared to the fictitious domain methods both with respect to accuracy and computational cost

fast direct solution of the helmholtz equation with a perfectly matched layer or an absorbing boundary condition we consider the efficient 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 finite 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 reflection of waves caused by an absorbing boundary and a pml and we optimize certain parameters of the layer to minimize the reflection copyright 2003 john wiley sons ltd

brief some applications of randomized algorithms for control system design in this paper a few difficult problems related to simultaneous stabilization of three plants equivalent to a certain problem related to unit interpolation in h have been addressed through the framework of randomized algorithms these problems which were proposed by blondel simultaneous stabilization of linear systems springer berlin 1994 and blondel and gevers math control signals systems 6 1994 135 concern the existence of a controller

domain imbedding methods for the stokes equations we study direct and iterative domain imbedding methods for the stokes equations on certain non rectangular domains in two space dimensions we analyze a continuous analog of numerical domain imbedding for bounded smooth domains and give an example of a simple numerical algorithm suggested by the continuous analysis this algorithms is applicable for simply connected domains which can be covered by rectangular grids with uniformly spaced grid lines in at least one coordinate direction we also discuss a related fft based fast solver for stokes problems with physical boundary conditions on rectangles and present some numerical results

improved homological stability for the mapping class group with integral or twisted coefficients in this paper we prove stability results for the homology of the mapping class group of a surface we get a stability range that is near optimal and extend the result to twisted coefficients

stability of the homology of the mapping class groups of orientable surfaces the mapping class group of f fgs r is f rgs wo a where a is the topological group of orientation preserving diffeomorphisms of f which are the identity on df and fix the s punctures when r 0 and 2g s 3 an important feature of f is that it acts properly discontinuously on the teichmiller space us the quotient is the moduli space f of isometry classes of complete hyperbolic metrics of finite area on f since us is homeomorphic to euclidean space and f is virtually torsion free we have

a damping preconditioner for time harmonic wave equations in fluid and elastic material a physical damping is considered as a preconditioning technique for acoustic and elastic wave scattering the earlier preconditioners for the helmholtz equation are generalized for elastic materials and three dimensional domains an algebraic multigrid method is used in approximating the inverse of damped operators several numerical experiments demonstrate the behavior of the method in complicated two dimensional and three dimensional domains

a parallel fictitious domain method for the three dimensional helmholtz equation the application of a fictitious domain domain embedding method to the three dimensional helmholtz equation with absorbing boundary conditions is considered the finite element discretization is performed by using locally fitted meshes and an algebraic fictitious domain method with a separable preconditioner is used in the iterative solution of the resultant linear systems such a method is based on embedding the original domain into a larger one with a simple geometry with this approach it is possible to realize the gmres iterations in a low dimensional subspace and use the partial solution method to solve the linear systems with the preconditioner an efficient parallel implementation of the iterative algorithm is introduced results of numerical experiments demonstrate good scalability properties on distributed memory parallel computers and the ability to solve high frequency acoustic scattering problems

an implementation of the qmr method based on coupled two term recurrences recently we proposed a new krylov subspace iteration the quasi minimal residual algorithm qmr 1 for solving general nonsingular non hermitian systems of linear equations ax b 1

spectral domain embedding for elliptic pdes in complex domains spectral methods are a class of methods for solving partial differential equations pdes when the solution of the pde is analytic it is known that the spectral solutions converge exponentially as a function of the number of modes used the basic spectral method works only for regular domains such as rectangles or disks domain decomposition methods spectral element methods extend the applicability of spectral methods to more complex geometries an alternative is to embed the irregular domain into a regular one this paper uses the spectral method with domain embedding to solve pdes on complex geometry the running time of the new algorithm has the same order as that for the usual spectral collocation method for pdes on regular geometry the algorithm is extremely simple and can handle dirichlet neumann boundary conditions as well as nonlinear equations

a finite element capacitance matrix method for exterior helmholtz problems we introduce an algorithm for the efficient numerical solution of exterior boundary value problems for the helmholtz equation the problem is reformulated as an equivalent one on a bounded domain using an exact non local boundary condition on a circular artificial boundary an fft based fast helmholtz solver is then derived for a finite element discretization on an annular domain the exterior problem for domains of general shape are treated using an imbedding or capacitance matrix method the imbedding is achieved in such a way that the resulting capacitance matrix has a favorable spectral distribution leading to mesh independent convergence rates when krylov subspace methods are used to solve the capacitance matrix equation

a fast algorithm for the electromagnetic scattering from a large cavity a fast algorithm is presented for solving electromagnetic scattering from a rectangular open cavity embedded in an infinite ground plane the medium inside the cavity is assumed to be vertically layered by introducing a transparent artificial boundary condition the problem in the open cavity is reduced to a bounded domain problem a simple finite difference method is then applied to solve the model helmholtz equation the fast algorithm is designed for solving the resulting discrete system in terms of the discrete fourier transform in the horizontal direction a gaussian elimination in the vertical direction and a preconditioning conjugate gradient method with a complex diagonal preconditioner for the indefinite interface system the existence and uniqueness of the finite difference solution are established for arbitrary wave numbers our numerical experiments for large numbers of mesh points up to 16 million unknowns and for large wave numbers e g between 100 and 200 wavelengths show that the algorithm is extremely efficient the cost for calculating the radar cross section which is of significant interest in practice is o m2 for an m times m mesh the proposed algorithm may be extended easily to solve discrete systems from other discretization methods of the model problem

a composite preconditioner for the electromagnetic scattering from a large cavity in this paper we propose a composite preconditioner for the electromagnetic scattering from a large cavity the electromagnetic cavity problem is described by the helmholtz equation with a nonlocal boundary condition on the aperture of the cavity and dirichlet or neumann boundary conditions on the walls of the cavity the preconditioner proposed here combines the optimal sine transform based approximation with a layered medium model using fast fourier transforms the computational cost of every iteration is o n 2logn on an nxn uniform partition of the unit square numerical results for a model problem show that the new preconditioner is more efficient than those recently considered in the literature for the cavity with a small portion of non layered media we propose a sparse preconditioned conjugate orthogonal conjugate gradient solver combined with the new preconditioner numerical results for a model problem are reported to demonstrate the efficiency of the sparse solver

on the tautological ring of mg we prove that any product of tautological classes of g of degreed in the chow ring of g with rational coefficients vanishes ford g 2 and is proportional to the class of the hyperelliptic locus in degreeg 2

solution to the champagne problem on the simultaneous stabilisation of three plants abstract in this paper the problem proposed in blondel et al siam j optim 32 2 572 590 as an illustration of the difficulty of the simultaneous stabilization problem has been solved the same problem was also mentioned in blondel and gevers math control signals systems 6 1994 135 145 where a bottle of good french champagne was offered for its solution a new open problem is proposed based on the solution of the problem in this paper

a domain decomposition method for the helmholtz equation in a multilayer domain the two dimensional helmholtz equation for problems where the physical domain consists of layers with different material properties is studied an efficient preconditioner for iterative solution of the problem is constructed the problem is discretized with fourth order accurate finite difference operators for the construction of the radiation boundary conditions a fourth order finite element method also is used the large sparse complex indefinite and ill conditioned system of equations that arises is solved with preconditioned restarted gmres a domain decomposition method is used in which the preconditioning is based on the schur complement algorithm with fast poisson type preconditioners for the subdomains the memory requirements for the preconditioner are nearly linear in the number of unknowns the arithmetic complexity for each iteration is low whereas the construction of the preconditioner is a bit more expensive electromagnetic wave propagation in a three layered waveguide is used as a model problem numerical experiments show that convergence is achieved in a few iterations compared with banded gaussian elimination which is a standard solution method the iterative method shows significant gain in both memory requirements and arithmetic complexity furthermore the relative gain grows when the problem size increases

a parallel fictitious domain multigrid preconditioner for the solution of poisson s equation in complex geometries abstract a parallel multilevel preconditioner based on domain decomposition and fictitious domain methods has been presented for the solution of the poisson equation in complicated geometries rectangular blocks with matching grids on interfaces on a structured rectangular mesh have been used for the decomposition of the problem domain sloping sides or curved boundary surfaces are approximated using stepwise surfaces formed by the grid cells a seven point stencil based on the central difference scheme has been used for the discretization of the laplacian for both interior and boundary grid points and this results in a symmetric linear algebraic system for any type of boundary condition the preconditioned conjugate gradient method has been used for the solution of this symmetric system the multilevel preconditioner for the cg is based on a v cycle multigrid applied to the poisson equation on a fictitious domain formed by the union of the rectangular blocks used for the domain decomposition numerical results are presented for two typical poisson problems in complicated geometries one related to heat conduction and the other one arising from the les dns of incompressible turbulent flow over a packed array of spheres these results clearly show the efficiency and robustness of the proposed approach

an algebraic multigrid based shifted laplacian preconditioner for the helmholtz equation a preconditioner defined by an algebraic multigrid cycle for a damped helmholtz operator is proposed for the helmholtz equation this approach is well suited for acoustic scattering problems in complicated computational domains and with varying material properties the spectral properties of the preconditioned systems and the convergence of the gmres method are studied with linear quadratic and cubic finite element discretizations numerical experiments are performed with two dimensional problems describing acoustic scattering in a cross section of a car cabin and in a layered medium asymptotically the number of iterations grows linearly with respect to the frequency while for lower frequencies the growth is milder the proposed preconditioner is particularly effective for low frequency and mid frequency problems

brief condition and algorithm for simultaneous stabilization of linear plants this paper is devoted to the problem of simultaneous stabilization of linear plants it is shown that four given plants p i s i 1 2 3 4 are simultaneously stabilizable if and only if there exist four stable polynomials h i s i 1 2 3 4 which simultaneously satisfy two polynomial equations if we can only find four stable h i s i 1 2 3 4 to satisfy one of the two polynomial equations then two controllers c 1 s and c 2 s having a common denominator or numerator can be obtained such that c 1 s simultaneously stabilizes p 1 s and p 3 s and c 2 s simultaneously stabilizes p 2 s and p 4 s which is called the simultaneous stabilization in groups based on such two controllers a design algorithm for the simultaneous stabilization of three plants is induced

simultaneous stabilization of three or more plants conditions on the positive real axis do not suffice the problem of the simultaneous stabilizability of a finite family of single input single output time invariant systems by a time invariant controller is studied the link between stabilization and avoidance is shown and is used to derive necessary conditions for the simultaneous stabilization of k plants these necessary conditions are proved to be in general not sufficient this result also disproves a long standing conjecture on the stabilizability condition of a single plant with a stable minimum phase controller the main result is to show that unlike the case of two plants the existence of a simultaneous stabilizing controller for more than two plants is not guaranteed by the existence of a controller such that the closed loops have no real unstable poles

error analysis of the dtn fem for the scattering problem in acoustics via fourier analysis in this paper we are concerned with the error analysis for the nite element solution of the two dimensional exterior neumann boundary value problem in acoustics in particular we establish an explicit priori error estimates in h 1 and l 2 norms including both the eect of the truncation of the dtn mapping and that of the numerical discretization to apply the nite element method fem to the exterior problem the original boundary value problem is reduced to an equivalent nonlocal boundary value problem via a dirichlet to neumann dtn mapping represented in terms of the fourier expansion series we discuss essential features of the corresponding variational equation and its modication due to the truncation of the dtn mapping in appropriate function spaces numerical tests are presented to validate our theoretical results

an operator splitting scheme with a distributed lagrange multiplier based fictitious domain method for wave propagation problems we propose a novel fictitious domain method based on a distributed lagrange multiplier technique for the solution of the time dependent problem of scattering by an obstacle we study discretizations that include a fully conforming approach as well as mixed finite element formulations utilizing the lowest order nedelec edge elements in 2d on rectangular grids we also present a symmetrized operator splitting scheme for the scattering problem which decouples the operator that propagates the wave from the operator that enforces the dirichlet condition on the boundary of an obstacle a new perfectly matched layer pml model is developed to model the unbounded problem of interest this model is based on a formulation of the wave equation as a system of first order equations and uses a change of variables approach that has been developed to construct pml s for maxwell s equations we present an analysis of our fictitious domain approach for a one dimensional wave problem based on calculations of reflection coefficients we demonstrate the advantages of our fictitious domain approach over the staircase approximation of the finite difference method we also demonstrate some important properties of the distributed multiplier approach that are not shared by a boundary multiplier fictitious domain approach for the same problem numerical results for two dimensional wave problems that validate the effectiveness of the different methods are presented

the gorenstein conjecture fails for the tautological ring of mathcal overline m _ 2 n we prove that for n equal to at least one of the integers 8 12 16 20 the tautological ring r bullet overline mathcal m 2 n is not gorenstein in fact our n equals the smallest integer such that there is a non tautological cohomology class of even degree on overline mathcal m 2 n by work of graber and pandharipande such a class exists on overline mathcal m 2 20 and we present some evidence indicating that n is in fact 20

incomplete factorization based preconditionings for solving the helmholtz equation preconditioning techniques based on incomplete factorization of matrices are investigated to solve highly indefinite complex symmetric linear systems a novel preconditioning is introduced the real part of the matrix is made positive definite or less indefinite by adding properly defined perturbations to the diagonal entries while the imaginary part is unaltered the resulting preconditioning matrix which is obtained by applying standard methods to the perturbed complex matrix turns out to perform significantly better than classical incomplete factorization schemes for realistic values of the gmres restart parameter spectacular reduction of iteration counts is observed a theoretical spectral analysis is provided in which the spectrum of the preconditioner applied to indefinite matrix is related to the spectrum of the same preconditioner applied to a stieltjes matrix extracted from the indefinite matrix results of numerical experiments are reported which display the efficiency of the new preconditioning copyright 2001 john wiley sons ltd

a finite element capacitance matrix method for the elliptic problem the capacitance matrix method for finite element approximation of the symmetric elliptic problem with the dirichlet condition on an arbitrary region is considered two variants of this method are presented which correspond to the situations when the single and double layer potential are used it is shown that the condition number of the capacitance matrix c is o h 1 for the first variant and the eigenvalues of c are positive and independent of h for the second variant where h is the mesh size of the triangulation

on finite element domain imbedding methods finite element domain imbedding methods for the helmholtz equation with neumann and dirichlet boundary conditions on nonrectangular domains in two space dimensions are considered a survey and comparison of known methods are given and a number of improvements suggested results of numerical experiments are also presented

numerical solution for exterior problems we solve the helmholtz equation in an exterior domain in the plane a perfect absorption condition is introduced on a circle which contains the obstacle this boundary condition is given explicitly by bessel functions we use a finite element method in the bounded domain an explicit formula is used to compute the solution out of the circle we give an error estimate and we present relevant numerical results

domain decomposition technique for the continuity equations of semiconductor device models abstract in this paper we use the pcg method to solve the linear systems obtained from the continuity equations of semiconductor models a new kind of preconditioner which is based on the domain decomposition technique is used this kind of preconditioner is suitable for parallel computing and efficient for the linear system with rapidly varied solutions since the class of the linear system we considered is a special kind of an unsymmetric matrix we modify the pcg to this case as well with our modified pcg method we compare the preconditioner which is constructed by using incomplete cholesky decomposition without fill in preconditioner to our preconditioners which are based on the domain decomposition technique we find that the latter is much more efficient than the former for our semiconductor model problem

error estimates of the dtn finite element method for the exterior helmholtz problem a priori error estimates are established for the dtn dirichlet to neumann finite element method applied to the exterior helmholtz problem the error estimates include the effect of truncation of the dtn boundary condition as well as that of the finite element discretization a property of the hankel functions which plays an important role in the proof of the error estimates is introduced

the κ ring of the moduli of curves of compact type the subalgebra of the tautological ring of the moduli of curves of compact type generated by the classes is studied in all genera relations constructed via the virtual geometry of the moduli of stable quotients are used to obtain minimal sets of generators bases and betti numbers of the rings are computed a universality property relating the higher genus rings to the genus 0 rings is proven using the virtual geometry of the moduli space of stable maps the g formula for hodge integrals arises as the simplest consequence