optimized schwarz method with complete radiation transmission conditions for the helmholtz equation in waveguides in this paper we present a nonoverlapping optimized schwarz algorithm with a high order transmission condition for the helmholtz equation posed in waveguides we introduce the new high order transmission conditions based on the complete radiation boundary conditions crbcs that have been developed for high order absorbing boundary conditions we obtain the convergence rate of the algorithm in terms of reflection coefficients of crbcs which decrease exponentially with the order of crbcs it will be shown that damping parameters involved in the transmission conditions can be selected in an optimal way for enhancing the convergence of the schwarz algorithm also this algorithm can be employed efficiently for a preconditioner in gmres implementations based on the substructured form as in m j gander f magoules and f nataf siam j sci comput 24 2002 pp 38 60 finally numerical examples confirming the theory will be presented

a new approach to inverse spectral theory i fundamental formalism we present a new approach distinct from gelfand levitan to the the orem of borg marchenko that the m function equivalently spectral measure for a finite interval or half line schrodinger operator determines the poten tial our approach is an analog of the continued fraction approach for the moment problem we prove there is a representation for the m function m 2 r b

an exact method for the calculation of certain titchmarsh weyl m functions we consider the second order linear differential equation where q is real valued in the case we calculate the titchmarsh weyl m function associated with

asymptotics of the titchmarsh weyl m coefficient for integrable potentials ii this article is concerned with the titchmarsh weyl m coefficient for the differential equation y q x y 0 on 0 whee the potential q is integrable near 0 the m coefficient which is a function of the complex variable c depends on an initialization parameter a uniform as well as various non uniform asymptotic expansions of m are derived which are valid as in any sector of the complex plane that does not contain the real axis the results reported here extend those of kaper and kwong proc roy soc edinburgh sect a to appear

optimized double sweep schwarz method by complete radiation boundary conditions we present an optimized double sweep nonoverlapping schwarz method for solving the helmholtz equation in semi infinite waveguides the domain is decomposed into nonoverlapped layered subdomains along the axis of the waveguide and local wave propagation problems equipped with complete radiation conditions for high order absorbing boundary conditions are solved forward and backward sequentially for communication between subdomains neumann data of local solutions in one domain are transferred to the neighboring subdomain in the forward direction and dirichlet data are exploited in the backward direction the complete radiation boundary conditions enable us to not only minimize reflection coefficients for most important modes in an optimal way but also find neumann data without introducing errors that would be produced if finite difference formulas were used for computing neumann data the convergence of the double sweep schwarz method is proved and numerical experiments using it as a preconditioner are presented to confirm the convergence theory

double sweep preconditioner for optimized schwarz methods applied to the helmholtz problem this paper presents a preconditioner for non overlapping schwarz methods applied to the helmholtz problem starting from a simple analytic example we show how such a preconditioner can be designed by approximating the inverse of the iteration operator for a layered partitioning of the domain the preconditioner works by propagating information globally by concurrently sweeping in both directions over the subdomains and can be interpreted as a coarse grid for the domain decomposition method the resulting algorithm is shown to converge very fast independently of the number of subdomains and frequency the preconditioner has the advantage that like the original schwarz algorithm it can be implemented as a matrix free routine with no additional preprocessing

domain decomposition method for harmonic wave propagation a general presentation in this paper we give a general presentation of non overlapping domain decomposition methods for harmonic wave propagation models our abstract framework leads to concise convergence proofs and contains some recent developments about the use of non local transmission conditions it also includes applications to acoustic electromagnetic or elastic waves as well as the treatment of space discretization

optimized schwarz methods without overlap for the helmholtz equation the classical schwarz method is a domain decomposition method to solve elliptic partial differential equations in parallel convergence is achieved through overlap of the subdomains we study in this paper a variant of the schwarz method which converges without overlap for the helmholtz equation we show that the key ingredients for such an algorithm are the transmission conditions we derive optimal transmission conditions which lead to convergence of the algorithm in a finite number of steps these conditions are however nonlocal in nature and we introduce local approximations which we optimize for performance of the schwarz method this leads to an algorithm in the class of optimized schwarz methods we present an asymptotic analysis of the optimized schwarz method for two types of transmission conditions robin conditions and transmission conditions with second order tangential derivatives numerical results illustrate the effectiveness of the optimized schwarz method on a model problem and on a problem from industry

non overlapping domain decomposition algorithm based on modified transmission conditions for the helmholtz equation abstract a square root based transmission conditions domain decomposition method was recently introduced for the helmholtz equation it produces an effective algorithm where the convergence is independent of the wavenumber and the mesh discretization we modify here these conditions in order to guarantee well posedness of local problems and further improve the efficiency of the whole method numerical results in particular in the three dimensional case show significant reduction of the computational time needed in the iterative procedure while preserving the iteration number when compared with the original algorithm

the method of polarized traces for the 2d helmholtz equation we present a solver for the 2d high frequency helmholtz equation in heterogeneous acoustic media with online parallel complexity that scales optimally as o n l where n is the number of volume unknowns and l is the number of processors as long as l grows at most like a small fractional power of n the solver decomposes the domain into layers and uses transmission conditions in boundary integral form to explicitly define polarized traces i e up and down going waves sampled at interfaces local direct solvers are used in each layer to precompute traces of local green s functions in an embarrassingly parallel way the offline part and incomplete green s formulas are used to propagate interface data in a sweeping fashion as a preconditioner inside a gmres loop the online part adaptive low rank partitioning of the integral kernels is used to speed up their application to interface data the method uses second order finite differences the complexity scalings are empirical but motivated by an analysis of ranks of off diagonal blocks of oscillatory integrals they continue to hold in the context of standard geophysical community models such as bp and marmousi 2 where convergence occurs in 5 to 10 gmres iterations while the parallelism in this paper stems from decomposing the domain we do not explore the alternative of parallelizing the systems solves with distributed linear algebra routines

weyl titchmarsh m function asymptotics local uniqueness results trace formulas and borg type theorems for dirac operators we explicitly determine the high energy asymptotics for weyl titchmarsh matrices associated with general dirac type operators on half lines and on r we also prove new local uniqueness results for dirac type operators in terms of exponentially small differences of weyl titchmarsh matrices as concrete applications of the asymptotic high energy expansion we derive a trace formula for dirac operators and use it to prove a borg type theorem

domain decomposition method for maxwell s equations scattering off periodic structures abstract we present a domain decomposition approach for the computation of the electromagnetic field within periodic structures we use a schwarz method with transparent boundary conditions at the interfaces of the domains transparent boundary conditions are approximated by the perfectly matched layer method pml an adaptive strategy to determine optimal pml parameters is developed thus we can treat wood anomalies appearing in periodic structures we focus on the application to typical euv lithography line masks light propagation within the multilayer stack of the euv mask is treated analytically this results in a drastic reduction of the computational costs and allows for the simulation of next generation lithography masks on a standard personal computer

a rapidly converging domain decomposition method for the helmholtz equation a new domain decomposition method is introduced for the heterogeneous 2 d and 3 d helmholtz equations transmission conditions based on the perfectly matched layer pml are derived that avoid artificial reflections and match incoming and outgoing waves at the subdomain interfaces we focus on a subdivision of the rectangular domain into many thin subdomains along one of the axes in combination with a certain ordering for solving the subdomain problems and a gmres outer iteration when combined with multifrontal methods the solver has near linear cost in examples due to very small iteration numbers that are essentially independent of problem size and number of subdomains it is to our knowledge only the second method with this property next to the moving pml sweeping method

recursive sweeping preconditioner for the three dimensional helmholtz equation this paper introduces the recursive sweeping preconditioner for the numerical solution of the helmholtz equation in three dimensions this is based on the earlier work of the sweeping preconditioner with the moving perfectly matched layers the key idea is to apply the sweeping preconditioner recursively to the quasi two dimensional auxiliary problems introduced in the three dimensional 3d sweeping preconditioner compared to the nonrecursive 3d sweeping preconditioner the setup cost of this new approach drops from o n 4 3 to o n the application cost per iteration drops from o n log n to o n and the iteration count increases only mildly when combined with the standard gmres solver several numerical examples are tested and the results are compared with the nonrecursive sweeping preconditioner to demonstrate the efficiency of the new approach

a convergent born series for solving the inhomogeneous helmholtz equation in arbitrarily large media we present a fast method for numerically solving the inhomogeneous helmholtz equation our iterative method is based on the born series which we modified to achieve convergence for scattering media of arbitrary size and scattering strength compared to pseudospectral time domain simulations our modified born approach is two orders of magnitude faster and nine orders of magnitude more accurate in benchmark tests in 1 2 and 3 dimensional systems

a quasi optimal non overlapping domain decomposition algorithm for the helmholtz equation this paper presents a new non overlapping domain decomposition method for the helmholtz equation whose effective convergence is quasi optimal these improved properties result from a combination of an appropriate choice of transmission conditions and a suitable approximation of the dirichlet to neumann operator a convergence theorem of the algorithm is established and numerical results validating the new approach are presented in both two and three dimensions

sampling and interpolation problems for vector valued signals in the paley wiener spaces a new approach to sampling interpolation problem for vector valued signals is proposed for a function signal f from the paley wiener space pw pi bbc n of entire vector valued functions the sampling data are defined as the inner products langf lambdan alphan rang of f lambdan and vectors alphan it is proved that alphan lambdan is a sampling and interpolating sequence for pw pi bbc n if and only if the family alphanei lambdan t forms a riesz basis in l 2 pi pi bbc n this relation allows to characterize all sampling and interpolating sequences in terms of a generating matrix function to produce a wide class of matrix functions generating sampling and interpolating sequences we consider the matrix sturm liouville problems and apply the boundary control theory for hyperbolic dynamical systems this approach allows to construct both asymptotically uniform and nonuniform sampling and interpolating sequences

subspace based multivariable system identification from frequency response data two noniterative subspace based algorithms which identify linear time invariant mimo multi input multioutput systems from frequency response data are presented the algorithms are related to the recent time domain subspace identification techniques the first algorithm uses equidistantly in frequency spaced data and is strongly consistent under weak noise assumptions the second algorithm uses arbitrary frequency spacing and is strongly consistent under more restrictive noise assumptions promising results are obtained when the algorithms are applied to real frequency data originating from a large flexible structure

the boundary control approach to the titchmarsh weyl m function i the response operator and the a amplitude we link the boundary control theory and the titchmarsh weyl theory this provides a natural interpretation of the a amplitude due to simon and yields a new efficient method to evaluate the titchmarsh weyl m function associated with the schrodinger operator h x 2 q x on l 2 0 with dirichlet boundary condition at x 0

a coarse space for heterogeneous helmholtz problems based on the dirichlet to neumann operator the helmholtz equation governing wave propagation and scattering phenomena is difficult to solve numerically its discretization with piecewise linear finite elements results in typically large linear systems of equations the inherently parallel domain decomposition methods constitute hence a promising class of preconditioners an essential element of these methods is a good coarse space here the helmholtz equation presents a particular challenge as even slight deviations from the optimal choice can be devastating in this paper we present a coarse space that is based on local eigenproblems involving the dirichlet to neumann operator our construction is completely automatic ensuring good convergence rates without the need for parameter tuning moreover it naturally respects local variations in the wave number and is hence suited also for heterogeneous helmholtz problems the resulting method is parallel by design and its efficiency is demonstrated on 2d homogeneous and heterogeneous numerical examples

model reduction via balanced realizations an extension and frequency weighting techniques two model reduction methods for discrete systems related to balanced realizations are described the first is a technique which utilizes the least controllable and observable subsystem in deriving a balanced discrete reduced order model for this technique as l sup infinity norm bound on the reduction error is given the second method is a frequency weighting technique for discrete and continuous time systems where the input normal or output normal realizations are modified to include a simple frequency weighting for this technique l sup infinity norm bounds on the weighted reduction errors are obtained

a dispersion minimizing scheme for the 3 d helmholtz equation based on ray theory we develop a new dispersion minimizing compact finite difference scheme for the helmholtz equation in 2 and 3 dimensions the scheme is based on a newly developed ray theory for difference equations a discrete helmholtz operator and a discrete operator to be applied to the source and the wavefields are constructed their coefficients are piecewise polynomial functions of hk chosen such that phase and amplitude errors are minimal the phase errors of the scheme are very small approximately as small as those of the 2 d quasi stabilized fem method and substantially smaller than those of alternatives in 3 d assuming the same number of gridpoints per wavelength is used in numerical experiments accurate solutions are obtained in constant and smoothly varying media using meshes with only five to six points per wavelength and wave propagation over hundreds of wavelengths when used as a coarse level discretization in a multigrid method the scheme can even be used with down to three points per wavelength tests on 3 d examples with up to 108 degrees of freedom show that with a recently developed hybrid solver the use of coarser meshes can lead to corresponding savings in computation time resulting in good simulation times compared to the literature

a new approach to inverse spectral theory ii general real potentials and the connection to the spectral measure we continue the study of the a amplitude associated to a half line schrodinger operator d 2 dx 2 q in l 2 0 b b a is related to the weyl titchmarsh m function via m k 2 k a 0 a e 2 k d o e 2 k for all 0 we discuss five issues here first we extend the theory to general q in l 1 0 for all a including q s which are limit circle at infinity second we prove the following relation between the a amplitude and the spectral measure p a 2 1 2 sin 2 dp since the integral is divergent this formula has to be properly interpreted third we provide a laplace transform representation for m without error term in the case b fourth we discuss m functions associated to other boundary conditions than the dirichlet boundary conditions associated to the principal weyl titchmarsh m function finally we discuss some examples where one can compute a exactly

algebraic multilevel preconditioner for the helmholtz equation in heterogeneous media an algebraic multilevel ml preconditioner is presented for the helmholtz equation in heterogeneous media it is based on a multilevel incomplete ldl t factorization and preserves the inherent complex symmetry of the helmholtz equation the ml preconditioner incorporates two key components for efficiency and numerical stability symmetric maximum weight matchings and an inverse based pivoting strategy the former increases the block diagonal dominance of the system whereas the latter controls l 1 for numerical stability when applied recursively their combined effect yields an algebraic coarsening strategy similar to algebraic multigrid methods even for highly indefinite matrices the ml preconditioner is combined with a krylov subspace method and applied as a black box solver to a series of challenging two and three dimensional test problems mainly from geophysical seismic imaging the numerical results demonstrate the robustness and efficiency of the ml preconditioner even at higher frequency regimes

a domain decomposition method for the helmholtz equation and related optimal control problems we present an iterative domain decomposition method to solve the helmholtz equation and related optimal control problems the proof of convergence of this method relies on energy techniques this method leads to efficient algorithms for the numerical resolution of harmonic wave propagation problems in homogeneous and heterogeneous media

fast alternating bidirectional preconditioner for the 2d high frequency lippmann schwinger equation this paper presents a fast iterative solver for the lippmann schwinger equation for high frequency waves scattered by a smooth medium with a compactly supported inhomogeneity the solver is based on the sparsifying preconditioner l ying multiscale model simul 13 2015 pp 644 660 and a domain decomposition approach similar to the method of polarized traces l zepeda nunez and l demanet j comput phys 308 2016 pp 347 388 the iterative solver has two levels the outer level in which a sparsifying preconditioner for the lippmann schwinger equation is constructed and the inner level in which the resulting sparsified system is solved quickly using an iterative solver preconditioned with a bidirectional matrix free variant of the method of polarized traces the complexity of the construction and application of the preconditioner is cal o n and cal o n log n respectively where n is the number of degrees of freedom numerical experiments in 2 dimensions indicate that t

a multigrid method for the helmholtz equation with optimized coarse grid corrections we study the convergence of multigrid schemes for the helmholtz equation focusing in particular on the choice of the coarse scale operators let g rm c denote the number of points per wavelength at the coarse level if the coarse scale solutions are to approximate the true solutions then the oscillatory nature of the solutions implies the requirement g rm c 2 however in examples the requirement is more like g rm c gtrsim 10 in a trade off involving also the amount of damping present and the number of multigrid iterations we conjecture that this is caused by the difference in phase speeds between the coarse and fine scale operators standard 5 point finite differences in two dimensions are our first example a new coarse scale 9 point operator is constructed to match the fine scale phase speeds we then compare phase speeds and multigrid performance of standard schemes with a scheme using the new operator the required g rm c is reduced from about 10 to about 3 5 with less dam

optimized schwarz methods for circular domain decompositions with overlap optimized schwarz methods are based on transmission conditions between subdo mains which are optimized for the problem class that is being solved such optimizations have been performed for many different types of partial differential equations but were almost exclusively based on the assumption of straight interfaces we study in this paper the influence of curvature on the optimization and we obtain four interesting new results first we show that the curvature does indeed enter the optimized parameters and the contraction factor estimates second we develop an asymptotically accurate approximation technique based on turan type inequalities in our case to solve the much harder optimization problem on the curved interface and this approximation tech nique will also be applicable to currently too complex best approximation problems in the area of optimized schwarz methods third we show that one can obtain transmission conditions from a simple circular model decomposition which have also been found using microlocal analysis but that these are not the best choices for the performance of the optimized schwarz method finally we find that in the case of curved interfaces optimized schwarz methods are not necessarily convergent for all admissible parameters our optimization leads however to parameter choices that give the same good performance for a circular decomposition as for a straight interface decomposition we illustrate our analysis with numerical experiments

a double sweeping preconditioner for the helmholtz equation a new preconditioner is developed to increase the efficiency of iterative solution of the helmholtz equation the key idea of the proposed preconditioner is to split the domain of interest into smaller subdomains and sequentially approximate the forward and backward components of the solution the sequential solution is facilitated by approximate interface conditions that ignore the effect of multiple reflections the efficiency of the proposed method is tested using various 2 d heterogeneous media we observe that the proposed preconditioner results in good convergence with number of iterations growing very slowly with increasing frequency we also note that the mesh size and number of subdomains do not affect the convergence rate finally we find that the overall computational time is much smaller than that of the sweeping preconditioner

compressed absorbing boundary conditions via matrix probing absorbing layers are sometimes required to be impractically thick in order to offer an accurate approximation of an absorbing boundary condition for the helmholtz equation in a heterogeneous medium it is always possible to reduce an absorbing layer to an operator at the boundary by layer stripping elimination of the exterior unknowns but the linear algebra involved is costly we propose bypassing the elimination procedure and directly fitting the surface to surface operator in compressed form from a few exterior helmholtz solves with random dirichlet data the result is a concise description of the absorbing boundary condition with a complexity that grows slowly often logarithmically in the frequency parameter

multigrid based shifted laplacian preconditioning for the time harmonic elastic wave equation we investigate the numerical performance of an iterative solver for a frequency domain finite difference discretization of the isotropic elastic wave equation the solver is based on the shifted laplacian preconditioner originally designed for the acoustic wave equation this preconditioner represents a discretization of a heavily damped wave equation and can be efficiently inverted by a multigrid iteration however the application of multigrid to the elastic case is not straightforward because standard methods such as point jacobi fail to smooth the s wave wavenumber components of the error when high p to s velocity ratios are present we consider line smoothers as an alternative and apply local mode analysis to evaluate the performance of the various components of the multigrid preconditioner numerical examples in 2 d demonstrate the efficacy of our method

m operators a generalisation of weyl titchmarsh theory the theory of the weyl titchmarsh m function for second order ordinary differential operators is generalized and applied to partial differential operators of the form q x acting in three space dimensions weyl operators m z are defined as maps from l2 s 1 to l2 s 1 s 1 unit sphere in r3 for exterior and interior boundary value problems and for the partial differential operator acting in l2 r 3 with the standard weyl titchmarsh m function recovered in the special case that q is spherically symmetric the analysis is carried out rather explicitly allowing for the determination of precise norm bounds for m operators and for the proof of higher dimensional analogues of a number of the fundamental results of standard weyl titchmarsh theory

natural domain decomposition algorithms for the solution of time harmonic elastic waves we study for the first time schwarz domain decomposition methods for the solution of the navier equations modeling the propagation of elastic waves these equations in the time harmonic regime are

an adaptive dpg method for high frequency time harmonic wave propagation problems abstract the discontinuous petrov galerkin dpg method for high frequency wave propagation problems is discussed the dpg method with its attractive uniform mesh and wavenumber independent pre asymptotic stability property allows for a fully automatic adaptive h p algorithm that can be initiated from very coarse meshes moreover dpg always delivers a hermitian positive definite system suggesting the use of the conjugate gradient algorithm for its solution we present a new iterative solution scheme which capitalizes on these attractive properties of dpg this novel solver is integrated within the adaptive procedure by constructing a two grid like preconditioner for the conjugate gradient method that exploits information from previous meshes the construction of our preconditioner is discussed and its efficacy is illustrated with an example of a 2d acoustics problem our results show that the proposed iterative algorithm converges at a rate independent of the mesh and the wavenumber