a chebyshev based rectangular polar integral solver for scattering by general geometries described by non overlapping patches this paper introduces a high order accurate strategy for integration of singular kernels and edge singular integral densities that appear in the context of boundary integral equation formulations of the problem of acoustic scattering in particular the proposed method is designed for use in conjunction with geometry descriptions given by a set of arbitrary non overlapping logically quadrilateral patches which makes the algorithm particularly well suited for treatment of cad generated geometries fejer s first quadrature rule is incorporated in the algorithm to provide a spectrally accurate method for evaluation of contributions from far integration regions while highly accurate precomputations of singular and near singular integrals over certain surface patches together with two dimensional chebyshev transforms and suitable surface varying rectangular polar changes of variables are used to obtain the contributions for singular and near singular interactions the overall integration method is then used in conjunction with the linear algebra solver gmres to produce solutions for sound soft open and closed surface scattering obstacles including an application to an aircraft described by means of a cad representation the approach is robust fast and highly accurate use of a few points per wavelength suffices for the algorithm to produce far field accuracies of a fraction of a percent and slight increases in the discretization densities give rise to significant accuracy improvements

analysis of multiple scattering iterations for high frequency scattering problems i the two dimensional case we present an analysis of a recently proposed integral equation method for the solution of high frequency electromagnetic and acoustic scattering problems that delivers error controllable solutions in frequency independent computational times within single scattering configurations the method is based on the use of an appropriate ansatz for the unknown surface densities and on suitable extensions of the method of stationary phase the extension to multiple scattering configurations in turn is attained through consideration of an iterative neumann series that successively accounts for further geometrical wave reflections as we show for a collection of two dimensional cylindrical convex obstacles this series can be rearranged into a sum of periodic orbits of increasing period each corresponding to contributions arising from waves that reflect off a fixed subset of scatterers when these are transversed sequentially in a periodic manner here we analyze the properties of these periodic orbits in the high frequency regime by deriving precise asymptotic expansions for the currents i e the normal derivative of the fields that they induce on the surface of the obstacles as we demonstrate these expansions can be used to provide accurate estimates of the rate at which their magnitude decreases with increasing number of reflections which defines the overall rate of convergence of the multiple scattering series moreover we show that the detailed asymptotic knowledge of these currents can be used to accelerate this convergence and thus to reduce the number of iterations necessary to attain a prescribed accuracy

convergence analysis of a high order nystrom integral equation method for surface scattering problems in this paper we present a convergence analysis for the nystrom method proposed in j comput phys 169 1 80 110 2001 for the solution of the combined boundary integral equation formulations of sound soft acoustic scattering problems in three dimensional space this fast and efficient scheme combines fft techniques and a polar change of variables that cancels out the kernel singularity we establish the stability of the algorithms in the l 2 norm and we derive convergence estimates in both the l 2 and l infty norms in particular our analysis establishes theoretically the previously observed super algebraic convergence of the method in cases in which the right hand side is smooth

analysis of multiple scattering iterations for high frequency scattering problems ii the three dimensional scalar case in this paper we continue our analysis of the treatment of multiple scattering effects within a recently proposed methodology based on integral equations for the numerical solution of scattering problems at high frequencies in more detail here we extend the two dimensional results in part i of this work to fully three dimensional geometries as in the former case our concern here is the determination of the rate of convergence of the multiple scattering iterations for a collection of three dimensional convex obstacles that are inherent in the aforementioned high frequency schemes to this end we follow a similar strategy to that we devised in part i first we recast the iterated neumann multiple scattering series in the form of a sum of periodic orbits of increasing period corresponding to multiple reflections that periodically bounce off a series of scattering sub structures then we proceed to derive a high frequency recurrence that relates the normal derivatives of the fields induced on these structures as the waves reflect periodically and finally we analyze this recurrence to provide an explicit rate of convergence associated with each orbit while the procedure is analogous to its two dimensional counterpart the actual analysis is significantly more involved and perhaps more interestingly it uncovers new phenomena that cannot be distinguished in two dimensional configurations e g the further dependence of the convergence rate on the relative orientation of interacting structures as in the two dimensional case and beyond their intrinsic interest we also explain here how the results of our analysis can be used to accelerate the convergence of the multiple scattering series and thus to provide significant savings in computational times

burton miller type singular boundary method for acoustic radiation and scattering abstract this paper proposes the singular boundary method sbm in conjunction with burton and miller s formulation for acoustic radiation and scattering the sbm is a strong form collocation boundary discretization technique using the singular fundamental solutions which is mathematically simple easy to program meshless and introduces the concept of source intensity factors sifs to eliminate the singularities of the fundamental solutions therefore it avoids singular numerical integrals in the boundary element method bem and circumvents the troublesome placement of the fictitious boundary in the method of fundamental solutions mfs in the present method we derive the sifs of exterior helmholtz equation by means of the sifs of exterior laplace equation owing to the same order of singularities between the laplace and helmholtz fundamental solutions in conjunction with the burton miller formulation the sbm enhances the quality of the solution particularly in the vicinity of the corresponding interior eigenfrequencies numerical illustrations demonstrate efficiency and accuracy of the present scheme on some benchmark examples under 2d and 3d unbounded domains in comparison with the analytical solutions the boundary element solutions and dirichlet to neumann finite element solutions

a fast directional algorithm for high frequency electromagnetic scattering this paper is concerned with the fast solution of high frequency electromagnetic scattering problems using the boundary integral formulation we extend the o nlogn directional multilevel algorithm previously proposed for the acoustic scattering case to the vector electromagnetic case we also detail how to incorporate the curl operator of the magnetic field integral equation into the algorithm when combined with a standard iterative method this results in an almost linear complexity solver for the combined field integral equations in addition the butterfly algorithm is utilized to compute the far field pattern and radar cross section with o nlogn complexity

non planar square orthogonal drawing with few bend edges we investigate square orthogonal drawings of non planar graphs with vertices represented as unit grid squares we present quadratic time algorithms to construct the square orthogonal drawings of 5 graphs 6 graphs and 8 graphs such that each edge in the drawing contains at most two two and three bends respectively in particular the novel analysis method we use to split a vertex so as to build some specific propagation channels in our algorithms is an interesting technique and may be of independent interest moreover we show that the decision problem of determining whether an 8 graph has a square orthogonal drawing without edge bends is np complete

an efficient high order nystrom scheme for acoustic scattering by inhomogeneous penetrable media with discontinuous material interface this text proposes a fast rapidly convergent nystrom method for the solution of the lippmann schwinger integral equation that mathematically models the scattering of time harmonic acoustic waves by inhomogeneous obstacles while allowing the material properties to jump across the interface the method works with overlapping coordinate charts as a description of the given scatterer in particular it employs partitions of unity to simplify the implementation of high order quadratures along with suitable changes of parametric variables to analytically resolve the singularities present in the integral operator to achieve desired accuracies in approximations to deal with the discontinuous material interface in a high order manner a specialized quadrature is used in the boundary region the approach further utilizes an fft based strategy that uses equivalent source approximations to accelerate the evaluation of large number of interactions that arise in the approximation of the volumetric integral operator and thus achieves a reduced computational complexity of o n log n for an n point discretization a detailed discussion on the solution methodology along with a variety of numerical experiments to exemplify its performance are presented in this paper

cost trade offs in graph embeddings with applications an embedding of the graph g in the graph h is a one to one association of the vertices of g with the vertices of h there are two natural measures of the cost of a graph embedding namely the dilation cost of the embedding the maximum distance in h between the images of vertices that are adjacent in g and the expansion cost of the embedding the ratio of the size of h to the size of g the main results of this paper illustrate three situations wherein one of these costs can be minimized only at the expense of a dramatic increase in the other cost the first result establishes the following there is an embedding of n node complete ternary trees in complete binary trees with dilation cost 2 and expansion cost n where log3 4 3 but any embedding of these ternary trees in binary trees that has expansion cost 2 must infinitely often have dilation cost const log log log n the second result provides a stronger but less easily stated example of the same type of tradeoff the third result concerns generic binary trees that is complete binary trees into which all n node binary trees are efficiently embeddable there is a generic binary tree into which all n node binary trees are embeddable with dilation cost o 1 and expansion cost o nc for some fixed constant c if one insists on embeddings whose dilation cost is exactly 1 then these embeddings must have expansion cost n log n 2 if one insists on embeddings whose expansion cost is 2 then these embeddings must infinitely often have dilation cost const log log log n an interesting application of the polynomial size generic binary tree in the first part of this three part result is to yield simplified proofs of several results concerning computational systems with an intrinsic notion of computation tree such as alternating and nondeterministic turing machines and context free grammars

multicommodity max flow min cut theorems and their use in designing approximation algorithms in this paper we establish max flow min cut theorems for several important classes of multicommodity flow problems in particular we show that for any n node multicommodity flow problem with uniform demands the max flow for the problem is within an o log n factor of the upper bound implied by the min cut the result which is existentially optimal establishes an important analogue of the famous 1 commodity max flow min cut theorem for problems with multiple commodities the result also has substantial applications to the field of approximation algorithms for example we use the flow result to design the first polynomial time polylog n times optimal approximation algorithms for well known np hard optimization problems such as graph partitioning min cut linear arrangement crossing number vlsi layout and minimum feedback arc set applications of the flow results to path routing problems network reconfiguration communication in distributed networks scientific computing and rapidly mixing markov chains are also described in the paper categories and subject descriptors f 2 2 analysis of algorithms and problem complexity

on sum coloring and sum multi coloring for restricted families of graphs we consider the sum coloring chromatic sum problem and the sum multi coloring problem for restricted families of graphs in particular we consider the graph classes of proper intersection graphs of axis parallel rectangles proper interval graphs and unit disk graphs all the above mentioned graph classes belong to a more general graph class of k 1 clawfree graphs respectively for k 4 2 5 we prove that sum coloring is np hard for penny graphs and unit square graphs which implies np hardness for unit disk graphs and proper intersection graphs of axis parallel rectangles we show a 2 approximation algorithm for unit square graphs with the assumption that the geometric representation of the graph is given for sum multi coloring we confirm that the greedy first fit coloring after ordering vertices by their demands achieves a k approximation for the preemptive version of sum multi coloring on k 1 clawfree graphs finally we study priority algorithms as a model for greedy algorithms for the sum coloring problem and the sum multi coloring problem we show various inapproximation results under several natural input representations

spectral elements for the integral equations of time harmonic maxwell problems we present a novel high order method of moments mom with interpolatory vector functions on quadrilateral patches the main advantage of this method is that the hdiv conforming property is enforced and at the same time it can be interpreted as a point based scheme we apply this method to field integral equations fies to solve time harmonic electromagnetic scattering problems our approach is applied to the first and second nedelec families of hdiv conforming elements it consists in a specific choice of the degrees of freedom dof made in order to allow a fast integral evaluation in this paper we describe these two sets of dof and their corresponding quadrature rules sample numerical results on fie confirm the good properties of our schemes faster convergence rate and cheap matrix calculation we also present observations on the choice of the discretization method depending on the fie selected

frequency adapted galerkin boundary element methods for convex scattering problems we introduce a class of hybrid boundary element methods for the solution of sound soft scattering problems in the exterior of two dimensional smooth convex obstacles to facilitate the applicability of our algorithms throughout the entire frequency spectrum we have enriched our galerkin approximation spaces through incorporation of oscillations in the incident field of radiation into the algebraic and trigonometric polynomial approximation spaces the resulting methodologies have three distinctive properties indeed from a theoretical point of view 1 they can be tuned to demand only an mathcal o k varepsilon o k increase for any varepsilon 0 0 in the number of degrees of freedom to maintain a fixed accuracy with increasing wavenumberk owing to the optimal adaptation of approximation spaces to asymptotic stretching shrinking of illuminated and deep shadow regions shadow boundaries and 2 they are convergent for each fixed wavenumberk thanks to the additional approximation spaces in the deep shadow region perhaps more importantly from a practical point of view 3 they give rise to linear systems with significantly enhanced condition numbers and this in turn allows for more accurate solutions if desired

a fast high order solver for em scattering from complex penetrable bodies te case we present a new high order integral algorithm for the solution of scattering problems by heterogeneous bodies under te radiation here a scatterer is represented by a continuously or discontinuously varying refractive index n x within a two dimensional 2 d bounded region solutions of the associated helmholtz equation under given incident fields are then obtained by high order inversion of the lippmann schwinger integral equation the algorithm runs in o n log n operations where n is the number of discretization points our method provides highly accurate solutions in short computing times even for problems in which the scattering bodies contain complex geometric singularities

on analyticity of linear waves scattered by a layered medium abstract the scattering of linear waves by periodic structures is a crucial phenomena in many branches of applied physics and engineering in this paper we establish rigorous analytic results necessary for the proper numerical analysis of a class of high order perturbation of surfaces methods for simulating such waves more specifically we prove a theorem on existence and uniqueness of solutions to a system of partial differential equations which model the interaction of linear waves with a multiply layered periodic structure in three dimensions this result provides hypotheses under which a rigorous numerical analysis could be conducted for recent generalizations to the methods of operator expansions field expansions and transformed field expansions

a nystrom flavored calderon calculus of order three for two dimensional waves time harmonic and transient in this paper we present and test a full discretization of all elements of the calderon calculus layer potentials and integral operators for the helmholtz equation in smooth closed curves in the plane the resulting integral equations provide approximations of order three for all variables involved tests are shown for a wide array of direct indirect and combined field integral equation at fixed frequency and for a convolution quadrature based approximation in the time domain

the planar k means problem is np hard in the k means problem we are given a finite set s of points in m and integer k 1 and we want to find k points centers so as to minimize the sum of the square of the euclidean distance of each point in s to its nearest center we show that this well known problem is np hard even for instances in the plane answering an open question posed by dasgupta 2007 7

time domain adaptive integral method for surface integral equations an efficient marching on in time mot scheme is presented for solving electric magnetic and combined field integral equations pertinent to the analysis of transient electromagnetic scattering from perfectly conducting surfaces residing in an unbounded homogenous medium the proposed scheme is the extension of the frequency domain adaptive integral pre corrected fast fourier transform fft method to the time domain fields on the scatterer that are produced by space time sources residing on its surface are computed 1 by locally projecting for each time step all sources onto a uniform auxiliary grid that encases the scatterer 2 by computing everywhere on this grid the transient fields produced by the resulting auxiliary sources via global multilevel blocked space time ffts 3 by locally interpolating these fields back onto the scatterer surface as this procedure is inaccurate when source and observer points reside close to each other and 4 near fields are computed classically albeit pre corrected for errors introduced through the use of global ffts the proposed scheme has a computational complexity and memory requirement of o n sub t n sub s log sup 2 n sub s and o n sub s sup 3 2 when applied to quasiplanar structures and of o n sub t n sub s sup 3 2 log sup 2 n sub s and o n sub s sup 2 when used to analyze scattering from general surfaces here n sub s and n sub t denote the number of spatial and temporal degrees of freedom of the surface current density these computational cost and memory requirements are contrasted to those of classical mot solvers which scale as o n sub t n sub s sup 2 and o n sub s sup 2 respectively a parallel implementation of the scheme on a distributed memory computer cluster that uses the message passing interface is described simulation results demonstrate the accuracy efficiency and the parallel performance of the implementation

a convergence theorem for the fast multipole method for 2 dimensional scattering problems the fast multipole method fmm designed by v rokhlin rapidly computes the field scattered from an obstacle this computation consists of solving an integral equation on the boundary of the obstacle the main result of this paper shows the convergence of the fmm for the two dimensional helmholtz equation before giving the theorem we give an overview of the main ideas of the fmm this is done following the papers of v rokhlin nevertheless the way we present the fmm is slightly different the fmm is finally applied to an acoustic problem with an impedance boundary condition the moment method is used to discretize this continuous problem

a high accuracy algorithm for 3d periodic electromagnetic scattering we develop a highly accurate numerical method for scattering of 3d electromagnetic waves by doubly periodic structures we approximate scattered fields using the muller boundary integral formulation of maxwell s equations the accuracy is achieved as singularities are isolated through the use of partitions of unity leaving smooth periodic integrands that can be evaluated with high accuracy using trapezoid sums the removed singularities are resolved through a transformation to polar coordinates the method relies on the ideas used in the free space scattering algorithm of bruno and kunyansky

a fast high order algorithm for the solution of surface scattering problems basic implementation tests and applications we present a new algorithm for the numerical solution of problems of acoustic scattering by surfaces in three dimensional space this algorithm evaluates scattered fields through fast high order solution of the corresponding boundary integral equation the high order accuracy of our solver is achieved through use of partitions of unity together with analytical resolution of kernel singularities the acceleration in turn results from use of a novel approach which based on high order two face equivalent source approximations reduces the evaluation of far interactions to evaluation of 3 d fast fourier transforms ffts this approach is faster and substantially more accurate and it runs on dramatically smaller memories than other fft and k space methods the present algorithm computes one matrix vector multiplication in o n6 5logn to o n4 3logn operations where n is the number of surface discretization points the latter estimate applies to smooth surfaces for which our high order algorithm provides accurate solutions with small values of n the former more favorable count is valid for highly complex surfaces requiring significant amounts of subwavelength sampling further our approach exhibits super algebraic convergence it can be applied to smooth and nonsmooth scatterers and it does not suffer from accuracy breakdowns of any kind in this paper we introduce the main algorithmic components in our approach and we demonstrate its performance with a variety of numerical results in particular we show that the present algorithm can evaluate accurately in a personal computer scattering from bodies of acoustical sizes of several hundreds

power consumption in packet radio networks in this paper we study the problem of assigning transmission ranges to the nodes of a multi hop packet radio network so as to minimize the total power consumed under the constraint that adequate power is provided to the nodes to ensure that the network is strongly connected i e each node can communicate along some path in the network to every other node such assignment of transmission ranges is called complete we also consider the problem of achieving strongly connected bounded diameter networks

separators for sphere packings and nearest neighbor graphs a collection of n balls in d dimensions forms a k ply system if no point in the space is covered by more than k balls we show that for every k ply system there is a sphere s that intersects at most o k 1 d n 1 1 d balls of and divides the remainder of into two parts those in the interior and those in the exterior of the sphere s respectively so that the larger part contains at most 1 1 d 2 n balls this bound of o k 1 d n 1 1 d is the best possible in both n and k we also present a simple randomized algorithm to find such a sphere in o n time our result implies that every k nearest neighbor graphs of n points in d dimensions has a separator of size o k 1 d n 1 1 d in conjunction with a result of koebe that every triangulated planar graph is isomorphic to the intersection graph of a disk packing our result not only gives a new geometric proof of the planar separator theorem of lipton and tarjan but also generalizes it to higher dimensions the separator algorithm can be used for point location and geometric divide and conquer in a fixed dimensional space

advances in the theory and practice of graph drawing the visualization of conceptual structures is a key component of support tools for complex applications in science and engineering foremost among the visual representations used are drawings of graphs and ordered sets in this talk we survey recent advances in the theory and practice of graph drawing specific topics include bounds and tradeoffs for drawing properties three dimensional representations methods for constraint satisfaction and experimental studies

theta functions gaussian series and spatially periodic solutions of the korteweg de vries equation it has been shown by novikov funct anal appl 8 236 1974 dubrovin et al russian math surveys 31 59 1976 lax commun pure appl math 28 141 1975 mckean and van moerbeke inv math 30 217 1975 and others that the nonlinear evolution equations which admit solitary waves also have spatially periodic exact solutions polycnoidal waves which can be expressed in terms of multidimensional riemann theta functions here it is shown that via poisson summation the fourier series that define the theta functions can be transformed into an infinite series of gaussian functions because the lowest terms of the gaussian series generate the usual solitary waves it is possible to intimately explore the relationship between solitary waves and these spatially periodic polycnoidal waves also by using the gaussian series one can perturbatively calculate phase velocities and wave structure for the polycnoidal wave even in the strongly nonlinear regime for which the soliton or mult

nonlinear stability of periodic traveling wave solutions to the schrodinger and the modified korteweg de vries equations abstract this work is concerned with stability properties of periodic traveling waves solutions of the focusing schrodinger equation i u t u x x u 2 u 0 posed in r and the modified korteweg de vries equation u t 3 u 2 u x u x x x 0 posed in r our principal goal in this paper is the study of positive periodic wave solutions of the equation 3 0 called dnoidal waves a proof of the existence of a smooth curve of solutions with a fixed fundamental period l 2 2 l 2 h per 0 l is given it is also shown that these solutions are nonlinearly stable in the energy space h per 1 0 l and unstable by perturbations with period 2l in the case of the schrodinger equation

rapidly convergent two dimensional quasi periodic green function throughout the spectrum including wood anomalies we introduce a new methodology based on new quasi periodic green functions which converge rapidly even at and around wood anomaly configurations for the numerical solution of problems of scattering by periodic rough surfaces in two dimensional space as is well known the classical quasi periodic green function ceases to exist at wood anomalies the approach introduced in this text produces fast green function convergence throughout the spectrum on the basis of a certain finite differencing approach and smooth windowing of the classical green function lattice sum the resulting green function convergence is super algebraically fast away from wood anomalies and it reduces to an arbitrarily high user prescribed algebraic order of convergence at wood anomalies

an efficient high order algorithm for acoustic scattering from penetrable thin structures in three dimensions this paper presents a high order accelerated algorithm for the solution of the integral equation formulation of volumetric scattering problems the scheme is particularly well suited to the analysis of thin structures as they arise in certain applications e g material coatings in addition it is also designed to be used in conjunction with existing low order fft based codes to upgrade their order of accuracy through a suitable treatment of material interfaces the high order convergence of the new procedure is attained through a combination of changes of parametric variables to resolve the singularities of the green function and partitions of unity to allow for a simple implementation of spectrally accurate quadratures away from singular points accelerated evaluations of the interaction between degrees of freedom on the other hand are accomplished by incorporating two face equivalent source approximations on cartesian grids a detailed account of the main algorithmic components of the scheme are presented together with a brief review of the corresponding error and performance analyses which are exemplified with a variety of numerical results

liar s domination in unit disk graphs abstract in this article we study a variant of the minimum dominating set problem known as the minimum liar s dominating set mlds problem we prove that the mlds problem is np hard in unit disk graphs we point out that the approximation guarantee of 11 2 for the approximation algorithm for unit disk graphs proposed by banerjee and bhore 2019 11 is not correct and we propose a simple o n m time 7 31 factor approximation algorithm where n and m are the number of vertices and edges respectively in the given unit disk graph finally we prove that the mlds problem admits a polynomial time approximation scheme

a fast and high order method for the three dimensional elastic wave scattering problem in this paper we present a fast and high order boundary integral equation method for the elastic scattering by three dimensional large penetrable obstacles the algorithm extends the method introduced in 5 for the acoustic surface scattering to the fully elastic case in our algorithm high order accuracy is achieved through the use of the partition of unity and a semi classical treatment of relevant singular integrals the computational efficiency associated with the nonsingular integrals is attained by the method of equivalent source representations on a cartesian grid and fast fourier transform fft the resulting algorithm computes one matrix vector product associated with the discretization of the integral equation with o n 4 3 log n operations and it shows algebraic convergence several numerical experiments are provided to demonstrate the efficiency of the method

sparse fourier transform via butterfly algorithm this paper introduces a fast algorithm for computing sparse fourier transforms with spatial and fourier data supported on curves or surfaces this problem appears naturally in several important applications of wave scattering digital signal processing and reflection seismology the main idea of the algorithm is that the interaction between a frequency region and a spatial region is approximately low rank if the product of their widths are bounded by the maximum frequency based on this property we can approximate the interaction between these two boxes accurately and compactly using a small number of equivalent sources the overall structure of the algorithm follows the butterfly algorithm the computation is further accelerated by exploiting the tensor product property of the fourier kernel in two and three dimensions the proposed algorithm is accurate and has the optimal complexity we present numerical results in both two and three dimensions

improved approximations of crossings in graph drawings and vlsi layout areas we give improved approximations for two classical embedding problems i minimizing the number of crossings in a drawing on the plane of a bounded degree graph and ii minimizing the vlsi layout area of a graph of maximum degree four these improved algorithms can be applied to improve a variety of vlsi layout problems our results are as follows i we compute a drawing on the plane of a bounded degree graph in which the sum of the numbers of vertices and crossings is o log3 n times the optimal minimum sum this is a logarithmic factor improvement relative to the best known result ii we compute a vlsi layout of a graph of maximum degree four in a square grid whose area is o log4 n times the minimum layout area this is an o log2 n improvement over the best known long standing result

a high order algorithm for obstacle scattering in three dimensions in this work we describe implement and analyse in detail a high order fully discrete spectral algorithm for solving the helmholtz equation exterior to a bounded sound soft sound hard or absorbing obstacle in three space dimensions with dirichlet neumann or robin impedance boundary conditions our algorithm may be thought of as a discrete glerkin method but it is also equivalent to a nystrom method after a simple transformation we test our algorithm with extensive computational experiments on a variety of three dimensional smooth and non smooth obstacles with conical singularities our tests include the computation of scattered and far fields induced by incident plane waves our method is shown to be very accurate for scattering from surfaces which are globally parameterised by spherical coordinates and tests show that it performs very much better than several of the well established fast algorithms for obstacle sattering on a range of such surfaces even some which are non smooth further we prove superalgebraic convergence of the scattered and far fields obtained using our algorithm in the case of smooth scatterers

second kind integral equations for the first kind dirichlet problem of the biharmonic equation in three dimensions a fredholm second kind integral equation skie formulation is constructed for the dirichlet problem of the biharmonic equation in three dimensions a fast numerical algorithm is developed based on the constructed skie its performance is illustrated via several numerical examples

a parallel directional fast multipole method this paper introduces a parallel directional fast multipole method fmm for solving n body problems with highly oscillatory kernels with a focus on the helmholtz kernel in three dimensions this class of oscillatory kernels requires a more restrictive low rank criterion than that of the low frequency regime and thus effective parallelizations must adapt to the modified data dependencies we propose a simple partition at a fixed level of the octree and show that if the partitions are properly balanced between p processes the overall runtime is essentially mathcal o n log n p p by the structure of the low rank criterion we are able to avoid communication at the top of the octree we demonstrate the effectiveness of our parallelization on several challenging models

rapid solution of integral equations of classical potential theory an algorithm is described for rapid solution of classical boundary value problems dirichlet an neumann for the laplace equation based on iteratively solving integral equations of potential theory cpu time requirements for previously published algorithms of this type are proportional to n2 where n is the number of nodes in the discretization of the boundary of the region the cpu time requirements for the algorithm of the present paper are proportional to n making it considerably more practical for large scale problems

an efficient and high order frequency domain approach for transient acoustic structure interactions in three dimensions this paper is concerned with numerical solution of the transient acoustic structure interaction problems in three dimensions an efficient and higher order method is proposed with a combination of the exponential window technique and a fast and accurate boundary integral equation solver in the frequency domain the exponential window applied to the acoustic structure system yields an artificial damping to the system which eliminates the wrap around errors brought by the discrete fourier transform the frequency domain boundary integral equation approach relies on accurate evaluations of relevant singular integrals and fast computation of nonsingular integrals via the method of equivalent source representations and the fast fourier transform numerical studies are presented to demonstrate the accuracy and efficiency of the method copyright 2016 john wiley sons ltd

a mixed hybrid formulation based on oscillated finite element polynomials for solving helmholtz problems a mixed hybrid type formulation is proposed for solving helmholtz problems this method is based on a a local approximation of the solution by oscillated finite element polynomials and b the use of lagrange multipliers to weakly enforce the continuity across element boundaries the computational complexity of the proposed discretization method is determined mainly by the total number of lagrange multiplier degrees of freedom introduced at the interior edges of the finite element mesh and the sparsity pattern of the corresponding system matrix preliminary numerical results are reported to illustrate the potential of the proposed solution methodology for solving efficiently helmholtz problems in the mid and high frequency regimes

orbital stability of double solitons for the benjamin ono equation in this paper we prove the orbital stability of double solitons for the benjamin ono equation in the case of the kdv equation this stability has been proved in 17 parts of the proof given there rely on the fact that the euler lagrange equations for the conserved quantities of the kdv equation are ordinary differential equations since this is not the case for the benjamin ono equation new methods are required our approach consists in using a new invariant for multi solitons and certain new identities motivated by the sylvester law of inertia

an accelerated surface discretization based bem approach for non homogeneous linear problems in 3 d complex domains for non homogeneous or non linear problems a major difficulty in applying the boundary element method bem is the treatment of the volume integrals that arise an accurate scheme that requires no volume discretization is highly desirable in this paper we describe an efficient approach based on the precorrected fft technique for the evaluation of volume integrals resulting from non homogeneous linear problems in this approach the 3 d uniform grid constructed initially to accelerate surface integration is used as the baseline mesh for the evaluation of volume integrals as such no volume discretization of the interior problem domain is necessary moreover with the uniform 3 d grid the matrix sparsification techniques such as the precorrected fft technique used in this work can be extended to accelerate volume integration in addition to surface integration thus greatly reducing the computational time the accuracy and efficiency of our approach are demonstrated through several examples a 3 d accelerated bem solver for poisson equations has been developed and has been applied to a 3 d multiply connected problem with complex geometries good agreement between simulation results and analytical solutions has been obtained copyright 2005 john wiley sons ltd

isoinertial family of operators and convergence of kdv cnoidal waves to solitons abstract in this paper we show the convergence of korteweg de vries cnoidal waves to the limit soliton it is proved that the convergence is uniform and in h 2 norm as the period of the solutions tends to infinity families of hill operators are also studied we obtain a condition under which families of operators are isoinertial this condition is satisfied for classes of hill operators that are obtained by linearization our application is to the family of linearized operators at the kdv cnoidal waves it is proved that this family is isoinertial and also the value of the inertial index is calculated

orbital stability of one parameter periodic traveling waves for dispersive equations and applications abstract this paper establishes sufficient conditions for the orbital stability of one parameter spatially periodic traveling wave solutions for one dimensional dispersive equations our method of proof combines known techniques with some new ideas as a consequence of our result we give several applications for well known dispersive equations extension of the theory to regularized equations is also established

on embedding rectangular grids in square grids the main results in this paper demonstrate that there exist pairs of integers e d for area expansion and edge dilation respectively such that any n vertex rectangular grid can be embedded into a square grid having at most en vertices in such a way that images in the square grid of vertices that are adjacent in the rectangular grid are at most distance d apart several techniques for squaring up rectangular grids are presented sample values for the parameter pair e d are e 1 2 d 15 e 1 45 d 9 e 1 8 d 3 note that these values of e and d hold for all rectangular grids independent of number of vertices the quest for these results was motivated by the question of whether or not one could automatically square up circuit layouts having aspect ratios very far from unity without compromising the efficiency of the layout in terms of area and length of the longest run of wire the results reported here yield an affirmative answer to this question at least in an idealized setting one corollary of the embeddings presented here is that the side 2n square king s move grid contains as a subgraph every n vertex rectangular grid another way to think of this result is that this embellished grid can be programmed or personalized by setting switches to represent any n vertex rectangular grid

harmonic density interpolation methods for high order evaluation of laplace layer potentials in 2d and 3d abstract we present an effective harmonic density interpolation method for the numerical evaluation of singular and nearly singular laplace boundary integral operators and layer potentials in two and three spatial dimensions the method relies on the use of green s third identity and local taylor like interpolations of density functions in terms of harmonic polynomials the proposed technique effectively regularizes the singularities present in boundary integral operators and layer potentials and recasts the latter in terms of integrands that are bounded or even more regular depending on the order of the density interpolation the resulting boundary integrals can then be easily accurately and inexpensively evaluated by means of standard quadrature rules a variety of numerical examples demonstrate the effectiveness of the technique when used in conjunction with the classical trapezoidal rule to integrate over smooth curves in two dimensions and with a chebyshev type quadrature rule to integrate over surfaces given as unions of non overlapping quadrilateral patches in three dimensions

a fast distributed shortest path algorithm for a class of hierarchically clustered data networks a distributed algorithm is presented that can be used to solve the single destination shortest path sdsp problem or the all pairs shortest path apsp problem for a class of clustered data networks the network graph is assumed to be characterized with a balanced hierarchically clustered bhc topology the bhc topology is introduced in this paper and is shown to be a realistic characterization for a large class of interconnected data networks for certain types of bhc topologies the sdsp problem can be solved with computation and communication time complexities of o log n assuming one processor is available at each of the n number of nodes assuming p processors are available at each node computation and communication time complexities of o n p log n and o n log n are achievable respectively for solving the apsp problem it is also shown that the algorithm converges in an asynchronous environment

universality considerations in vlsi circuits the problem of embedding the interconnection pattern of a circuit into a two dimensional surface of minimal area is discussed since even for some natural patterns graphs containing m connections may require m2 area in order to achieve compact embeddings restricted classes of graphs have to be considered for example arbitrary trees of bounded degree can be embedded in linear area without edges crossing over planar graphs can be embedded efficiently only if crossovers are allowed in the embedding

stability and convergence of collocation schemes for retarded potential integral equations time domain boundary integral formulations of transient scattering problems involve retarded potential integral equations solving such equations numerically is both complicated and computationally intensive and numerical methods often prove to be unstable collocation schemes are easier to implement than full finite element formulations but little appears to be known about their stability and convergence here we derive and analyze some new stable collocation schemes for the single layer equation for transient acoustic scattering and use spatial fourier and temporal laplace transform techniques to demonstrate that such stable schemes are second order convergent

stability theory of solitary waves in the presence of symmetry ii abstract consider an abstract hamiltonian system which is invariant under a group of operators we continue to study the effect of the group invariance on the stability of solitary waves applications are given to bound states and traveling wave solutions of nonlinear wave equations

orbital stability of periodic traveling wave solutions for the log kdv equation abstract in this paper we establish the orbital stability of periodic waves related to the logarithmic korteweg de vries equation our motivation is inspired in the recent work 3 in which the authors established the well posedness and the linear stability of gaussian solitary waves by using the approach put forward recently in 20 to construct a smooth branch of periodic waves as well as to get the spectral properties of the associated linearized operator we apply the abstract theories in 13 and 25 to deduce the orbital stability of the periodic traveling waves in the energy space

local solutions to high frequency 2d scattering problems we consider the solution of high frequency scattering problems in two dimensions modeled by an integral equation on the boundary of a smooth scattering object we devise a numerical method to obtain solutions on only parts of the boundary with little computational effort the method incorporates asymptotic properties of the solution and can therefore attain particularly good results for high frequencies we show that the integral equation in this approach reduces to an ordinary differential equation

fast directional algorithms for the helmholtz kernel this paper presents a new directional multilevel algorithm for solving n body or n point problems with highly oscillatory kernels we address the problem by first proving that the interaction between a ball of radius r and a well separated region has an approximate low rank representation as long as the well separated region belongs to a cone with a spanning angle of o 1 r and is at a distance which is at least o r 2 away from the ball based on this representation our algorithm organizes the high frequency computation using a multidirectional and multiscale strategy our algorithm is proved to have an optimal o nlogn computational complexity for any given accuracy when the points are sampled from a two dimensional surface

3d numerical simulations of vesicle and inextensible capsule dynamics vesicles are locally inextensible fluid membranes capsules are endowed with in plane shear elasticity mimicking the cytoskeleton of red blood cells rbcs but are extensible while rbcs are inextensible we use boundary integral bi methods based on the green function techniques to model and solve numerically their dynamics we regularize the single layer integral by subtraction of exact identities for the terms involving the normal and the tangential components of the force the stability and precision of bi calculation is enhanced by taking advantage of additional quadrature nodes located in vertices of an auxiliary mesh constructed by a standard refinement procedure from the main mesh we extend the partition of unity technique to boundary integral calculation on triangular meshes the proposed algorithm offers the same treatment of near singular integration regardless whether the source and the target points belong to the same surface or not bending forces are calculated by using expressions derived from differential geometry membrane incompressibility is handled by using two penalization parameters per suspended entity one for deviation of the global area from prescribed value and another for the sum of squares of local strains defined on each vertex extensible or inextensible capsules a model of rbc are studied by storing the position in the reference configuration for each vertex the elastic force is then calculated by direct variation of the elastic energy various nonequilibrium physical examples on vesicles and capsules will be presented and the convergence and precision tests highlighted overall a good convergence is observed with numerical error inversely proportional to the number of vertices used for surface discretization the highest order of convergence allowed by piece wise linear interpolation of the surface

a ptas for the minimum weight connected vertex cover p 3 problem on unit disk graphs let g v e be a weighted graph i e with a vertex weight function w v r we study the problem of determining a minimum weight connected subgraph of g that has at least one vertex in common with all paths of length two in g it is known that this problem is np hard for general graphs we first show that it remains np hard when restricted to unit disk graphs our main contribution is a polynomial time approximation scheme for this problem if we assume that the problem is c local and the unit disk graphs have minimum degree of at least two

a polynomial algorithm for the min cut linear arrangement of trees an algorithm is presented that finds a min cut linear arrangement of a tree in o n log n time an extension of the algorithm determines the number of pebbles needed to play the black and white pebble game on a tree

embedding meshes and tori on double loop networks of the same size double loop networks are extensions of ring networks and are widely used in the design and implementation of local area networks and parallel processing architectures however embedding of other types of networks on double loop networks has not been well studied due to the topological complexity of double loop networks the traditional l shape designed to compute the diameter of double loop networks is not effective to solve the embedding problem we propose a novel tessellation approach to partition the geometric plane of double loop networks into a set of parallelogram tiles called p shape based on the characteristics of p shape we design a simple embedding scheme namely p shape embedding that embeds meshes and tori on double loop networks in a systematic way under p shape embedding we evaluate the embedding metrics of dilation average dilation and congestion which depend heavily on the parameters of p shape a main merit of p shape embedding is that a large fraction of embedded mesh torus edges have edge dilation 1 resulting in a low average dilation these are the first results to our knowledge for embedding meshes and tori on double loop networks which is of great significance due to the popularity of these architectures our p shape construction bridges between regular graphs and double loop networks and provides a powerful tool for studying double loop networks

numerical solution of the helmholtz equation in 2d and 3d using a high order nystrom discretization we show how to solve time harmonic scattering problems by means of a high order nystrom discretization of the boundary integral equations of wave scattering in 2d and 3d the novel aspect of our new method is its use of local corrections to the discretized kernel in the vicinity of the kernel singularity enhanced by local corrections the new algorithm has the simplicity and speed advantages of the traditional nystrom method but also enjoys the advantages of high order convergence for controlling solution error we explain the practical details of implementing a scattering code based on a high order nystrom discretization and demonstrate by numerical example that a scattering code based on this algorithm can achieve high order convergence to the correct answer we also demonstrate its performance advantages over a high order galerkin code

spirality and optimal orthogonal drawings we deal with the problem of constructing the orthogonal drawing of a graph with the minimum number of bends along the edges the problem has been recently shown to be np complete in the general case in this paper we introduce and study the new concept of spirality which is a measure of how an orthogonal drawing is rolled up and develop a theory on the interplay between spirality and number of bends of orthogonal drawings we exploit this theory to present polynomial time algorithms for two significant classes of graphs series parallel graphs and 3 planar graphs series parallel graphs arise in a variety of problems such as scheduling electrical networks data flow analysis database logic programs and circuit layout also they play a central role in planarity problems furthermore drawings of 3 planar graphs are a classical field of investigation

high order discretization of a stable time domain integral equation for 3d acoustic scattering abstract we develop a high order explicit method for acoustic scattering in three space dimensions based on a combined field time domain integral equation the spatial discretization of nystrom type uses gaussian quadrature on panels combined with a special treatment of the weakly singular kernels arising in near neighbor interactions in time a new class of convolution splines is used in a predictor corrector algorithm experiments on a torus and a perturbed torus are used to explore the stability and accuracy of the proposed scheme this involved around one thousand solver runs at up to 8th order and up to around 20 000 spatial unknowns demonstrating 5 9 digits of accuracy in addition we show that parameters in the combined field formulation chosen on the basis of analysis for the sphere and other convex scatterers work well in these cases

a fast algorithm for three dimensional potential fields calculation fast fourier transform on multipoles in this paper we present a fast algorithm for rapid calculation of the potential fields in three dimensions this method arises from an observation that potential evaluation using the multipole to local expansion translation operator can be expressed as a series discrete convolutions of the multipole moments with their associated spherical harmonics functions the high efficiency of the algorithm is primarily due to the use of fft algorithms to evaluate the numerous discrete convolutions we refer to it as the fast fourier transform on multipoles fftm method it is demonstrated that fftm is all accurate method it is significantly more accurate than fmm for a given order of expansion it is also shown that the algorithm has computational complexity of o na where a ranges from 1 0 to 1 3

orbital stability of periodic standing waves for the logarithmic klein gordon equation abstract the main goal of this paper is to present orbital stability results of periodic standing waves for the one dimensional logarithmic klein gordon equation to do so we first use compactness arguments and a non standard analysis to obtain the existence and uniqueness of weak solutions for the associated cauchy problem in the energy space second we prove the orbital stability of standing waves using a stablity analysis of conservative systems

numerical accuracy of a pade type non reflecting boundary condition for the finite element solution of acoustic scattering problems at high frequency the present text deals with the numerical solution of two dimensional high frequency acoustic scattering problems using a new high order and asymptotic pade type artificial boundary condition the pade type condition is easy to implement in a galerkin least squares iterative finite element solver for arbitrarily convex shaped boundaries the method accuracy is investigated for different model problems and for the scattering problem by a submarine shaped scatterer as a result relatively small computational domains optimized according to the shape of the scatterer can be considered while yielding accurate computations for high frequencies copyright 2005 john wiley sons ltd

the computational complexity of the self avoiding walk on random lattices it is proved that finding a minimum energy n step self avoiding walk on a 2d square lattice with assigned site energies is an np complete problem however if the fraction of lowest energy sites exceeds the percolation threshold then it is argued the problem can almost always be solved in polynomial time

on the computational complexity of upward and rectilinear planarity testing a directed graph is upward planar if it can be drawn in the plane such that every edge is a monotonically increasing curve in the vertical direction and no two edges cross an undirected graph is rectilinear planar if it can be drawn in the plane such that every edge is a horizontal or vertical segment and no two edges cross testing upward planarity and rectilinear planarity are fundamental problems in the effective visualization of various graph and network structures for example upward planarity is useful for the display of order diagrams and subroutine call graphs while rectilinear planarity is useful for the display of circuit schematics and entity relationship diagrams we show that upward planarity testing and rectilinear planarity testing are np complete problems we also show that it is np hard to approximate the minimum number of bends in a planar orthogonal drawing of an n vertex graph with an o n 1 epsilon error for any epsilon 0

a fast higher order solver for scattering by penetrable bodies in three dimensions in this paper we introduce a new fast higher order solver for scattering by inhomogeneous media in three dimensions as in previously existing methods the low complexity of our integral equation method o nlogn operations for an n point discretization is obtained through extensive use of the fast fourier transform fft for the evaluation of convolutions however the present approach obtains significantly higher order accuracy than these previous approaches yielding at worst third order far field accuracy or substantially better for smooth scatterers even for discontinuous and complex refractive index distributions possibly containing severe geometric singularities such as corners and cusps the increased order of convergence of our method results from i a partition of unity decomposition of the green s function into a smooth part with unbounded support and a singular part with compact support and ii replacement of the possibly discontinuous scatterer by an appropriate fourier smoothed scatterer the resulting convolutions can then be computed with higher order accuracy by means of o nlogn ffts we present a parallel implementation of our approach and demonstrate the method s efficiency and accuracy through a variety of computational examples for a very large scatterer considered earlier in the literature with a volume of 3648 l 3 where l is the wavelength using the same number of points per wavelength and in computing times comparable to those required by the previous approach the present algorithm produces far field values whose errors are two orders of magnitude smaller than those reported previously

computing the ground state and dynamics of the nonlinear schrodinger equation with nonlocal interactions via the nonuniform fft we present efficient and accurate numerical methods for computing the ground state and dynamics of the nonlinear schrodinger equation nlse with nonlocal interactions based on a fast and accurate evaluation of the long range interactions via the nonuniform fast fourier transform nufft we begin with a review of the fast and accurate nufft based method in 29 for nonlocal interactions where the singularity of the fourier symbol of the interaction kernel at the origin can be canceled by switching to spherical or polar coordinates we then extend the method to compute other nonlocal interactions whose fourier symbols have stronger singularity at the origin that cannot be canceled by the coordinate transform many of these interactions do not decay at infinity in the physical space which adds another layer of complexity since it is more difficult to impose the correct artificial boundary conditions for the truncated bounded computational domain the performance of our method against other existing methods is illustrated numerically with particular attention on the effect of the size of the computational domain in the physical space finally to study the ground state and dynamics of the nlse we propose efficient and accurate numerical methods by combining the nufft method for potential evaluation with the normalized gradient flow using backward euler fourier pseudospectral discretization and time splitting fourier pseudospectral method respectively extensive numerical comparisons are carried out between these methods and other existing methods for computing the ground state and dynamics of the nlse with various nonlocal interactions numerical results show that our scheme performs much better than those existing methods in terms of both accuracy and efficiency

an accurate spectral discontinuous finite element formulation of a phase space based level set approach to geometrical optics in this paper we introduce a new numerical procedure for simulations in geometrical optics that based on the recent development of eulerian phase space formulations of the model can deliver very accurate uniformly resolved solutions which can be made to converge with arbitrarily high orders in general geometrical configurations following previous treatments the scheme is based on the evolution of a wavefront in phase space defined as the intersection of level sets satisfying the relevant liouville equation in contrast with previous work however our numerical approximation is specifically designed i to take full advantage of the smoothness of solutions ii to facilitate the treatment of scattering obstacles all while retaining high order convergence characteristics indeed to incorporate the full regularity of solutions that results from the unfolding of singularities our method is based on their spectral representation to enable a simple high order treatment of scattering boundaries on the other hand we resort to a discontinuous galerkin finite element method for the solution of the resulting system of equations the procedure is complemented with the use of a recently derived strong stability preserving runge kutta ssp rk scheme for the time integration that as we demonstrate allows for overall approximations that are rapidly convergent

rapid solution of integral equations of scattering theory in two dimensions abstract the present paper describes an algorithm for rapid solution of boundary value problems for the helmholtz equation in two dimensions based on iteratively solving integral equations of scattering theory cpu time requirements of previously published algorithms of this type are of the order n 2 where n is the number of nodes in the discretization of the boundary of the scatterer the cpu time requirements of the algorithm of the present paper are n 4 3 and can be further reduced making it considerably more practical for large scale problems

superalgebraically convergent smoothly windowed lattice sums for doubly periodic green functions in three dimensional space this work part i in a two part series presents i a simple and highly efficient algorithm for evaluation of quasi periodic green functions as well as ii an associated boundary integral equation method for the numerical solution of problems of scattering of waves by doubly periodic arrays of scatterers in three dimensional space except for certain wood frequencies at which the quasi periodic green function ceases to exist the proposed approach which is based on smooth windowing functions gives rise to tapered lattice sums which converge superalgebraically fast to the green function that is faster than any power of the number of terms used this is in sharp contrast to the extremely slow convergence exhibited by the lattice sums in the absence of smooth windowing the wood frequency problem is treated in part ii this paper establishes rigorously the superalgebraic convergence of the windowed lattice sums a variety of numerical results demonstrate the practical efficiency of the proposed approach

stability properties of periodic waves for the klein gordon equation with quintic nonlinearity in this paper we prove results of existence and orbital instability of periodic waves for the klein gordon equation with quintic nonlinearity the main tool used here is a recent computational approach in order to decide about the nonpositive spectrum of the linearized operator combined with the grillakis shatah and strauss stability theory