treatment of a sloping fluid solid interface and sediment layering with the seismo acoustic parabolic equation the parabolic equation method is extended to handle problems in seismo acoustics that have multiple fluid and solid layers continuous depth dependence within layers and sloping interfaces between layers the medium is approximated in terms of a series of range independent regions and a single scattering approximation is used to compute transmitted fields across the vertical interfaces between regions the approach is implemented in terms of a set of dependent variables that is well suited to piecewise continuous depth dependence in the elastic parameters but one of the fluid solid interface conditions in that formulation involves a second derivative that complicates the treatment of sloping interfaces this issue is resolved by using a non centered four point difference formula for the second derivative the approach is implemented using a matrix decomposition that is efficient when the parameters of the medium have a general dependence within the upper layers of the sediment but only depend on depth in the water column and deep within the sediment

multipliers paramultipliers and weak strong uniqueness for the navier stokes equations abstract in this article we describe spaces p such that if u is a weak in the sense of leray j leray sur le mouvement d un fluide visqueux remplissant l espace acta math 63 1934 193 248 solution of the navier stokes system for some initial data u 0 and if u belongs to p then u is unique in the class of weak solutions we say then that weak strong uniqueness holds it turns out that the proof of such results relies on the boundedness of a trilinear functional f l 2 h l 2 h p r where belong to 0 1 in order to find optimal conditions for the boundedness of f we are led to describing spaces of multipliers and of paramultipliers that is functions which map by classical pointwise product or by paraproduct a given sobolev spaces in another given sobolev space the study of these spaces enables us to give conditions for weak strong uniqueness which generalise all previously known results from the famous serrin criterion j serrin the initial value problem for the navier stokes equations in r e langer ed nonlinear problems univ of wisconsin press 1963 pp 69 98 to the recent conditions formulated by lemarie rieusset p g lemarie rieusset recent developments in the navier stokes problem chapman and hall 2003

net and prune a linear time algorithm for euclidean distance problems we provide a general framework for getting expected linear time constant factor approximations and in many cases fptas s to several well known problems in computational geometry such as k center clustering and farthest nearest neighbor the new approach is robust to variations in the input problem and yet it is simple elegant and practical in particular many of these well studied problems which fit easily into our framework either previously had no linear time approximation algorithm or required rather involved algorithms and analysis a short list of the problems we consider include farthest nearest neighbor k center clustering smallest disk enclosing k points kth largest distance kth smallest m nearest neighbor distance kth heaviest edge in the mst and other spanning forest type problems problems involving upward closed set systems and more finally we show how to extend our framework such that the linear running time bound holds with high probability

navier stokes flow in the weighted hardy space with applications to time decay problem abstract the asymptotic expansions of the navier stokes flow in r n and the rates of decay are studied with aid of weighted hardy spaces fujigaki and miyakawa 12 miyakawa 28 proved the nth order asymptotic expansion of the navier stokes flow if initial data decays like 1 x n 1 and if nth moment of initial data is finite in the present paper it is clarified that the moment condition for initial data is essential in order to obtain higher order asymptotic expansion of the flow and to consider the rapid time decay problem the second author 39 established the weighted estimates of the strong solutions in the weighted hardy spaces with small initial data which belongs to l n and a weighed hardy space firstly the refinement of the previous work 39 is achieved with alternative proof then the existence time of the solution in the weighted hardy spaces is characterized without any hardy norm as a result in two dimensional case the smallness condition on initial data is completely removed as an application the rapid time decay of the flow is investigated with aid of asymptotic expansions and of the symmetry conditions introduced by brandolese 3

fitting a cm smooth function to data iii suppose we are given a finite subset e rn and a function f e r how to extend f to a cm function f rn r with cm norm of the smallest possible order of magnitude in this paper and in 20 we tackle this question from the perspective of theoretical computer science we exhibit algorithms for constructing such an extension function f and for computing the order of magnitude of its cm norm the running time of our algorithms is never more than cn log n where n is the cardinality of e and c is a constant depending only on m and n

l3 solutions of the liquid crystals system abstract in this article we study the simplified system of the original ericksen leslie equations for the flow of liquid crystals it is shown that the l 3 solutions of the cauchy problem for the three dimensional incompressible liquid crystal system are smooth

electrostatic fields without singularities theory algorithms and error analysis the following problems that arise in the computation of electrostatic forces and in the boundary element method are considered given two convex interior disjoint polyhedra in 3 space endowed with a volume charge density which is a polynomial in the cartesian coordinates of r 3 compute the coulomb force acting on them given two interior disjoint polygons in 3 space endowed with a surface charge density which is polynomial in the cartesian coordinates of r 3 compute the normal component of the coulomb force acting on them for both problems adaptive gaussian approximation algorithms are given which for n gaussian points in time o n achieve absolute error o c n for a constant c 1 such a result improves upon previously known best asymptotic bounds this result is achieved by blending techniques from integral geometry computational geometry and numerical analysis in particular integral geometry is used in order to represent the forces as integrals whose kernal is free from singularities

extension criterion via two components of vorticity on strong solutions to the 3 d navier stokes equations we shall show that only two components of vorticity play an essential role to determine possibility of extension of the time interval for the local strong solution to the navier stokes equations then we shall apply our extension theorem to regularity criterion on weak solutions due to serrin and beirao da veiga chae choe proved the same criterion as beirao da veiga only by means of the two components of vorticity we deal with the critical case which they excluded our criterion may be regarded as the generalization of the result of beal kato majda from l to bmo

fast approximate k nn graph construction for high dimensional data via recursive lanczos bisection nearest neighbor graphs are widely used in data mining and machine learning a brute force method to compute the exact knn graph takes dn2 time for n data points in the d dimensional euclidean space we propose two divide and conquer methods for computing an approximate knn graph in dnt time for high dimensional data large d the exponent t 1 2 is an increasing function of an internal parameter which governs the size of the common region in the divide step experiments show that a high quality graph can usually be obtained with small overlaps that is for small values of t a few of the practical details of the algorithms are as follows first the divide step uses an inexpensive lanczos procedure to perform recursive spectral bisection after each conquer step an additional refinement step is performed to improve the accuracy of the graph finally a hash table is used to avoid repeating distance calculations during the divide and conquer process the combination of these techniques is shown to yield quite effective algorithms for building knn graphs

sobolev extension by linear operators let l r be the sobolev space of functions with mth derivatives lying in l r assume that n p for e r let l e denote the space of restrictions to e of functions in l r we show that there exists a bounded linear map t lm p e l r such that for any f l e we have tf f on e we also give a formula for the order of magnitude of f lm p e for a given f e r when e is finite

a mapping approach for handling sloping interfaces a mapping approach for handling sloping interfaces in parabolic equation solutions is developed and tested at each range the medium is rigidly translated vertically so that a sloping interface becomes horizontal to simplify the approach the slope is assumed to be small and the extra terms that arise in the wave equation under the mapping are neglected the effects of these terms can be approximately accounted for by applying a leading order correction to the phase the mapping introduces variations in topography which are relatively easy to handle for the case of a pressure release boundary condition the accuracy of the approach is demonstrated for problems involving fluid sediments the approach should also be accurate for problems involving elastic sediments and should be useful for solving three dimensional problems involving variable topography

homogeneous statistical solutions and local energy inequality for 3d navier stokes equations we are interested in space time spatially homogeneous statistical solutions of navier stokes equations in space dimension three we first review the construction of such solutions and introduce convenient tools to study the pressure gradient then we show that given a spatially homogeneous initial measure with finite energy density one can construct a homogeneous statistical solution concentrated on weak solutions which satisfy the local energy inequality

existence and positivity of solutions of a fourth order nonlinear pde describing interface fluctuations we study the partial differential equation which arose originally as a scaling limit in the study of interface fluctuations in a certain spin system in that application x lies in r but here we study primarily the periodic case x e s we establish existence uniqueness and regularity of solutions locally in time for positive initial data in h s and prove the existence of several families of lyapunov functions for the evolution from the latter we establish a sharp connection between existence globally in time and positivity preservation if o t is a maximal half open interval of existence for a positive solution of the equation with t 1 16r2 then t 03 and lim m w r exists and is constant we discuss also some explicit solutions and propose a generalization to higher dimensions 0 1994 john wiley

comparison of algorithms for solving parabolic wave equations the three types of algorithms that have been developed for solving parabolic wave equations are compared until recently it was necessary to choose between the finite difference and split step fourier algorithms and make a trade off between efficiency and capability test problems are presented to illustrate the efficiency of the split step pade algorithm which provides the capability of the finite difference algorithm for deep water problems the split step pade algorithm provides efficiency comparable to the split step fourier algorithm for the shallow water problems that are currently of interest the split step pade algorithm can be more than an order of magnitude faster than the split step fourier algorithm

impedance matched absorbers for finite difference parabolic equation algorithms in this paper a perfectly matched layer pml absorber recently introduced into the electromagnetic propagation literature by berenger j comput phys 114 185 200 1994 is adapted for use with both paraxial and wide angle acoustic parabolic equations pes our procedure incorporates an imaginary component into the transverse coordinate that mimics the introduction of a fictitious absorber on the edge of the computational grid use of such an impedance matched layer can significantly reduce spurious reflections compared to physical absorbing layer methods and thus allows a smaller number of boundary points to be employed in pe calculations numerical results obtained with several higher order propagator approximations confirm that such impedance matched absorbers efficiently eliminate reflections

finite difference solution to the parabolic wave equation in this article an implicit finite difference ifd scheme is introduced for solving underwater acoustic wave propagation problems this scheme is applied to the parabolic wave equation model for the acoustic problem with a free surface and arbitrary bottom and bottom boundary conditions the mathematical theory of the implicit finite difference method as applied to the parabolic equation is developed and a computer model to implement the ifd is introduced representative problems are presented to demonstrate the capability of the ifd model advantages and limitations of the ifd are discussed

an optimal algorithm for approximate nearest neighbor searching fixed dimensions consider a set of s of n data points in real d dimensional space r d where distances are measured using any minkowski metric in nearest neighbor searching we preprocess s into a data structure so that given any query point q r d is the closest point of s to q can be reported quickly given any positive real e data point p is a 1 e approximate nearest neighbor of q if its distance from q is within a factor of 1 e of the distance to the true nearest neighbor we show that it is possible to preprocess a set of n points in r d in o dn log n time and o dn space so that given a query point q r d and e 0 a 1 e approximate nearest neighbor of q can be computed in o c d e log n time where c d e d 1 6d e d is a factor depending only on dimension and e in general we show that given an integer k 1 1 e approximations to the k nearest neighbors of q can be computed in additional o kd log n time

implementing the fast multipole method in three dimensions the rokhlin greengard fast multipole algorithm for evaluating coulomb and multipole potentials has been implemented and analyzed in three dimensions the implementation is presented for bounded charged systems and systems with periodic boundary conditions the results include timings and error characterizations

the derivative of a waveguide acoustic field with respect to a three dimensional sound speed perturbation semianalytic expressions are derived for the first order derivative of a pressure field in a laterally homogeneous waveguide with respect to an arbitrary three dimensional refractive index perturbation in either the water column or ocean bottom these expressions for the environmental derivative derived using an adjoint method require a three dimensional spatial correlation between two green s functions weighted by an environmental parameter basis function with the green s functions expressed in terms of normal modes when a particular set of orthogonal spatial basis functions is chosen the three dimensional spatial integral can be converted into a set of one dimensional integrations over depth and azimuth the use of the orthogonal basis permits environmental derivatives to be computed for an arbitrary sound speed perturbation to illustrate the formulas a simple sensitivity study is presented that explores under what circumstances three dimensional plane wave and cylindrical perturbations produ

the initial boundary value problem for thin plate flow of incompressible non newtonian viscous fluids abstract in this paper the particular model that describes the thin plate flow of incompressible non newtonian viscous fluids is investigated this model which arises from polymer processing concerns the initial boundary problem of nonstationary thin plate flow of non newtonian viscous incompressible fluids by using monotone operator theory and schauder s fixed point theorem an existence and uniqueness theorem for a generalized solution is obtained and the regularity for this solution is tested

on a larger class of stable solutions to the navier stokes equations in exterior domains where w w x w1 x wn x and x denote the unknown velocity vector and the unknown pressure of the fluid at point x respectively while f f x f i j x i j 1 n is the given tensor with div f n j 1 f 1 j xj nj 1 f n j xj denoting the external force in our previous paper 23 we established the theory of existence uniqueness and regularity for solutions to n s0 in the class w ln

a cost optimal parallel algorithm for computing force field in n body simulations on a crew pram abstract we consider the following force field computation problem given a cluster of n particles in three dimensional space compute the force exerted on each particle by the other particles depending on different applications the pairwise interaction could be either gravitational or lennard jones in both cases the force between two particles vanishes as the distance between them approaches to infinity since there are n n 1 2 pairs direct method requires n 2 time for force evaluation which is very expensive for astronomical simulations in 1985 and 1986 two famous n log n time hierarchical tree algorithms were published by appel siam j sci statist comput 6 1985 85 103 and by barnes and hut nature 324 1980 446 449 respectively in a recent paper we presented a linear time algorithm which builds the oct tree bottom up and showed that appel s algorithm can be implemented in n sequential time in this paper we present an algorithm which computes the force field in log n time on a n log n processor crew pram a key to this optimal parallel algorithm is replacing a recursive top down force calculation procedure of appel by an equivalent non recursive bottom up procedure our parallel algorithm also yields a new n time sequential algorithm for force field computation

a scalable parallel wideband mlfma for efficient electromagnetic simulations on large scale clusters the development of the multilevel fast multipole algorithm mlfma and its multiscale variants have enabled the use of integral equation ie based solvers to compute scattering from complicated structures development of scalable parallel algorithms to extend the reach of these solvers has been a topic of intense research for about a decade in this paper we present a new algorithm for parallel implementation of ie solver that is augmented with a wideband mlfma and scalable on large number of processors the wideband mlfma employed here to handle multiscale problems is a hybrid combination of the accelerated cartesian expansion ace and the classical mlfma the salient feature of the presented parallel algorithm is that it is implicitly load balanced and exhibits higher performance this is achieved by developing a strategy to partition the mlfma tree and hence the associated computations in a self similar fashion among the parallel processors as detailed in the paper the algorithm employs both spatial and direction partitioning approaches in a flexible manner to ensure scalable performance plethora of results are presented here to exhibit the scalability of this algorithm on 512 and more processors

global stability of large solutions to the 3d navier stokes equations we prove the stability of mildly decaying global strong solutions to the navier stokes equations in three space dimensions combined with previous results on the global existence of large solutions with various symmetries this gives the first global existence theorem for large solutions with approximately symmetric initial data the stability of unforced 2d flow under 3d perturbations is also obtained

asymptotic stability of large solutions with large perturbation to the navier stokes equations abstract consider weak solutions w of the navier stokes equations in serrin s class w l 0 l q for 2 3 q 1 with 3 q where is a general unbounded domain in r 3 we shall show that although the initial and external disturbances from w are large every perturbed flow v with the energy inequality converges asymptotically to w as v t w t l 2 0 v t w t l 2 o t 1 2 as t

long time decay to the lei lin solution of 3d navier stokes equations abstract in this paper we prove if u c 0 x 1 r 3 is a global solution of 3d navier stokes equations then u t x 1 decays to zero as time goes to infinity fourier analysis and standard techniques are used

a parallel version of the fast multipole method abstract this paper presents a parallel version of the fast multipole method fmm the fmm is a recently developed scheme for the evaluation of the potential and force fields in systems of particles whose interactions are coulombic or gravitational in nature the sequential method requires o n operations to obtain the fields due to n charges rather than the o n2 operations required by the direct calculation here we describe the modifications necessary for implementation of the method on parallel architectures and show that the expected time requirements grow as log n when using n processors numerical results are given for a shared memory machine the encore multimax 320

a higher order parabolic equation for wave propagation in an ocean overlying an elastic bottom a higher order elastic parabolic equation hepe is derived for wave propagation in depth dependent and weakly range dependent fluid solid media galerkin s method is used to discretize the depth operators in the hepe within layers in which depth variations in the lame constants and density are continuous discontinuities in material properties are handled with centered differences for the interface conditions between layers the numerical solution of the hepe also involves the method of alternating directions and crank nicolson integration the hepe is applied to underwater wave propagation problems involving a water column over an elastic bottom including a weakly range dependent problem the accuracy of the hepe is demonstrated with benchmark calculations

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

on weighted norm estimates for nonstationary incompressible navier stokes flows in a 3d exterior domain abstract we study the time decay of weighted norms of weak and strong solutions to the navier stokes equations in a 3d exterior domain moment estimates for weak solutions and weighted l q estimates for strong solutions are deduced both of which seem to be optimal the relation is discussed between the space time decay and the vanishing of the total net force exerted by the fluid to the body a class of initial data is given so that the total net force associated to the corresponding fluid flows does not vanish

generalization of the split step pade solution the split step pade solution j acoust soc am 93 1736 1742 1993 is generalized to simultaneously permit arbitrarily large range steps in range independent regions and dense sampling in range although the range step size is limited by the rate of range dependence range steps of many wavelengths are possible for many range dependent ocean acoustics problems at each range point in a dense grid the acoustic pressure is defined over depth in terms of a linear combination of solutions of tridiagonal systems on a multiprocessor computer each of the tridiagonal systems is solved on a different processor and the acoustic field can be generated throughout a large range independent region in the time required by standard finite difference solutions to take one range step the generalized split step pade solution is also applicable to elastic media

fast construction of nets in low dimensional metrics and their applications we present a near linear time algorithm for constructing hierarchical nets in finite metric spaces with constant doubling dimension this data structure is then applied to obtain improved algorithms for the following problems approximate nearest neighbor search well separated pair decomposition spanner construction compact representation scheme doubling measure and computation of the approximate lipschitz constant of a function in all cases the running preprocessing time is near linear and the space being used is linear

dynamic autotuning of adaptive fast multipole methods on hybrid multicore cpu and gpu systems dynamic autotuning of adaptive fast multipole methods on hybrid multicore cpu and gpu systems

nonnegative c2 r2 interpolation abstract in this paper we prove two improved versions of the finiteness principle for nonnegative c 2 r 2 interpolation previously proven by fefferman israel and luli the first version sharpens the finiteness constant to 64 and the second version carries better computational practicality along the way we also provide a detailed construction of nonnegative c 2 interpolants in one dimension and prove the nonexistence of a bounded linear c 2 extension operator that preserves nonnegativity

space time tradeoffs for approximate nearest neighbor searching nearest neighbor searching is the problem of preprocessing a set of n point points in d dimensional space so that given any query point q it is possible to report the closest point to q rapidly in approximate nearest neighbor searching a parameter e 0 is given and a multiplicative error of 1 e is allowed we assume that the dimension d is a constant and treat n and e as asymptotic quantities numerous solutions have been proposed ranging from low space solutions having space o n and query time o log n 1 ed 1 to high space solutions having space roughly o n log n ed and query time o log n e we show that there is a single approach to this fundamental problem which both improves upon existing results and spans the spectrum of space time tradeoffs given a tradeoff parameter where 2 1 e we show that there exists a data structure of space o n d 1 log 1 e that can answer queries in time o log n 1 e d 1 2 when e 2 this yields a data structure of space o n log 1 e that can answer queries in time o log n 1 e d 1 2 when e 1 e it provides a data structure of space o n ed 1 log 1 e that can answer queries in time o log n e our results are based on a data structure called a t e avd which is a hierarchical quadtree based subdivision of space into cells each cell stores up to t representative points of the set such that for any query point q in the cell at least one of these points is an approximate nearest neighbor of q we provide new algorithms for constructing avds and tools for analyzing their total space requirements we also establish lower bounds on the space complexity of avds and show that up to a factor of o log 1 e our space bounds are asymptotically tight in the two extremes e 2 and e 1 e

a scaling and energy equality preserving approximation for the 3d navier stokes equations in the finite energy case abstract we discuss a new model inspired by the work of vishik and fursikov approximating the 3d navier stokes equations which preserves the scaling as in the navier stokes equations and thus allows the study of self similar solutions using some energy estimates and leray s limiting process we show the existence of a solution of this model in the finite energy case and the energy equality and inequality fulfilled by it this approximation can be shown to converge to the navier stokes equations using a mild approach based on the approximated pressure and the solution satisfies scheffer s local energy inequality an essential tool for proving caffarelli kohn and nirenberg s regularity criterion we also give a partial result of self similarity satisfied by the approximated solution in the infinite energy case

global smooth solutions of mhd equations with large data abstract in this paper we establish the global existence of smooth solutions of the three dimensional mhd system for a class of large initial data both the initial velocity and magnetic field can be arbitrarily large in the critical norm

nonuniform depth grids in parabolic equation solutions the parabolic wave equation is solved using a finite difference solution in depth that involves a nonuniform grid the depth operator is discretized using galerkin s method with asymmetric hat functions examples are presented to illustrate that this approach can be used to improve efficiency for problems in ocean acoustics and seismo acoustics for shallow water problems accuracy is sensitive to the precise placement of the ocean bottom interface this issue is often addressed with the inefficient approach of using a fine grid spacing over all depth efficiency may be improved by using a relatively coarse grid with nonuniform sampling to precisely position the interface efficiency may also be improved by reducing the sampling in the sediment and in an absorbing layer that is used to truncate the computational domain nonuniform sampling may also be used to improve the implementation of a single scattering approximation for sloping fluid solid interfaces

improving error bounds for multipole based treecodes rapid evaluation of potentials in particle systems is an important time consuming step in many physical simulations over the past decade the development of treecodes such as the fast multipole method and barnes hut method has enabled large scale simulations in astrophysics molecular dynamics material science etc these methods use fixed degree polynomial approximations of the potential at a point due to a set of particles in a hierarchical representation of the particle system in this paper we present analysis and experiments to illustrate that fixed degree multipole approximations can lead to large aggregate errors we describe an alternate strategy based on careful selection of the multipole degree that leads to asymptotically lower errors while incurring minimal computation overhead first we estimate the error associated with each particle cluster interaction and the aggregate error for each particle then we describe a technique for computing the multipole degree of each interaction with a view to reducing aggregate error and establish the computational complexity of the new method following this we discuss numerical experiments that demonstrate favorable error properties of the new method at the expense of marginal increase in computation finally we report a parallel implementation on the sgi origin 2000 that achieves excellent speedup for problems from domains such as astrophysics and boundary element solvers

existence and generalized gevrey regularity of solutions to the kuramoto sivashinsky equation in rn abstract motivated by the work of foias and temam c foias r temam gevrey class regularity for the solutions of the navier stokes equations j funct anal 87 1989 359 369 we prove the existence and gevrey regularity of local solutions to the kuramoto sivashinsky equation in r n with initial data in the space of distributions the control on the gevrey norm provides an explicit estimate of the analyticity radius in terms of the initial data in the particular case when n 1 our analysis allows for initial data that are less smooth than that considered by grujic and kukavica z grujic i kukavica space analyticity for the navier stokes and related equations with initial data in l p j funct anal 152 1998 447 466

initial and boundary values for lαq lp solution of the navier stokes equations in the half space abstract in this paper we study the initial and boundary value problem of the navier stokes equations in the half space we prove the existence of weak solution u l q 0 l p r n 1 2 1 n p 2 q 0 n p with u l loc q 2 0 l loc p 2 r n for the solenoidal initial data h b p q 1 n p r n and the boundary data g l q 0 b p p 1 p r n 1 when h b p q 1 n p r n g l q 0 b p p 1 p r n 1 is small enough moreover the solution is unique in the class l q 0 t l p r n for any t if 0 and for some t if 0

forward self similar solutions of the fractional navier stokes equations abstract we study forward self similar solutions to the 3 d navier stokes equations with the fractional diffusion first we construct a global time forward self similar solutions to the fractional navier stokes equations with 5 6 1 for arbitrarily large self similar initial data by making use of the so called blow up argument moreover we prove that this solution is smooth in r 3 0 in particular when 1 we prove that the solution constructed by korobkov and tsai 2016 16 satisfies the decay estimate by establishing regularity of solution for the corresponding elliptic system which implies this solution has the same properties as a solution which was constructed in jia and sverak 2014 13

delaunay triangulations in o sort n time and more we present several results about delaunay triangulations dts and convex hulls in transdichotomous and hereditary settings i the dt of a planar point set can be computed in expected time o sort n on a word ram where sort n is the time to sort n numbers we assume that the word ram supports the shuffle operation in constant time ii if we know the ordering of a planar point set in x and in y direction its dt can be found by a randomized algebraic computation tree of expected linear depth iii given a universe u of points in the plane we construct a data structure d for delaunay queries for any p s u d can find the dt of p in expected time o vpv log log vuv iv given a universe u of points in 3 space in general convex position there is a data structure d for convex hull queries for any p s u d can find the convex hull of p in expected time o vpv log log vuv 2 v given a convex polytope in 3 space with n vertices which are colored with g 2 colors we can split it into the convex hulls of the individual color classes in expected time o n log log n 2 the results i iii generalize to higher dimensions where the expected running time now also depends on the complexity of the resulting dt we need a wide range of techniques most prominently we describe a reduction from dts to nearest neighbor graphs that relies on a new variant of randomized incremental constructions using dependent sampling

suitable solutions for the navier stokes problem with an homogeneous initial value this paper is devoted to the study of strong or weak solutions of the navier stokes equations in the case of an homogeneous initial data the case of small initial data is discussed for large initial data an approximation is developed in the spirit of a paper of vishik and fursikov qualitative convergence is obtained by use of the theory of muckenhoupt weights

approximate complex polynomial evaluation in near constant work per point given the n complex coefficients of a degree n 1 complex polynomial we wish to evaluate the polynomial at a large number m ge n of points on the complex plane this problem is required by many algebraic computations and so is considered in most basic algorithm texts e g a v aho j e hopcroft and j d ullman the design and analysis of computer algorithms addison wesley 1974 we assume an arithmetic model of computation where on each step we can execute an arithmetic operation which is computed exactly all previous exact algorithms c m fiduccia proceeding 4th annual acm symposium on theory of computing 1972 pp 88 93 h t kung fast evaluation and interpolation carnegie mellon 1973 a b borodin and i munro the computational complexity of algebraic and numerical problems american elsevier 1975 v pan a sadikou e landowne and o tiga comput math appl 25 1993 pp 25 30 cost at least work omega log 2 n per point and previously there were no known approximation algorithms for complex polynomial evaluation within the unit circle with work bounds better than the fastest known exact algorithms there are known approximation algorithms v rokhlin j complexity 4 1988 pp 12 32 v y pan j h reif and s r tate in proceedings 32nd annual ieee symposium on foundations of computer science 1992 pp 703 713 for polynomial evaluation at real points but these do not extend to evaluation at general points on the complex plane we provide approximation algorithms for complex polynomial evaluation that cost in many cases near constant amortized work per point let k log p epsilon where p is the sum of the moduli of the coefficients of the input polynomial p z let it tilde p z j be an epsilon approx of p z if epsilon upper bounds the modulus of the error of the approximation tilde p z j at each evaluation point zj that is p z j tilde p z j le epsilon note that epsilon is an absolute error bound rather than a relative error bound in many applications particularly in signal processing the evaluation points zj are fixed and require only polylogarithmic k log p epsilon o log o 1 n for these cases we get a surprising reduction in work by use of approximation algorithms as compared to the fastest known exact algorithms we epsilon approx complex degree n 1 polynomial evaluation at m ge n log n log 2 k fixed points on or within the unit disk in the complex plane in amortized work o log2 k per point which is o log2 log n for polylogarithmic k if the m points are not fixed then we have increased amortized work o log2 k log m per point which is o log m for polylogarithmic k and m ge n log n log k and is still substantially below the previous bound of omega log 2 m for known exact algorithms we further reduce our amortized bounds for special sets of evaluation points widely used in signal processing applications the chirp transform is equivalent to evaluating a complex degree n 1 polynomial at the chirp points which are zeta j j 0 dots m 1 for some fixed complex number zeta we epsilon approx complex degree n 1 polynomial evaluation at these m chirp points where m ge n log n log 2 k and zeta le 1 or ii m ge n and zeta le a function that limits to 1 for k o n and large n in amortized work o log k per point whereas the previous best bounds for exact evaluation via the chirp transform were omega log m per point a v aho k steiglitz and j d ullman siam j comput 4 1975 pp 533 539 using instead a reduction to approximate real polynomial evaluation by interpolation at the chebyshev points in total work o n log k we epsilon approx the evaluation of a degree n polynomial at the first n powers of the n th root of unity where n ge omega n 2 k and epsilon approx the n point dft for certain inputs with descending coefficient magnitude all of our results require polylogarithmic that is logo 1 n depth with the same work bounds we also provide a lower bound for a wide class of schemes for approximate evaluation of a degree n 1 polynomial on the unit circle namely we prove that if a scheme uses an approximation polynomial of degree k 1 then it can be convergent only over a small fraction o k n of the unit circle we believe this is the first lower bound of this sort proved and the proof uses an interesting reduction to the approximation of a matrix product by a matrix of reduced rank

a variational data assimilation system for the range dependent acoustic model using the representer method theoretical derivations this study presents the theoretical framework for variational data assimilation of acoustic pressure observations into an acoustic propagation model namely the range dependent acoustic model ram ram uses the split step pade algorithm to solve the parabolic equation the assimilation consists of minimizing a weighted least squares cost function that includes discrepancies between the model solution and the observations the minimization process which uses the principle of variations requires the derivation of the tangent linear and adjoint models of the ram the mathematical derivations are presented here and for the sake of brevity a companion study presents the numerical implementation and results from the assimilation simulated acoustic pressure observations

a single scattering correction for the seismo acoustic parabolic equation an efficient single scattering correction that does not require iterations is derived and tested for the seismo acoustic parabolic equation the approach is applicable to problems involving gradual range dependence in a waveguide with fluid and solid layers including the key case of a sloping fluid solid interface the single scattering correction is asymptotically equivalent to a special case of a single scattering correction for problems that only have solid layers kusel et al j acoust soc am 121 808 813 2007 the single scattering correction has a simple interpretation conservation of interface conditions in an average sense that facilitated its generalization to problems involving fluid layers promising results are obtained for problems in which the ocean bottom interface has a small slope

adjoint modeling for acoustic inversion the use of adjoint modeling for acoustic inversion is investigated an adjoint model is derived from a linearized forward propagation model to propagate data model misfit at the observation points back through the medium to the medium perturbations not being accounted for in the model this adjoint model can be used to aid in inverting for these unaccounted medium perturbations adjoint methods are being applied to a variety of inversion problems but have not drawn much attention from the underwater acoustic community this paper presents an application of adjoint methods to acoustic inversion inversions are demonstrated in simulation for both range independent and range dependent sound speed profiles using the adjoint of a parabolic equation model sensitivity and error analyses are discussed showing how the adjoint model enables calculations to be performed in the space of observations rather than the often much larger space of model parameters using an adjoint model enables directions of steepest descent in the model parameters what we invert for to be calculated using far fewer modeling runs than if a forward model only were used

a decomposition of multidimensional point sets with applications to k nearest neighbors and n body potential fields we define the notion of a well separated pair decomposition of points in d dimensional space we then develop efficient sequential and parallel algorithms for computing such a decomposition we apply the resulting decomposition to the efficient computation of k nearest neighbors and n body potential fields

approximating the stretch factor of euclidean graphs there are several results available in the literature dealing with efficient construction of t spanners for a given set s of n points in ir d t spanners are euclidean graphs in which distances between vertices in g are at most t times the euclidean distances between them in other words distances in g are stretched by a factor of at most t we consider the interesting dual problem given a euclidean graph g whose vertex set corresponds to the set s compute the stretch factor of g i e the maximum ratio between distances in g and the corresponding euclidean distances it can trivially be solved by solving the all pairs shortest path problem however if an approximation to the stretch factor is sufficient then we show it can be efficiently computed by making only o n approximate shortest path queries in the graph g we apply this surprising result to obtain efficient algorithms for approximating the stretch factor of euclidean graphs such as paths cycles trees planar graphs and general graphs the main idea behind the algorithm is to use callahan and kosaraju s well separated pair decomposition

fast greedy algorithms for constructing sparse geometric spanners given a set v of n points in ir d and a real constant t 1 we present the first o nlog n time algorithm to compute a geometric t spanner on v a geometric t spanner on v is a connected graph g v e with edge weights equal to the euclidean distances between the endpoints and with the property that for all u v in v the distance between u and v in g is at most t times the euclidean distance between u and v the spanner output by the algorithm has o n edges and weight o 1 cdot wt mst and its degree is bounded by a constant

greengard s n body algorithm is not order n greengard s n body algorithm claims to compute the pairwise interactions in a system of n particles in o n time for a fixed precision in this paper we show that the choice of precision is not independent of n and has a lower bound of log n we use this result to show that greengard s algorithm is not o n

optimal nonlocal boundary control of the wide angle parabolic equation for inversion of a waveguide acoustic field this paper applies the concept of optimal boundary control for solving inverse problems in shallow water acoustics to treat the controllability problem a continuous analytic adjoint model is derived for the claerbout wide angle parabolic equation pe using a generalized nonlocal impedance boundary condition at the water bottom interface while the potential of adjoint methodology has been recently demonstrated for ocean acoustic tomography this approach combines the advantages of exact transparent boundary conditions for the wide angle pe with the concept of adjoint based optimal control in contrast to meta heuristic approaches the inversion procedure itself is directly controlled by the waveguide physics and in a numerical implementation based on conjugate gradient optimization many fewer iterations are required for assessment of an environment that is supported by the underlying subbottom model furthermore since regularization schemes are particularly important to enhance the performance of full field acoustic inversion special attention is devoted to the application of penalization methods to the adjoint optimization formalism regularization incorporates additional information about the desired solution in order to stabilize ill posed inverse problems and identify useful solutions a feature that is of particular importance for inversion of field data sampled on a vertical receiver array in the presence of measurement noise and modeling uncertainty results with test data show that the acoustic field and the bottom properties embedded in the control parameters can be efficiently retrieved

a wide angle split step algorithm for the parabolic equation in this paper we describe a new wide angle parabolic equation based on an operator splitting that permits the use of a marching type fourier transform solution method the equation was first presented by feit and fleck appl opt 17 3990 3998 1978 for studying propagation within optical fibers existing computer codes which numerically solve the standard parabolic equation of ocean acoustics by the split step algorithm of tappert and hardin are easily modified to accommodate the wide angle capability of the new equation in addition since the new wide angle equation is less sensitive to the value of the reference wavenumber the effects of phase errors are greatly reduced the results of a simple error analysis indicate that improved accuracy can be achieved by the new wide angle equation for propagation conditions typical of deep ocean environments this is supported by our numerical experience a summary of which is presented in the paper for test cases where the variation of the acoustic index

a split step pade solution for the parabolic equation method a split step pade solution is derived for the parabolic equation pe method higher order pade approximations are used to reduce both numerical errors and asymptotic errors e g phase errors due to wide angle propagation this approach is approximately two orders of magnitude faster than solutions based on pade approximations that account for asymptotic errors but not numerical errors in contrast to the split step fourier solution which achieves similar efficiency for some problems the split step pade solution is valid for problems involving very wide propagation angles large depth variations in the properties of the waveguide and elastic ocean bottoms the split step pade solution is practical for global scale problems

ill posedness for the navier stokes equations in critical besov spaces b q 1 abstract we study the cauchy problem for the incompressible navier stokes equations in two and higher spatial dimensions 0 1 u t u u u p 0 div u 0 u 0 x u 0 for arbitrarily small 0 we show that the solution map u 0 u in critical besov spaces b q 1 q 1 2 is discontinuous at origin it is known that the navier stokes equation is globally well posed for small data in bmo 1 20 taking notice of the embedding b q 1 bmo 1 q 2 we see that for sufficiently small 0 u 0 b q 1 q 2 can guarantee that 0 1 has a unique global solution in bmo 1 however this solution is instable in b q 1 and the solution can have an inflation in b q 1 for certain initial data

note on prodi serrin ladyzhenskaya type regularity criteria for the navier stokes equations in this article we prove new regularity criteria of the prodi serrin ladyzhenskaya type for the cauchy problem of the three dimensional incompressible navier stokes equations our results improve the classical lr 0 t ls regularity criteria for both velocity and pressure by factors of certain negative powers of the scaling invariant norms u l 3 and u h 1 2

minimal initial data for potential navier stokes singularities abstract assuming some initial data u 0 h 1 2 r 3 lead to a singularity for the 3d navier stokes equations we show that there are also initial data with the minimal h 1 2 norm which will produce a singularity

an o n time hierarchical tree algorithm for computing force field in n body simulations abstract we consider the following force field computation problem given a cluster of n particles in three dimensional space compute the force exerted on each particle by the other particles depending on different applications the pairwise interaction could be either gravitational or lennard jones in both cases the force between two particles vanishes as the distance between them approaches to infinity since there are n n 1 2 pairs direct method requires n 2 time for force evaluation which is very expensive for astronomical simulations in 1985 and 1986 two famous o n log n time hierarchical tree algorithms were published by appel 1985 and by barnes and hut 1986 respectively in this paper we show that appel s algorithm can be implemented in o n time

space time regularity of the koch tataru solutions to navier stokes equations abstract in koch tataru 2001 the authors have proved local and global well posedness of the navier stokes equations with small initial data u 0 b m o 1 and then the spatial analyticity of the koch tataru solution stated as t k 2 k u x t for any positive integer k has been presented by germain pavlovic staffilani 2007 where x t is the space defined by koch tataru 2001 see also definition 1 2 in this paper we shall present the space time regularity more precisely for any positive integer m k we have t m k 2 t m k u x t

multiple order derivatives of a waveguide acoustic field with respect to sound speed density and frequency an adjoint perturbative method is used to derive expressions for the first through third order derivatives of a pressure field with respect to sound speed density and frequency for the restricted case of a laterally homogenous waveguide in which environmental parameters are only a function of depth by using a normal mode green s function the three dimensional spatial correlation required by the standard acoustic adjoint equation can be reduced to a set of one dimensional depth integrals the resulting expressions for the first order derivative are similar to those obtained by previous perturbative approaches based on the depth separated wave equation but the approach followed here permits straightforward extension to higher order derivatives explicit evaluations of the expressions for a representative shallow water waveguide model are in excellent agreement with numerical finite difference computations an analysis of the expressions as a function of source receiver range finds the contributions t

well posedness for the navier stokes nernst planck poisson system in triebel lizorkin space and besov space with negative indices abstract this paper is concerned with the well posedness of the navier stokes nerst planck poisson system nsnpp let s p 2 n p we prove that the nsnpp has a unique local solution u v w e u t e v t e v t for u 0 v 0 w 0 in a subspace i e v u 1 v v 1 v v 1 of f 1 2 b p s p b p s p with u 0 0 we also prove that there exists a unique small global solution u v w e u e v e v for any small initial data u 0 v 0 w 0 f 1 2 b p s p b p s p with u 0 0