new examples of willmore tori in s4 using the generalized darboux transformation in the case of the clifford torus we construct for all pythagorean triples a family of willmore tori in s4 with willmore energy 2 n 2

reconstructing hyperbolic cross trigonometric polynomials by sampling along rank 1 lattices with given fourier coefficients the evaluation of multivariate trigonometric polynomials at the nodes of a rank 1 lattice leads to a one dimensional discrete fourier transform in many applications one is also interested in the reconstruction of the fourier coefficients from samples in the spatial domain we present necessary and sufficient conditions on rank 1 lattices allowing a stable reconstruction of trigonometric polynomials supported on hyperbolic crosses in addition we suggest approaches for determining suitable rank 1 lattices using a component by component algorithm we present numerical results for reconstructing trigonometric polynomials up to spatial dimension 100

large deviations for stochastic pde with levy noise abstract we prove a large deviation principle result for solutions of abstract stochastic evolution equations perturbed by small levy noise we use general large deviations theorems of varadhan and bryc coupled with the techniques of feng and kurtz 2006 15 viscosity solutions of integro partial differential equations in hilbert spaces and deterministic optimal control methods the laplace limit is identified as a viscosity solution of a hamilton jacobi bellman equation of an associated control problem we also establish exponential moment estimates for solutions of stochastic evolution equations driven by levy noise general results are applied to stochastic hyperbolic equations perturbed by subordinated wiener process

fast wave computation via fourier integral operators this paper presents a numerical method for time upscaling wave equations i e performing time steps not limited by the courant friedrichs lewy cfl condition the proposed method leverages recent work on fast algorithms for pseudodierential and fourier integral operators fio this algorithmic approach is not asymptotic it is shown how to construct an exact fio propagator by 1 solving hamilton jacobi equations for the phases and 2 sampling rows and columns of low rank matrices at random for the amplitudes the setting of interest is that of scalar waves in two dimensional smooth periodic media of class c 1 over the torus where the bandlimit n of the waves goes to innity in this setting it is demonstrated that the algorithmic complexity for solving the wave equation to xed time t 1 can be as low as o n 2 logn with controlled accuracy numerical experiments show that the time complexity can be lower than that of a spectral method in certain situations of physical interest

direct inversion of the nonequispaced fast fourier transform abstract various applications such as mri solution of pdes etc need to perform an inverse nonequispaced fast fourier transform nfft i e compute m fourier coefficients from given n nonequispaced data in the present paper we consider direct methods for the inversion of the nfft we introduce algorithms for the setting m n as well as for the underdetermined and overdetermined cases for the setting m n a direct method of complexity o n log n is presented which utilizes lagrange interpolation and the fast summation for the remaining cases we use the matrix representation of the nfft to deduce our algorithms thereby we are able to compute an inverse nfft up to a certain accuracy by means of a modified adjoint nfft in o m log m n arithmetic operations finally we show that these approaches can also be explained by means of frame approximation

a solution to certain polynomial equations with applications to nonlinear fitting we present a combinatorial method for solving a certain system of polynomial equations of vandermonde type in 2n variables by reducing it to the problem of solving two special linear systems of size n and rooting a single univariate polynomial of degree n over c all solutions can be found with fixed precision using up to polylogarithmic factors o n 2 bitwise operations in the worst case furthermore if the data is well conditioned then this can be reduced to o n bit operations up to polylogarithmic factors as an application we show how this can be used to fit data to a complex exponential sum with n terms in the same nearly optimal time

3d finite difference synthetic acoustic log in cylindrical coordinates a finite difference method of numerical simulation of sonic logging has been developed and implemented the very general statement is considered the surrounding medium is allowed to be 3d heterogeneous and a source can be located at any point inside or outside the well to provide a maximally precise description of the sharpest interface of the problem the interface between the well and surrounding formations we use cylindrical coordinates with the axis directed along the well in order to avoid an excessive azimuth inflation of grid cells with an increase of radius we perform periodical refinement of the grid step in the azimuth direction in order to truncate the area of computations perfectly matched layer pml for cylindrical coordinates is developed and implemented its main advantage in comparison with other approaches is an extremely low level of artificial reflections and the absence of necessity to perform splitting of variables in the azimuth direction based on this numerical method a software oriented for the use of parallel computations is developed and implemented under message passing interface library results of numerical experiments for a well with completion embedded within layered elastic background with a vertical planar crack are presented and discussed

rapid computation of the discrete fourier transform algorithms for the rapid computation of the forward and inverse discrete fourier transform for points which are nonequispaced or whose number is unrestricted are presented the computational procedure is based on approximation using a local taylor series expansion and the fast fourier transform fft the forward transform for nonequispaced points is computed as the solution of a linear system involving the inverse fourier transform this latter system is solved using the iterative method gmres with preconditioning numerical results are given to confirm the efficiency of the algorithms

a high order solver for the heat equation in 1d domains with moving boundaries we describe a fast high order accurate method for the solution of the heat equation in domains with moving dirichlet or neumann boundaries and distributed forces we assume that the motion of the boundary is prescribed our method extends the work of greengard and strain comm pure appl math xliii 1990 pp 949 963 our scheme is based on a time space chebyshev pseudo spectral collocation discretization which is combined with a recursive product quadrature rule to accurately and efficiently approximate convolutions with green s function for the heat equation we present numerical results that exhibit up to eighth order convergence rates assuming n time steps and m spatial discretization points the evaluation of the solution of the heat equation at the same number of points in space time requires mathcal o n m log m work thus our scheme can be characterized as fast that is it is work optimal up to a logarithmic factor

fast and accurate evaluation of nonlocal coulomb and dipole dipole interactions via the nonuniform fft we present a fast and accurate algorithm for the evaluation of nonlocal long range coulomb and dipole dipole interactions in free space the governing potential is simply the convolution of an interaction kernel u mathbf x and a density function rho mathbf x psi mathbf x 2 for some complex valued wave function psi mathbf x permitting the formal use of fourier methods these are hampered by the fact that the fourier transform of the interaction kernel widehat u mathbf k has a singularity and or widehat rho mathbf k ne0 at the origin mathbf k bf 0 in fourier phase space thus accuracy is lost when using a uniform cartesian grid in mathbf k which would otherwise permit the use of the fft for evaluating the convolution here we make use of a high order discretization of the fourier integral accelerated by the nonuniform fast fourier transform nufft by adopting spherical and polar phase space discretizations in three and two dimensions respectively

fast elliptic solvers in cylindrical coordinates and the coulomb collision operator in this paper we describe a new class of fast solvers for separable elliptic partial differential equations in cylindrical coordinates r q z with free space radiation conditions by combining integral equation methods in the radial variable r with fourier methods in q and z we show that high order accuracy can be achieved in both the governing potential and its derivatives a weak singularity arises in the fourier transform with respect to z that is handled with special purpose quadratures we show how these solvers can be applied to the evaluation of the coulomb collision operator in kinetic models of ionized gases

nonequispaced hyperbolic cross fast fourier transform a straightforward discretization of problems in d spatial dimensions often leads to an exponential growth in the number of degrees of freedom thus even efficient algorithms like the fast fourier transform fft have high computational costs hyperbolic cross approximations allow for a severe decrease in the number of used fourier coefficients to represent functions with bounded mixed derivatives we propose a nonequispaced hyperbolic cross fft based on one hyperbolic cross fft and a dedicated interpolation by splines on sparse grids analogously to the nonequispaced fft for trigonometric polynomials with fourier coefficients supported on the full grid this allows for the efficient evaluation of trigonometric polynomials with fourier coefficients supported on the hyperbolic cross at arbitrary spatial sampling nodes

the exponentially convergent trapezoidal rule it is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line the mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity it is linked with computational methods all across scientific computing including algorithms related to inverse laplace transforms special functions complex analysis rational approximation integral equations and the computation of functions and eigenvalues of matrices and operators

an efficient calculation of the far field radiated by non uniformly sampled planar fields complying nyquist theorem an efficient and accurate method to calculate the far field radiated by non uniformly sampled planar apertures using the non uniform fast fourier transform nufft is shown the sampled fields in the planar aperture have to comply the nyquist theorem the method takes into account the amplitude of the unit cell radiation pattern which allows to compute more accurately the copolar and crosspolar components of the far field due to the generality of the nufft it also allows to overcome some limitations of the fft when computing the radiated fields of reflectarrays with periodicity larger than half a wavelength because the nufft is precision dependent a discussion of how its accuracy can affect the computed radiated fields is carried out numerical examples are provided to show the accuracy and performance of the nufft with regard to the fft and direct evaluation of the fields finally a study of computing times comparing the fft nufft and direct evaluation is presented

time domain calculation of acoustical wave propagation in discontinuous media using acoustical wave propagator with mapped pseudospectral method the acoustical wave propagator awp scheme involves an effective time domain calculation of sound propagation using the combination of chebyshev polynomial expansion and the fourier pseudospectral method the accuracy of this scheme degrades when the media has discontinuities due to the well known gibbs phenomenon in this paper several issues concerning awp are addressed including an analysis of the effect of gibbs phenomenon on the accuracy a mapped pseudospectral method is proposed wherein the grid points are redistributed with the emphasis across the media discontinuities by a pre determined smooth mapping curve then the spatial derivatives are calculated through a modified fourier pseudospectral method using this method the influence of the gibbs phenomenon is effectively alleviated while the computational efficiency of awp is still maintained the superiority of this improved awp scheme is illustrated by three one dimensional 1 d numerical examples

strang splitting for the time dependent schrodinger equation on sparse grids the time dependent schrodinger equation is discretized in space by a sparse grid pseudospectral method the strang splitting for the resulting evolutionary problem features first or second order convergence in time depending on the smoothness of the potential and of the initial data in contrast to the full grid case where the frequency domain is the working place the proof of the sufficient conditions for the convergence is done in the space realm

efficient thermal field computation in phase field models we solve the phase field equations in two dimensions to simulate crystal growth in the low undercooling regime the novelty is the use of a fast solver for the free space heat equation to compute the thermal field this solver is based on the efficient direct evaluation of the integral representation of the solution to the constant coefficient free space heat equation with a smooth source term the computational cost and memory requirements of the new solver are reasonable and no artificial boundary conditions are needed this allows one to solve for the thermal field in a computational domain whose size depends only on the size of the growing crystal and not on the extent of the thermal field which can result in significant computational savings in the low undercooling regime

smooth approximation of potential fields from noisy scattered data geophysical potential field data generally are observed at scattered sampling points and are contaminated by measurement and preprocessing errors analysis and further treatment require a representation of the underlying potential fields on a regular grid in the presence of noise an approximation of the data is more appropriate than an exact interpolation analysis of the smoothness of potential fields indicates an approximation by band limited functions the resulting least squares problem can be formulated as a linear system of equations in which the system matrix is of block toeplitz type this special structure allows a computationally attractive and robust solution via the fast fourier transform and the conjugate gradient algorithm on this basis we present a gridding method that incorporates additional knowledge about the physical properties of potential fields and the statistics of the data experimental results for synthetic and real world gravity data demonstrate good performance for highly sc

a multivariate generalization of prony s method abstract prony s method is a prototypical eigenvalue analysis based method for the reconstruction of a finitely supported complex measure on the unit circle from its moments up to a certain degree in this note we give a generalization of this method to the multivariate case and prove simple conditions under which the problem admits a unique solution provided the order of the moments is bounded from below by the number of points on which the measure is supported as well as by a small constant divided by the separation distance of these points stable reconstruction is guaranteed in its simplest form the reconstruction method consists of setting up a certain multilevel toeplitz matrix of the moments compute a basis of its kernel and compute by some method of choice the set of common roots of the multivariate polynomials whose coefficients are given in the second step all theoretical results are illustrated by numerical experiments

computing steerable principal components of a large set of images and their rotations we present here an efficient algorithm to compute the principal component analysis pca of a large image set consisting of images and for each image the set of its uniform rotations in the plane we do this by pointing out the block circulant structure of the covariance matrix and utilizing that structure to compute its eigenvectors we also demonstrate the advantages of this algorithm over similar ones with numerical experiments although it is useful in many settings we illustrate the specific application of the algorithm to the problem of cryo electron microscopy

on the stability of the hyperbolic cross discrete fourier transform a straightforward discretisation of problems in high dimensions often leads to an exponential growth in the number of degrees of freedom sparse grid approximations allow for a severe decrease in the number of used fourier coefficients to represent functions with bounded mixed derivatives and the fast fourier transform fft has been adapted to this thin discretisation we show that this so called hyperbolic cross fft suffers from an increase of its condition number for both increasing refinement and increasing spatial dimension

fast iterative image reconstruction for mri in the presence of field inhomogeneities in magnetic resonance imaging magnetic field inhomogeneities cause distortions in images that are reconstructed by conventional fast fourier transform fft methods several noniterative image reconstruction methods are used currently to compensate for field inhomogeneities but these methods assume that the field map that characterizes the off resonance frequencies is spatially smooth recently iterative methods have been proposed that can circumvent this assumption and provide improved compensation for off resonance effects however straightforward implementations of such iterative methods suffer from inconveniently long computation times this paper describes a tool for accelerating iterative reconstruction of field corrected mr images a novel time segmented approximation to the mr signal equation we use a min max formulation to derive the temporal interpolator speedups of around 60 were achieved by combining this temporal interpolator with a nonuniform fast fourier transform with normalized root mean squared approximation errors of 0 07 the proposed method provides fast accurate field corrected image reconstruction even when the field map is not smooth

non uniform fft for the finite element computation of the micromagnetic scalar potential we present a quasi linearly scaling first order polynomial finite element method for the solution of the magnetostatic open boundary problem by splitting the magnetic scalar potential the potential is determined by solving a dirichlet problem and evaluation of the single layer potential by a fast approximation technique based on fourier approximation of the kernel function the latter approximation leads to a generalization of the well known convolution theorem used in finite difference methods we address it by a non uniform fft approach overall our method scales o m n nlogn for n nodes and m surface triangles we confirm our approach by several numerical tests

fast convolution with radial kernels at nonequispaced knots we develop a new algorithm for the fast evaluation of linear combinations of radial functions based on the recently developed fast fourier transform at nonequispaced knots for smooth kernels e g the gaussian our algorithm requires arithmetic operations in case of singular kernels an additional regularization procedure must be incorporated and the algorithm has the arithmetic complexity if either the points yj or the points xk are reasonably uniformly distributed we prove error estimates to obtain clues about the choice of the involved parameters and present numerical examples for various singular and smooth kernels in two dimensions

nonuniform fast fourier transforms using min max interpolation the fast fourier transform fft is used widely in signal processing for efficient computation of the ft of finite length signals over a set of uniformly spaced frequency locations however in many applications one requires nonuniform sampling in the frequency domain i e a nonuniform ft several papers have described fast approximations for the nonuniform ft based on interpolating an oversampled fft this paper presents an interpolation method for the nonuniform ft that is optimal in the min max sense of minimizing the worst case approximation error over all signals of unit norm the proposed method easily generalizes to multidimensional signals numerical results show that the min max approach provides substantially lower approximation errors than conventional interpolation methods the min max criterion is also useful for optimizing the parameters of interpolation kernels such as the kaiser bessel function

fourier series approximation of separable models the approximation of a function affected by noise in several dimensions suffers from the so called curse of dimensionality in this paper a fourier series method based on regularization is developed both for uniform and random design when a restriction on the complexity of the curve such as additivity is considered in order to circumvent the problem optimal convergence theorems are stated and numerical experiments are shown on several test problems available in the literature together with comparisons with alternative methods

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

a global local synthesis approach for large non regular arrays a method is proposed for the synthesis of large planar non regular arrays assuming constant magnitude for the excitation of the elements the approach which combines global and local optimization is based on distinguishing aperture type and non coherent parts of the array factor the following constraints are considered minimal separation of the antennas maximum size of the array and fixed mainbeam width the level of the sidelobes is reduced via the minimization of a new type of flexible averaging cost function based on an lp norm possible applications of this model lie in the field of radio astronomy satellite communications and radar systems the proposed optimization strategy consists of three successive steps designed to be independent of each other starting with an equivalent continuous aperture the first two steps act as global transformations of the radial and azimuthal positions of the elements in the initial array while the third step performs a local optimization of individual elements of the array this last step heavily relies on computations of the gradient of the cost function which can be done quickly using an fft based procedure the method is illustrated for several large scale examples by considering as inputs four different types of non regular arrays

fouriertransformation auf dunnen gittern mit hierarchischen basen an algorithm for the trigonometric interpolation of functions ofn variables on a sparse grid is described this discrete fourier transform based on fft has a greatly reduced complexity in comparison to the fourier transform on a regular grid whereas the approximation quality is only slightly reduced if the function belongs to a korobov space the transformation is easily invertible es wird ein auf der schnellen fouriertransformation beruhender algorithmus zur trigonometrischen interpolation von multivariaten funktionen auf einem dunnen gitter beschrieben bei nur geringfugig schlechterer approximationsgenauigkeit fur funktionen aus korobovraumen hat dieser algorithmus eine viel kleinere komplexitat als die algorithmen die auf gewohnlichen gittern arbeiten die transformation ist auf einfache weise umkehrbar

fast tomographic reconstruction via rotation based hierarchical backprojection we introduce a family of fast algorithms for tomographic backprojection in the parallel beam geometry the algorithms reduce the computational cost of backprojecting p projections onto an n times n pixel image from the conventional o n 2 p to o n 2 log p the new algorithms aggregate the projections in a hierarchical structure with images in the hierarchy formed by the rotation and addition of other images made up of fewer projections while these algorithms are related to existing fast algorithms this work places them within the signal processing framework providing a systematic means to optimize and adjust the trade off between computational cost and accuracy rotations are performed separably in order that higher order interpolators may be used with low computational cost the same ideas are applied to create a tomographic projection algorithm which computes projections of an n times n pixel image onto p view angles at a cost of o n 2 log p

on the numerical solution of the heat equation i we describe a fast solver for the inhomogeneous heat equation in free space following the time evolution of the solution in the fourier domain it relies on a recently developed spectral approximation of the free space heat kernel coupled with the non uniform fast fourier transform unlike finite difference and finite element techniques there is no need for artificial boundary conditions on a finite computational domain the method is explicit unconditionally stable and requires an amount of work of the order o nm log n where n is the number of discretization points in physical space and m is the number of time steps we refer to the approach as the fast recursive marching frm method

on otsuki tori and their willmore energy abstract otsuki tori in the three dimensional unit sphere are studied from calculations of the willmore energy the lower and upper bound estimates for the otsuki tori are established invariance of the otsuki tori under antipodal maps is also considered and in such cases better lower bound estimates of their willmore energies are obtained

rotationally invariant quadratures for the sphere we construct near optimal quadratures for the sphere that are invariant under the icosahedral rotation group these quadratures integrate all n 1 2 linearly independent functions in a rotationally invariant subspace of maximal order and degree n the nodes of these quadratures are nearly uniformly distributed and the number of nodes is only marginally more than the optimal n 1 2 3 nodes using these quadratures we discretize the reproducing kernel on a rotationally invariant subspace to construct an analogue of lagrange interpolation on the sphere this representation uses function values at the quadrature nodes in addition the representation yields an expansion that uses a single function centred and mostly concentrated at nodes of the quadrature thus providing a much better localization than spherical harmonic expansions we show that this representation may be localized even further we also describe two algorithms of complexity for using these grids and representations finally we note that our approach is also applicable to other discrete rotation groups

the black box fast multipole method a new o n fast multipole formulation is proposed for non oscillatory kernels this algorithm is applicable to kernels k x y which are only known numerically that is their numerical value can be obtained for any x y this is quite different from many fast multipole methods which depend on analytical expansions of the far field behavior of k for x y large other black box or kernel independent fast multipole methods have been devised our approach has the advantage of requiring a small pre computation time even for very large systems and uses the minimal number of coefficients to represent the far field for a given l 2 tolerance error in the approximation this technique can be very useful for problems where the kernel is known analytically but is quite complicated or for kernels which are defined purely numerically

approximate factorizations of fourier matrices with nonequispaced knots abstract we propose a new algorithm for the multiplication of vectors with fourier matrices of the form a tf e 2 i x k v j n n 2 1 j k n 2 where both x k and v j are arbitrary knots in n 2 n 2 the algorithm is based on an approximate factorization of the transform matrix a tf into sparse matrices which entries are chosen to minimize the frobenius norm of certain error matrices numerical experiments demonstrate that the approximation error introduced by our new algorithm is about 10 2 times smaller than the approximation error of previously reported algorithms with the gaussian or b splines as window functions and as good as a previous algorithm with the kaiser bessel function as window function

on the smallest singular value of multivariate vandermonde matrices with clustered nodes abstract we prove lower bounds for the smallest singular value of rectangular multivariate vandermonde matrices with nodes on the complex unit circle the nodes are off the grid groups of nodes cluster and the studied minimal singular value is bounded below by the product of inverted distances of a node to all other nodes in the specific cluster by using known and new upper bounds for the smallest singular value this completely settles the univariate case and pairs of nodes in the multivariate case both including reasonable sharp constants for larger clusters we show that the smallest singular value depends also on the geometric configuration within a cluster

simple factor dressing and the lopez ros deformation of minimal surfaces in euclidean 3 space the aim of this paper is to investigate a new link between integrable systems and minimal surface theory the dressing operation uses the associated family of flat connections of a harmonic map to construct new harmonic maps since a minimal surface in 3 space is a willmore surface its conformal gauss map is harmonic and a dressing on the conformal gauss map can be defined we study the induced transformation on minimal surfaces in the simplest case the simple factor dressing and show that the well known lopez ros deformation of minimal surfaces is a special case of this transformation we express the simple factor dressing and the lopez ros deformation explicitly in terms of the minimal surface and its conjugate surface in particular we can control periods and end behaviour of the simple factor dressing this allows to construct new examples of doubly periodic minimal surfaces arising as simple factor dressings of scherk s first surface

fast sparse nonlinear fourier expansions of high dimensional functions abstract the nonlinear fourier basis has shown its advantages over the classical fourier basis in the time frequency analysis the need of processing large amount of high dimensional data motivates the extension of the methods based upon the nonlinear fourier basis to high dimensions we consider the multi dimensional nonlinear fourier basis which is the tensor product of univariate nonlinear fourier basis we investigate the convergence order in norm and also the almost everywhere convergence of the nonlinear fourier expansions in order to compute fast and efficiently the nonlinear fourier expansions of d dimensional functions we introduce the sparse nonlinear fourier expansion and develop a fast algorithm for evaluating it we also prove that the fast sparse nonlinear fourier expansions enjoy the optimal convergence order and reduce the computational costs to o n log 2 d 1 n numerical experiments are presented to demonstrate the efficiency and accuracy of the proposed method

fast fourier transforms of piecewise polynomials abstract we introduce an efficient algorithm for evaluating the fourier transform of piecewise polynomial data on d dimensional simplices in d dimensional euclidean space r d it generalizes butterfly algorithms for pointwise d 0 nonuniform fast fourier transforms with new dimensional recurrences for exponential polynomial moments error analysis and numerical comparisons with direct evaluation validate the efficiency and accuracy of the algorithm

accelerating the nonuniform fast fourier transform the nonequispaced fourier transform arises in a variety of application areas from medical imaging to radio astronomy to the numerical solution of partial differential equations in a typical problem one is given an irregular sampling of n data in the frequency domain and one is interested in reconstructing the corresponding function in the physical domain when the sampling is uniform the fast fourier transform fft allows this calculation to be computed in o n log n operations rather than o n 2 operations unfortunately when the sampling is nonuniform the fft does not apply over the last few years a number of algorithms have been developed to overcome this limitation and are often referred to as nonuniform ffts nuffts these rely on a mixture of interpolation and the judicious use of the fft on an oversampled grid a dutt and v rokhlin siam j sci comput 14 1993 pp 1368 1383 in this paper we observe that one of the standard interpolation or gridding schemes based on gaussians can be accelerated by a significant factor without precomputation and storage of the interpolation weights this is of particular value in two and three dimensional settings saving either 10 d n in storage in d dimensions or a factor of about 5 10 in cputime independent of dimension

frequency warping biorthogonal frames frequency warping is theoretically designed to be a unitary operator of infinite input and output dimensions thus performing the resolution of identity in real implementations finite dimensions have to be considered then perfect reconstruction cannot be fulfilled the accuracy of reconstruction is particularly compromised in case of non smooth warping maps which are more useful for practical applications in order to overcome this limitation a new frequency warping biorthogonal frame operator for non smooth warping maps is introduced in this work the proposed transformation is based on a mathematical model which has been previously introduced for computational purposes by adding some redundancy with respect to the truncation of the infinite dimensions operator the effect of an infinite output dimension can be taken into account in a compressed way based on an analytical factorization in the reconstruction process the additional redundant samples are expanded thus guaranteeing near perfect reconstruction

generalized mehler semigroups and applications we construct and study generalized mehler semigroups p t t 0 and their associated markov processesm the construction methods for p t t 0 are based on some new purely functional analytic results implying in particular that any strongly continuous semigroup on a hilbert spaceh can be extended to some larger hilbert spacee with the embeddingh e being hilbert schmidt the same analytic extension results are applied to construct strong solutions to stochastic differential equations of typedx t c dw t ax t dt with possibly unbounded linear operatorsa andc onh on a suitably chosen larger spacee for gaussian generalized mehler semigroups p t t 0 with corresponding markov processm the associated non symmetric dirichlet forms ed e are explicitly calculated and a necessary and sufficient condition for path regularity ofm in terms of e d e is proved then using dirichlet form methods it is shown thatm weakly solves the above stochastic differential equation if the state spacee is chosen appropriately finally we discuss the differences between these two methods yielding strong resp weak solutions

fast spherical fourier algorithms spherical fourier series play an important role in many applications a numerically stable fast transform analogous to the fast fourier transform is of great interest for a standard grid of o n2 points on the sphere a direct calculation has computational complexity of o n4 but a simple separation of variables reduces the complexity to o n3 here we improve well known fast algorithms for the discrete spherical fourier transform with a computational complexity of o n2 log2 n furthermore we present for the first time a fast algorithm for scattered data on the sphere for arbitrary o n2 points on the sphere a direct calculation has a computational complexity of o n4 but we present an approximate algorithm with a computational complexity of o n2 log2 n

efficient spectral sparse grid methods and applications to high dimensional elliptic problems we develop in this paper some efficient algorithms which are essential to implementations of spectral methods on the sparse grid by smolyak s construction based on a nested quadrature more precisely we develop a fast algorithm for the discrete transform between the values at the sparse grid and the coefficients of expansion in a hierarchical basis and by using the aforementioned fast transform we construct two very efficient sparse spectral galerkin methods for a model elliptic equation in particular the chebyshev legendre galerkin method leads to a sparse matrix with a much lower number of nonzero elements than that of low order sparse grid methods based on finite elements or wavelets and can be efficiently solved by a suitable sparse solver ample numerical results are presented to demonstrate the efficiency and accuracy of our algorithms

nonlinear approximation by sums of exponentials and translates in this paper we discuss the numerical solution of two nonlinear approximation problems many applications in electrical engineering signal processing and mathematical physics lead to the following problem let h be a linear combination of exponentials with real frequencies determine all frequencies all coefficients and the number of summands if finitely many perturbed uniformly sampled data of h are given we solve this problem by an approximate prony method apm and prove the stability of the solution in the square and uniform norm further an apm for nonuniformly sampled data is proposed too the second approximation problem is related to the first one and reads as follows let varphi be a given 1 periodic window function as defined in section 4 further let f be a linear combination of translates of varphi determine all shift parameters all coefficients and the number of translates if finitely many perturbed uniformly sampled data of f are given using fourier technique this problem is transferred into the above parameter estimation problem for an exponential sum which is solved by apm the stability of the solution is discussed in the square and uniform norm too numerical experiments show the performance of our approximation methods

convolution based trigonometric interpolation of band limited signals this paper presents a method to obtain a trigonometric polynomial that accurately interpolates a given band limited signal from a finite sequence of samples the polynomial delivers accurate approximations in the range covered by the sequence except for a short frame close to the range limits besides its accuracy increases exponentially with the frame width the method is based on using a band limited window in order to reduce the truncation error of a convolution series it is shown that the polynomial can be efficiently constructed and evaluated using algorithms designed for the discrete fourier transform dft specifically two basic procedures are presented one based on the fast fourier transform fft and another based on a recursive update algorithm for the short time fft the paper contains three applications the first is a variable fractional delay vfd filter which consists of a short time fft combined with the evaluation of a trigonometric polynomial this filter has low complexity and can be implemented using cordic rotations the second is the interpolation of nonuniform fourier summations where the proposed method eliminates the need to interpolate any kernel sample finally the third can be viewed as a generalization of the fft convolution algorithm and makes it possible to interpolate the output of an finite impulse response fir filter efficiently

subordinated α stable ornstein uhlenbeck process as a tool for financial data description abstract the classical financial models are based on the standard brownian diffusion type processes however in the exhibition of some real market data like interest or exchange rates we observe characteristic periods of constant values moreover in the case of financial data the assumption of normality is often unsatisfied in such cases the popular vasicek model that is a mathematical system describing the evolution of interest rates based on the ornstein uhlenbeck process seems not to be applicable therefore we propose an alternative approach based on a combination of the popular ornstein uhlenbeck process with a stable distribution and subdiffusion systems that demonstrate such characteristic behavior the probability density function of the proposed process can be described by a fokker planck type equation and therefore it can be examined as an extension of the basic ornstein uhlenbeck model in this paper we propose the parameters estimation method and calibrate the subordinated vasicek model to the interest rate data

reconstruction algorithms in irregular sampling a constructive solution of the irregular sampling problem for band limited functions is given we show how a band limited function can be com pletely reconstructed from any random sampling set whose density is higher than the nyquist rate and give precise estimates for the speed of convergence of this iteration method variations of this algorithm allow for irregular sampling with derivatives reconstruction of band limited functions from local averages and irregular sampling of multivariate band limited functions in the irregular sampling problem one is asked whether and how a band limited function can be completely reconstructed from its irregularly sam pled values f xi this has many applications in signal and image processing seismology meteorology medical imaging etc finding constructive solutions of this problem has received considerable attention among mathematicians and engineers the mathematical literature provides several uniqueness results 1 2 17 18 19 it is now part of the folklore that for stable sampling the sampling rate must be at least the nyquist rate 18 these results as deep as they are have had little impact for the applied sciences because they were not constructive if the sampling set is just a perturbation of the regular oversampling then a reconstruction method has been obtained in a seminal paper by duffin and schaeffer 6 see also 29 if for some l 0 a 0 and o 0 the sampling points xk k e z satisfy a xk ok a k i then the norm equivalence a ir f x 2dx with w n o this norm equivalence implies that it is possible to reconstruct through an iterative procedure the so called frame method most of the later work on constructive methods consists of variations of this method 3 21 22 26 the above conditions on the sampling set exclude random irregular sampling sets e g sets with regions of higher sampling density a partial but undesirable remedy to handle highly irregular sampling sets would be to force the above conditions by throwing away information on part of the points and accept a very slow convergence of the iteration

efficient numerical methods in non uniform sampling theory summary we present a new second generation reconstruction algorithm for irregular sampling i e for the problem of recovering a band limited function from its non uniformly sampled values the efficient new method is a combination of the adaptive weights method which was developed by the two first named authors and the method of conjugate gradients for the solution of positive definite linear systems the choice of adaptive weights can be seen as a simple but very efficient method of preconditioning further substantial acceleration is achieved by utilizing the toeplitztype structure of the system matrix this new algorithm can handle problems of much larger dimension and condition number than have been accessible so far furthermore if some gaps between samples are large then the algorithm can still be used as a very efficient extrapolation method across the gaps

fast computation of fourier integral operators we introduce a general purpose algorithm for rapidly computing certain types of oscillatory integrals which frequently arise in problems connected to wave propagation general hyperbolic equations and curvilinear tomography the problem is to numerically evaluate a so called fourier integral operator fio of the form int e 2 pi i phi x xi a x xi hat f xi d xi at points given on a cartesian grid here xi is a frequency variable hat f xi is the fourier transform of the input f a x xi is an amplitude and phi x xi is a phase function which is typically as large as xi hence the integral is highly oscillatory because a fio is a dense matrix a naive matrix vector product with an input given on a cartesian grid of size n by n would require o n 4 operations this paper develops a new numerical algorithm which requires o n 2 5 log n operations and as low as o sqrt n in storage space the constants in front of these estimates are small it operates by localizing the integral over polar wedges with small angular aperture in the frequency plane on each wedge the algorithm factorizes the kernel e 2 pi i phi x xi a x xi into two components 1 a diffeomorphism which is handled by means of a nonuniform fft and 2 a residual factor which is handled by numerical separation of the spatial and frequency variables the key to the complexity and accuracy estimates is the fact that the separation rank of the residual kernel is provably independent of the problem size several numerical examples demonstrate the numerical accuracy and low computational complexity of the proposed methodology we also discuss the potential of our ideas for various applications such as reflection seismology

short note the type 3 nonuniform fft and its applications the nonequispaced or nonuniform fast fourier transform nufft arises in a variety of application areas including imaging processing and the numerical solution of partial differential equations in its most general form it takes as input an irregular sampling of a function and seeks to compute its fourier transform at a nonuniform sampling of frequency locations this is sometimes referred to as the nufft of type 3 like the fast fourier transform the amount of work required is of the order o nlogn where n denotes the number of sampling points in both the physical and spectral domains in this short note we present the essential ideas underlying the algorithm in simple terms we also illustrate its utility with application to problems in magnetic resonance imagin and heat flow

fast transform from an adaptive multi wavelet representation to a partial fourier representation we present a fast algorithm to compute the partial transformation of a function represented in an adaptive pseudo spectral multi wavelet representation to a partial fourier representation such fast transformations are useful in many contexts in physics and engineering where changes of representation from a piece wise polynomial basis to a fourier basis the algorithm is demonstrated for a gaussian in one and in three dimensions for 2d we apply this approach to a gaussian in a periodic domain the accuracy and the performance of this method is compared with direct summation

the chebyshev fast gauss and nonuniform fast fourier transforms and their application to the evaluation of distributed heat potentials we present a method for the fast and accurate computation of distributed heat potentials in two dimensions the distributed source is assumed to be given in terms of piecewise space time chebyshev polynomials we discretize uniformly in time whereas in space the polynomials are defined on the leaf nodes of a quadtree the quadtree can vary at each time step we combine a product integration rule with fast algorithms fast heat potentials nonuniform fft fast gauss transform to obtain a high order accurate method with optimal complexity if n is the number of time steps m is the maximum number of leaf nodes over all the time steps and the input contains a qth order polynomial representation of f then our method requires o q 3mnlogm work to evaluate the heat potential at arbitrary mn space time target locations the overall convergence rate of the method is of order q we present numerical experiments for q 4 8 and 16 and we verify the theoretical convergence rate of the method when the solution is sufficiently smooth the 16th order variant results in significant computational savings even in the case in which we require only a few digits of accuracy

sequences of willmore surfaces in this paper we develop the theory of willmore sequences for willmore surfaces in the 4 sphere we show that under appropriate conditions this sequence has to terminate in this case the willmore surface either is the twistor projection of a holomorphic curve into mathbb c mathbb p 3 or the inversion of a minimal surface with planar ends in mathbb r 4 these results give a unified explanation of previous work on the characterization of willmore spheres and willmore tori with non trivial normal bundles by various authors

fast high dimensional approximation with sparse occupancy trees this paper is concerned with scattered data approximation in high dimensions given a data set email protected r d of n data points x i along with values y email protected r d i 1 n and viewing the y i as values y i f x i of some unknown function f we wish to return for any query point email protected r d an approximation email protected x to y f x here the spatial dimension d should be thought of as large we emphasize that we do not seek a representation of email protected in terms of a fixed set of trial functions but define email protected through recovery schemes which are primarily designed to be fast and to deal efficiently with large data sets for this purpose we propose new methods based on what we call sparse occupancy trees and piecewise linear schemes based on simplex subdivisions

on numerical methods for discrete least squares approximation by trigonometric polynomials fast efficient and reliable algorithms for discrete least squares approximation of a real valued function given at arbitrary distinct nodes in 0 2 by trigonometric polynomials are presented the algorithms are based on schemes for the solution of inverse unitary eigenproblems and require only o mn arithmetic operations as compared to o mn 2 operations needed for algorithms that ignore the structure of the problem an algorithm which solves this problem with real valued data and real valued solution using only real arithmetic is given numerical examples are presented that show that the proposed algorithms produce consistently accurate results that are often better than those obtained by general qr decomposition methods for the least squares problem

fourier pseudospectral method on generalized sparse grids for the space fractional schrodinger equation abstract in this paper the time splitting fourier pseudospectral method on the generalized sparse grids is applied to solve the space fractional schrodinger equation we give a containment relation between different level index sets of the generalized sparse grids and it can be used in designing the reference generalized sparse grids which are finer than other considered grids thus the numerical solution on the reference generalized sparse grids can be used as the reference true solution of the equation then the fully discrete algorithm is obtained in the numerical experiments we compare the numerical results on the generalized sparse grids with those on the full grids for the interpolation of the gaussian multiplied by a factor and for the computation of the schrodinger equation with two kinds of non smooth potentials the advantages of the fourier pseudospectral method on the generalized sparse grids with the level index set of parameter k 1 2 3 are manifest in the approximation with high resolution here the sparsity of the generalized sparse grids will become weak when the parameter k becomes large moreover the advantage of the generalized sparse grids is more pronounced in solving the schrodinger equation with the higher dimension the square well potential or the fractional laplacian

stability results for scattered data interpolation by trigonometric polynomials a fast and reliable algorithm for the optimal interpolation of scattered data on the torus mathbb t d by multivariate trigonometric polynomials is presented the algorithm is based on a variant of the conjugate gradient method in combination with the fast fourier transforms for nonequispaced nodes the main result is that under mild assumptions the total complexity for solving the interpolation problem at m arbitrary nodes is of order cal o m log m this result is obtained by the use of localized trigonometric kernels where the localization is chosen in accordance with the spatial dimension d numerical examples show the efficiency of the new algorithm

combined complex ridgelet shrinkage and total variation minimization a new algorithm for the characterization of engineering surface topographies with line singularities is proposed it is based on thresholding complex ridgelet coefficients combined with total variation tv minimization the discrete ridgelet transform is designed by first using a discrete radon transform based on the nonequispaced fast fourier transform nfft and then applying a dual tree complex wavelet transform dt cwt the nfft based approach of the radon transform completely avoids linear interpolations of the cartesian to polar grid and requires only o n2 log n arithmetic operations for n by n arrays while its inverse preserves the good reconstruction quality of the filtered backprojection the dt cwt in the second step of the ridgelet transform provides approximate shift invariance on the projections of the radon transform after hard thresholding the ridgelet coefficients they are restored using tv minimization to eliminate the pseudo gibbs artifacts near the discontinuities numerical experiments demonstrate the remarkable ability of the methodology to extract line scratches

on the cauchy problem for non local ornstein uhlenbeck operators abstract we study the cauchy problem involving non local ornstein uhlenbeck operators in finite and infinite dimensions we prove classical solvability without requiring that the levy measure corresponding to the large jumps part has a first finite moment moreover we determine a core of regular functions which is invariant for the associated transition markov semigroup such a core allows to characterize the marginal laws of the ornstein uhlenbeck stochastic process as unique solutions to fokker planck kolmogorov equations for measures

fast computation of frequency warping transforms in this paper we introduce an analytical approach for the frequency warping transform criteria for the design of operators based on arbitrary warping maps are provided and an algorithm carrying out a fast computation is defined such operators can be used to shape the tiling of time frequency tf plane in a flexible way moreover they are designed to be inverted by the application of their adjoint operator according to the proposed model the frequency warping transform is computed by considering two additive operators the first one represents its nonuniform fourier transform approximation and the second one suppresses aliasing the first operator is fast computable by various interpolation approaches a factorization of the second operator is found for arbitrary shaped nonsmooth warping maps by properly truncating the operators involved in the factorization the computation turns out to be fast without compromising accuracy

fast fourier transforms for nonequispaced data a group of algorithms is presented generalizing the fast fourier transform to the case of noninteger frequencies and nonequispaced nodes on the interval pi pi the schemes of this paper are based on a combination of certain analytical considerations with the classical fast fourier transform and generalize both the forward and backward ffts each of the algorithms requires o n cdot log n n cdot log 1 varepsilon arithmetic operations where varepsilon is the precision of computations and n is the number of nodes the efficiency of the approach is illustrated by several numerical examples

non gaussian ornstein uhlenbeck based models and some of their uses in financial economics non gaussian processes of ornstein uhlenbeck ou type offer the possibility of capturing important distributional deviations from gaussianity and for flexible modelling of dependence structures this paper develops this potential drawing on and extending powerful results from probability theory for applications in statistical analysis their power is illustrated by a sustained application of ou processes within the context of finance and econometrics we construct continuous time stochastic volatility models for financial assets where the volatility processes are superpositions of positive ou processes and we study these models in relation to financial data and theory

kaiser bessel basis for particle mesh interpolation in this work we introduce the kaiser bessel interpolation basis for the particle mesh interpolation in the fast ewald method a reliable a priori error estimate is developed to measure the accuracy of the force computation in correlated charge systems and is shown to be effective in optimizing the shape parameter of the kaiser bessel basis in terms of accuracy by comparing the optimized kaiser bessel basis with the traditional b spline basis we demonstrate that the former is more accurate than the latter in part of the working parameter space say a relatively small real space cutoff a relatively small reciprocal space mesh and a relatively large truncation of basis in some cases the kaiser bessel basis is found to be more than one order of magnitude more accurate

ornstein uhlenbeck cauchy process we combine earlier investigations of linear systems subject to levy fluctuations with recent attempts to give meaning to so called levy flights in external force fields we give a complete construction of the ornstein uhlenbeck cauchy process as a fully computable paradigm example of doob s stable noise supported ornstein uhlenbeck process despite the nonexistence of all moments we determine local characteristics forward drift of the process generators of forward and backward dynamics and relevant pseudodifferential evolution equations the induced nonstationary spatial process is proved to be markovian and quite apart from its inherent discontinuity defines an associated velocity process in a probabilistic sense

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

diffusion semigroups in spaces of continuous functions with mixed topology abstract we study transition semigroups and kolmogorov equations corresponding to stochastic semilinear equations on a hilbert space h it is shown that the transition semigroup is strongly continuous and locally equicontinuous in the space of polynomially increasing continuous functions on h when endowed with the so called mixed topology as a result we characterize cores of certain second order differential operators in such spaces and show that they have unique extensions to generators of strongly continuous semigroups

numerical stability of nonequispaced fast fourier transforms this paper presents some new results on numerical stability for multivariate fast fourier transform of nonequispaced data nfft in contrast to fast fourier transform of equispaced data the nfft is an approximate algorithm in a worst case study we show that both approximation error and roundoff error have a strong influence on the numerical stability of nfft numerical tests confirm the theoretical estimates of numerical stability

sparse spectral approximations of high dimensional problems based on hyperbolic cross hyperbolic cross approximations by some classical orthogonal polynomials functions in both bounded and unbounded domains are considered in this paper optimal error estimates in proper anisotropic weighted korobov spaces for both regular hyperbolic cross approximations and optimized hyperbolic cross approximations are established these fundamental approximation results indicate that spectral methods based on hyperbolic cross approximations can be effective for treating certain high dimensional problems and will serve as basic tools for analyzing sparse spectral methods in high dimensions

an exact and fast computation of discrete fourier transform for polar and spherical grid numerous applied problems of two dimensional 2 d and 3 d imaging are formulated in continuous domain they place great emphasis on obtaining and manipulating the fourier transform in polar and spherical coordinates however the translation of continuum ideas with the discrete sampled data on a cartesian grid is problematic there exists no exact and fast solution to the problem of obtaining discrete fourier transform for polar and spherical grids in the literature in this paper we develop exact algorithms to the above problem for 2 d and 3 d which involve only 1 d equispaced fast fourier transform with no interpolation or approximation at any stage the result of the proposed approach leads to a fast solution with very high accuracy we describe the computational procedure to obtain the solution in both 2 d and 3 d which includes fast forward and inverse transforms we find the nested multilevel matrix structure of the inverse process and we propose a hybrid grid and use a preconditioned conjugate gradient method that exhibits a drastic improvement in the condition number

effective beam forming networks for large arbitrary array of antennas a deterministic and effective strategy to reduce as much as possible the number of amplitude controls points in large arrays is herein proposed and applied to cases of practical interest the proposed strategy exploits two important concepts the first one coming from a special class of aperiodic tiling and the second one based on nufft non uniform fast fourier transform techniques the first one allows designing innovative front end architectures made of a reduced number of control points while nufft algorithms lead to computationally effective synthesis procedures the overall results and design guidelines allow to achieve simplified beamforming networks derived from aperiodic tilings and to know the performances that can be achieved using this approach particularly in case of low sidelobe requirements to confirm the usefulness of the proposed tools numerical examples are discussed conclusion follows

fast direct methods for gaussian processes a number of problems in probability and statistics can be addressed using the multivariate normal gaussian distribution in the one dimensional case computing the probability for a given mean and variance simply requires the evaluation of the corresponding gaussian density in the n dimensional setting however it requires the inversion of an n times n covariance matrix c as well as the evaluation of its determinant det c in many cases such as regression using gaussian processes the covariance matrix is of the form c sigma 2 i k where k is computed using a specified covariance kernel which depends on the data and additional parameters hyperparameters the matrix c is typically dense causing standard direct methods for inversion and determinant evaluation to require mathcal o n 3 work this cost is prohibitive for large scale modeling here we show that for the most commonly used covariance functions the matrix c can be hierarchically factored into a product of block low rank updates of the identity matrix yielding an mathcal o n log 2 n algorithm for inversion more importantly we show that this factorization enables the evaluation of the determinant det c permitting the direct calculation of probabilities in high dimensions under fairly broad assumptions on the kernel defining k our fast algorithm brings many problems in marginalization and the adaptation of hyperparameters within practical reach using a single cpu core the combination of nearly optimal scaling in terms of problem size with high performance computing resources will permit the modeling of previously intractable problems we illustrate the performance of the scheme on standard covariance kernels