fast summation at nonequispaced knots by nfft we develop a new algorithm for the fast computation of discrete sums f y j sum k 1 n alpha k k y j x k j 1 m based on the recently developed fast fourier transform fft at nonequispaced knots our algorithm in particular our regularization procedure is simply structured and can be easily adapted to different kernels k our method utilizes the widely known fft and can consequently incorporate advanced fft implementations in summary it requires cal o n log n m arithmetic operations we prove error estimates to obtain clues about the choice of the involved parameters and present numerical examples in one and two dimensions

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

boundedness of inhomogeneous journe s type operators on multi parameter local hardy spaces abstract in this paper we establish a boundedness criterion of a class of inhomogeneous journe s type of multi parameter singular integral operators on multi parameter local hardy spaces h p r n 1 r n k recently introduced in ding et al 2019 the consideration of such inhomogeneous journe s type of multi parameter singular integral operators is motivated by the study of the multi parameter pseudo differential operators on multi parameter local hardy spaces the lack of the vanishing moments of both the corresponding functions in the multi parameter local calderon reproducing formula and the multi parameter local littlewood paley stein functions as well as the lack of vanishing moments of atoms in the multi parameter local hardy spaces h p r n 1 r n k with large size all create substantial difficulties in our proofs

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

trigonometric interpolation and quadrature in perturbed points the trigonometric interpolants to a periodic function f in equispaced points converge if f is dini continuous and the associated quadrature formula the trapezoidal rule converges if f is continuous what if the points are perturbed with equispaced grid spacing h let each point be perturbed by an arbitrary amount le alpha h where alpha in kern 5pt 0 1 2 is a fixed constant the kadec 1 4 theorem of sampling theory suggests there may be trouble for alpha ge 1 4 we show that convergence of both the interpolants and the quadrature estimates is guaranteed for all alpha 1 2 if f is twice continuously differentiable with the convergence rate depending on the smoothness of f more precisely it is enough for f to have 4 alpha derivatives in a certain sense and we conjecture that 2 alpha derivatives are enough connections with the fejer kalmar theorem are discussed

some comments on fourier analysis uncertainty and modeling investigation of the problem of simultaneously concentrating a function and its fourier transform has led to some interesting special functions that have widespread applications in engineering they provide a means of proving a rigorous version of an engineering folk theorem called the 2wt theorem many generalizations of these ideas seem to possess a similar elegant mathematical structure a brief descriptive review is given of these developments

calderon zygmund operators on product hardy spaces let t be a product calderon zygmund singular integral introduced by journe using an elegant rectangle atomic decomposition of hp rn rm and journe s geometric covering lemma r fefferman proved the remarkable hp rn rm lp rn rm boundedness of t in this paper we apply vector valued singular integral calderon s identity littlewood paley theory and the almost orthogonality together with fefferman s rectangle atomic decomposition and journe s covering lemma to show that t is bounded on product hp rn rm for max nn e mm e p 1 if and only if t1 1 t2 1 0 where e is the regularity exponent of the kernel of t

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

nonuniform sampling and reconstruction in shift invariant spaces this article discusses modern techniques for nonuniform sampling and reconstruction of functions in shift invariant spaces it is a survey as well as a research paper and provides a unified framework for uniform and nonuniform sampling and reconstruction in shift invariant subspaces by bringing together wavelet theory frame theory reproducing kernel hilbert spaces approximation theory amalgam spaces and sampling inspired by applications taken from communication astronomy and medicine the following aspects will be emphasized a the sampling problem is well defined within the setting of shift invariant spaces b the general theory works in arbitrary dimension and for a broad class of generators c the reconstruction of a function from any sufficiently dense nonuniform sampling set is obtained by efficient iterative algorithms these algorithms converge geometrically and are robust in the presence of noise d to model the natural decay conditions of real signals and images the sampling theory is developed in weighted l p spaces

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

nonuniform fast fourier transform the nonuniform discrete fourier transform ndft can be computed with a fast algorithm referred to as the nonuniform fast fourier transform nfft in l dimensions the nfft requires o n lne l x1d4c1 1lm x1d4c1 x1d4c1 1llogm x1d4c1 operations where m x1d4c1 is the number of fourier components along dimension x1d4c1 n is the number of irregularly spaced samples and e is the required accuracy this is a dramatic improvement over the o n x1d4c1 1lm x1d4c1 operations required for the direct evaluation ndft the performance of the nfft depends on the lowpass filter used in the algorithm a truncated gauss pulse proposed in the literature is optimized a newly proposed filter a gauss pulse tapered with a hanning window performs better than the truncated gauss pulse and the b spline also proposed in the literature for small filter length a numerically optimized filter shows the best results numerical experiments for 1 d and 2 d implementations confirm the theoretically predicted accuracy and efficiency properties of the algorithm

bilayer low density parity check codes for decode and forward in relay channels this paper describes an efficient implementation of binning for decode and forward df in relay channels using low density parity check ldpc codes bilayer ldpc codes are devised to approach the theoretically promised rate of the df relaying strategy by incorporating relay generated parity bits in specially designed bilayer graphical code structures while conventional ldpc codes are sensitively tuned to operate efficiently at a certain channel parameter the proposed bilayer ldpc codes are capable of working at two different channel parameters and two different rates that at the relay and at the destination to analyze the performance of bilayer ldpc codes bilayer density evolution is devised as an extension of the standard density evolution algorithm based on bilayer density evolution a design methodology is developed for the bilayer codes in which the degree distribution is iteratively improved using linear programming further in order to approach to the theoretical df rate for a wide range of channel parameters this paper proposes two different forms of bilayer codes the bilayer expurgated and bilayer lengthened codes it is demonstrated that the rate of a properly designed bilayer ldpc code can closely approach the theoretical df limit finally it is shown that a generalized version of the proposed bilayer code construction is applicable to relay networks with multiple relays

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

gaussian orthogonal relay channels optimal resource allocation and capacity a gaussian orthogonal relay model is investigated where the source transmits to the relay and destination in channel 1 and the relay transmits to the destination in channel 2 with channels 1 and 2 being orthogonalized in the time frequency plane in order to satisfy practical constraints the total available channel resource time and bandwidth is split into the two orthogonal channels and the resource allocation to the two channels is considered to be a design parameter that needs to be optimized the main focus of the analysis is on the case where the source to relay link is better than the source to destination link which is the usual scenario encountered in practice a lower bound on the capacity achievable rate is derived and optimized over the parameter spl theta which represents the fraction of the resource assigned to channel 1 it is shown that the lower bound achieves the max flow min cut upper bound at the optimizing spl theta the common value thus being the capacity of the channel at the optimizing spl theta furthermore it is shown that when the relay to destination signal to noise ratio snr is less than a certain threshold the capacity at the optimizing spl theta is also the maximum capacity of the channel over all possible resource allocation parameters spl theta finally the achievable rates for optimal and equal resource allocations are compared and it is shown that optimizing the resource allocation yields significant performance gains

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

optimized least square nonuniform fast fourier transform the main focus of this paper is to derive a memory efficient approximation to the nonuniform fourier transform of a support limited sequence we show that the standard nonuniform fast fourier transform nufft scheme is a shift invariant approximation of the exact fourier transform based on the theory of shift invariant representations we derive an exact expression for the worst case mean square approximation error using this metric we evaluate the optimal scale factors and the interpolator that provides the least approximation error we also derive the upper bound for the error component due to the lookup tablebased evaluation of the interpolator we use this metric to ensure that this component is not the dominant one theoretical and experimental comparisons with standard nufft schemes clearly demonstrate the significant improvement in accuracy over conventional schemes especially when the size of the uniform fast fourier transform fft is small since the memory requirement of the algorithm is dependent on the size of the uniform fft the proposed developments can lead to iterative signal reconstruction algorithms with significantly lower memory demands

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

field inhomogeneity correction based on gridding reconstruction for magnetic resonance imaging spatial variations of the main field give rise to artifacts in magnetic resonance images if disregarded in reconstruction with non cartesian k space sampling they often lead to unacceptable blurring data from such acquisitions are usually reconstructed with gridding methods and optionally restored with various correction methods both types of methods essentially face the same basic problem of adequately approximating an exponential function to enable an efficient processing with fast fourier transforms nevertheless they have commonly addressed it differently so far in the present work a unified approach is pursued the principle behind gridding methods is first generalized to nonequispaced sampling in both domains and then applied to field inhomogeneity correction three new algorithms which are compatible with a direct conjugate phase and an iterative algebraic reconstruction are derived in this way from a straightforward embedding of the data into a higher dimensional space their evaluation in simulations and phantom experiments with spiral k space sampling shows that one of them promises to provide a favorable compromise between fidelity and complexity compared with existing algorithms moreover it allows a simple choice of key parameters involved in approximating an exponential function and a balance between the accuracy of reconstruction and correction

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

on the optimality of the gridding reconstruction algorithm gridding reconstruction is a method to reconstruct data onto a cartesian grid from a set of nonuniformly sampled measurements this method is appreciated for being robust and computationally fast however it lacks solid analysis and design tools to quantify or minimize the reconstruction error least squares reconstruction lsr on the other hand is another method which is optimal in the sense that it minimizes the reconstruction error this method is computationally intensive and in many cases sensitive to measurement noise hence it is rarely used in practice despite their seemingly different approaches the gridding and lsr methods are shown to be closely related the similarity between these two methods is accentuated when they are properly expressed in a common matrix form it is shown that the gridding algorithm can be considered an approximation to the least squares method the optimal gridding parameters are defined as the ones which yield the minimum approximation error these parameters are calculated by minimizing the norm of an approximation error matrix this problem is studied and solved in the general form of approximation using linearly structured matrices this method not only supports more general forms of the gridding algorithm it can also be used to accelerate the reconstruction techniques from incomplete data the application of this method to a case of two dimensional 2 d spiral magnetic resonance imaging shows a reduction of more than 4 db in the average reconstruction error

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

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

harmonic analysis on product spaces rn x r and commute with the dilations p33 8 x2 x2 61x1 62x2 for x1 e rn x2 e rm and 61 82 0 then the corresponding operators are related to many problems in the theory of multiple fourier integrals see zygmund 21 the maximal operator which commutes with the p31382 is the jessen marcinkiewicz and zygmund 14 strong maximal operator ms defined by 1 ms f x1 x2 sup ri a y1 y2 1 dy1 dy2 xl x2 gr jr1

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

decorrelation of neutral vector variables theory and applications in this paper we propose novel strategies for neutral vector variable decorrelation two fundamental invertible transformations namely serial nonlinear transformation and parallel nonlinear transformation are proposed to carry out the decorrelation for a neutral vector variable which is not multivariate gaussian distributed the conventional principal component analysis cannot yield mutually independent scalar variables with the two proposed transformations a highly negatively correlated neutral vector can be transformed to a set of mutually independent scalar variables with the same degrees of freedom we also evaluate the decorrelation performances for the vectors generated from a single dirichlet distribution and a mixture of dirichlet distributions the mutual independence is verified with the distance correlation measurement the advantages of the proposed decorrelation strategies are intensively studied and demonstrated with synthesized data and practical application evaluations

boundedness criterion of journeʼs class of singular integrals on multiparameter hardy spaces abstract this article concerns nonconvolutional type operators also known as journe s type operators associated with a multiparameter family of dilations given by x 1 x 2 x m 1 x 1 2 x 2 m x m where x 1 r n 1 x 2 r n 2 x m r n m and m 3 we are especially interested in the boundedness of such operators on the multiparameter hardy spaces this work is motivated by pipher s result on the boundedness of these operators from the multiparameter h p spaces to l p spaces for 0 p 1 and journe s counter example which shows that the number of parameter plays a crucial role in boundedness of singular integral operators on multiparameter hardy spaces journe s work shows that there is a sharp difference between the situations for two and three or more parameters we establish in this paper the necessary and sufficient conditions under which the singular integral operators in journe s class are bounded on multiparameter h p spaces 0 p 1 with arbitrary number of parameters

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

a nonuniform fast fourier transform based on low rank approximation by viewing the nonuniform discrete fourier transform nudft as a perturbed version of a uniform discrete fourier transform we propose a fast and quasi optimal algorithm for computing the nudft based on the fast fourier transform fft our key observation is that an nudft and dft matrix divided entry by entry is often well approximated by a low rank matrix allowing us to express a nudft matrix as a sum of diagonally scaled dft matrices our algorithm is simple to implement automatically adapts to any working precision and is competitive with state of the art algorithms in the fully uniform case our algorithm is essentially the fft we also describe quasi optimal algorithms for the inverse nudft and two dimensional nudfts

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 regular fourier matrices and nonuniform fast fourier transforms for any triple of positive integers m n q the matrix f m n q called the m n q regular fourier matrix is defined the regular fourier matrices f m n q are then applied to set up new algorithms for nonuniform fast fourier transforms numerical results show that the accuracies obtained by our algorithms are much better than previously reported results with the same computation complexity the algorithms require o n cdot log 2n arithmetic operations where n is the number of data points

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

iterative decoding threshold analysis for ldpc convolutional codes an iterative decoding threshold analysis for terminated regular ldpc convolutional ldpcc codes is presented using density evolution techniques the convergence behavior of an iterative belief propagation decoder is analyzed for the binary erasure channel and the awgn channel with binary inputs it is shown that for a terminated ldpcc code ensemble the thresholds are better than for corresponding regular and irregular ldpc block codes

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

selection of a convolution function for fourier inversion using gridding computerised tomography application in the technique known as gridding the data samples are weighted for sampling density and convolved with a finite kernel then resampled on a grid preparatory to a fast fourier transform the authors compare the artifact introduced into the image for various convolving functions of different sizes including the kaiser bessel window and the zero order prolate spheroidal wave function pswf they also show a convolving function that improves upon the pswf in some circumstances

a practical guide to the recovery of wavelet coefficients from fourier measurements in a series of recent papers b adcock a c hansen and c poon appl comput harmon anal 36 2014 pp 387 415 b adcock m gataric and a c hansen siam j imaging sci 7 2014 pp 1690 1723 adcock et al siam j math anal 47 2015 pp 1196 1233 it was shown that one can optimally recover the wavelet coefficients of an unknown compactly supported function from pointwise evaluations of its fourier transform via the method of generalized sampling while these papers focused on the optimality of generalized sampling in terms of its stability and error bounds the current paper explains how this optimal method can be implemented to yield a computationally efficient algorithm in particular we show that generalized sampling has a computational complexity of mathcal o left m n log n right when recovering the first n boundary corrected wavelet coefficients of an unknown compactly supported function from m n fourier samples therefore due to the linear correspondences bet

fast approximate fourier transforms for irregularly spaced data several algorithms for efficiently evaluating trigonometric polynomials at irregularly spaced points are presented and analyzed the algorithms can be viewed as approximate generalizations of the fast fourier transform fft and they are compared with regard to their accuracy and their computational efficiency

bilayer ldpc convolutional codes for decode and forward relaying in this paper we present bilayer ldpc convolutional codes for half duplex relay channels two types of codes bilayer expurgated ldpc convolutional codes and bilayer lengthened ldpc convolutional codes are proposed for decode and forward df relaying in the case of the binary erasure relay channel we prove analytically that both code constructions achieve the capacities of the source relay link and the source destination link simultaneously provided that the channel conditions are known when designing the codes meanwhile both codes enable the highest transmission rate possible with df relaying for a wide range of channel parameters in addition the regular degree distributions can easily be computed from the channel parameters which significantly simplifies the code optimization the code construction and performance analysis are extended to the general binary memoryless symmetric channel where a capacity achieving performance is conjectured numerical results are provided for both types of codes with finite node degrees over binary erasure channels and binary input additive white gaussian noise channels which verify the aforementioned theoretical analysis

relative perturbation theory iii more bounds on eigenvalue variation abstract classically the relative error in ga 1 as an approximation to is measured by relative error in ga a the quantity log10 is usually used for the number of correct decimal digits in numerical results although this measure is clearly not a metric since it lacks symmetry between and ga in part i of this series two other kinds of relative distances which have much better mathematical properties have been introduced and employed to establish theories it is shown that these different measurements are topologically equivalent however the measure is more convenient to use in practice in this part we established relative perturbation bounds directly using the classical measure the new bounds for diagonalizable matrices are cleaner than the corresponding ones in part i and yield nice bounds for hermitian matrices too but when applied to nonnegative definite hermitian matrices the new bounds are weaker than those in part i

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

a frame theoretic approach to the nonuniform fast fourier transform nonuniform fourier data are routinely collected in applications such as magnetic resonance imaging synthetic aperture radar and synthetic imaging in radio astronomy to acquire a fast reconstruction that does not require an online inverse process the nonuniform fast fourier transform nfft also called convolutional gridding is frequently employed while various investigations have led to improvements in accuracy efficiency and robustness of the nfft not much attention has been paid to the fundamental analysis of the scheme and in particular its convergence properties this paper analyzes the convergence of the nfft by casting it as a fourier frame approximation in so doing we are able to design parameters for the method that satisfy conditions for numerical convergence our so called frame theoretic convolutional gridding algorithm can also be applied to detect features such as edges from nonuniform fourier samples of piecewise smooth functions

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

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

direct inversion of the three dimensional pseudo polar fourier transform the pseudo polar fourier transform is a specialized nonequally spaced fourier transform which evaluates the fourier transform on a near polar grid known as the pseudo polar grid the advantage of the pseudo polar grid over other nonuniform sampling geometries is that the transformation which samples the fourier transform on the pseudo polar grid can be inverted using a fast and stable algorithm for other sampling geometries even if the nonequally spaced fourier transform can be inverted the only known algorithms are iterative the convergence speed of these algorithms and their accuracy are difficult to control as they depend both on the sampling geometry and on the unknown reconstructed object in this paper a direct inversion algorithm for the three dimensional pseudo polar fourier transform is presented the algorithm is based only on one dimensional resampling operations and is shown to be significantly faster than existing iterative inversion algorithms

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

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 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

a framework for discrete integral transformations i the pseudopolar fourier transform the fourier transform of a continuous function evaluated at frequencies expressed in polar coordinates is an important conceptual tool for understanding physical continuum phenomena an analogous tool suitable for computations on discrete grids could be very useful however no exact analogue exists in the discrete case in this paper we present the notion of pseudopolar grid pp grid and the pseudopolar fourier transform ppft which evaluates the discrete fourier transform at points of the pp grid the pp grid is a type of concentric squares grid in which the radial density of squares is twice as high as usual the pp grid consists of equally spaced samples along rays where different rays are equally spaced in slope rather than angle we develop a fast algorithm for the ppft with the same complexity order as the cartesian fast fourier transform the algorithm is stable invertible requires only one dimensional operations and uses no approximate interpolations we prove that the ppft is invertible and develop two algorithms for its inversion iterative and direct both with complexity o n 2 log n where n times n is the size of the reconstructed image the iterative algorithm applies conjugate gradients to the gram operator of the ppft since the transform is ill conditioned we introduce a preconditioner which significantly accelerates the convergence the direct inversion algorithm utilizes the special frequency domain structure of the transform in two steps first it resamples the pp grid to a cartesian frequency grid and then recovers the image from the cartesian frequency grid

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

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

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

threshold saturation via spatial coupling why convolutional ldpc ensembles perform so well over the bec convolutional low density parity check ldpc ensembles introduced by felstrom and zigangirov have excellent thresholds and these thresholds are rapidly increasing functions of the average degree several variations on the basic theme have been proposed to date all of which share the good performance characteristics of convolutional ldpc ensembles we describe the fundamental mechanism that explains why convolutional like or spatially coupled codes perform so well in essence the spatial coupling of individual codes increases the belief propagation bp threshold of the new ensemble to its maximum possible value namely the maximum a posteriori map threshold of the underlying ensemble for this reason we call this phenomenon threshold saturation this gives an entirely new way of approaching capacity one significant advantage of this construction is that one can create capacity approaching ensembles with an error correcting radius that is increasing in the blocklength although we prove the threshold saturation only for a specific ensemble and for the binary erasure channel bec empirically the phenomenon occurs for a wide class of ensembles and channels more generally we conjecture that for a large range of graphical systems a similar saturation of the dynamical threshold occurs once individual components are coupled sufficiently strongly this might give rise to improved algorithms and new techniques for analysis

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

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

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

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

an 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

calderon zygmund theory for product domains hp spaces this article deals with the boundedness properties of singular integrals that generalize the double hilbert transform on product domains the action of these operators on hardy spaces and on l log l is discussed

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