randomized algorithms for the low rank approximation of matrices we describe two recently proposed randomized algorithms for the construction of low rank approximations to matrices and demonstrate their application inter alia to the evaluation of the singular value decompositions of numerically low rank matrices being probabilistic the schemes described here have a finite probability of failure in most cases this probability is rather negligible 10 17 is a typical value in many situations the new procedures are considerably more efficient and reliable than the classical deterministic ones they also parallelize naturally we present several numerical examples to illustrate the performance of the schemes

scattering resonances of convex obstacles for general boundary conditions we study the distribution of resonances for smooth strictly convex obstacles under general boundary conditions we show that under a pinched curvature condition for the boundary of the obstacle the resonances are separated into cubic bands and the distribution in each band satisfies weyl s law

complex scaling and the distribution of scattering poles the purpose of this paper is to establish sharp polynomial bounds on the number of scattering poles for a general class of compactly supported self adjoint perturbations of the laplacian in r n odd we also consider more general types of bounds that give sharper estimates in certain situations the general conclusion can be stated as follows the order of growth of the poles is the same as the order of growth of eigenvalues for corresponding compact problems from the few known cases however the exact asymptotics are expected to be different the scattering poles for compactly supported perturbations were rigorously defined by lax and phillips 13 in a more general setting they correspond to resonances the study of which has a long tradition in mathematical physics in the lax phillips theory they appear as the poles of the meromorphic continuation of the scattering matrix and coincide with the poles of the meromorphic continuation of the resolvent of the perturbed operator because of the latter characterization they can be considered as the analogue of the discrete spectral data for problems on noncompact domains the problem of estimating the counting function

geometric bounds on the density of resonances for semiclassical problems in this paper we shall give upper bounds on the number of resonances in certain regions in the complex plane close to the real axis for semiclassical operators like h 2 v x when h is small

iterative deblending of simultaneous source seismic data using seislet domain shaping regularization abstractwe used a novel iterative estimation scheme for separation of blended seismic data from simultaneous sources the scheme is based on an augmented estimation problem that can be solved by iteratively constraining the deblended data using shaping regularization in the seislet domain we formulated the forward modeling operator in the common receiver domain in which two sources were assumed to be blended using a random time shift dithering approach the nonlinear shaping regularization framework offered some freedom in designing a shaping operator to constrain the model in an underdetermined inverse problem we designed the backward operator and the shaping operator for the shaping regularization framework the backward operator can be optimally chosen as half of the identity operator in the two source case and the shaping operator can be chosen as coherency promoting operator the high performance deblending effect of the iterative framework was tested on three numerically blended synthetic data s

entropy of semiclassical measures in dimension 2 we study the high energy asymptotic properties of eigenfunctions of the laplacian in the case of a compact riemannian surfacem of anosov type to do this we look at families of distributions associated to them on the cotangent bundle t m and we derive entropic properties on their accumulation points in the high energy limit the so called semiclassical measures we show that the kolmogorov sinai entropy of a semiclassical measure for the geodesic ow gt is bounded from below by half of the ruelle upper bound i e hks g 1 2 s m d where is the upper lyapunov exponent at point

scattering asymptotics for riemann surfaces in this article we prove the optimal polynomial lower bound for the number of resonances of a surface with hyperbolic ends we also give weyl asymptotics for the relative scattering phase of such a surface the proofs are based on trace formulae analogous to those of the euclidean odd dimensional scattering the main technical ingredient is a new proof of the poisson formula theorem 5 7 which is applicable in the euclidean case as well our lower bound seems to be the first example of an optimal polynomial lower bound for the number of resonances holding for a general class of higher dimensional elliptic operators with no symmetries the previous general lower bounds or asymptotics were either nonoptimal 25 58 9 one dimensional or radial 65 67 and 54 41 1 or they required some degeneracy of the

finding structure with randomness probabilistic algorithms for constructing approximate matrix decompositions low rank matrix approximations such as the truncated singular value decomposition and the rank revealing qr decomposition play a central role in data analysis and scientific computing this work surveys and extends recent research which demonstrates that randomization offers a powerful tool for performing low rank matrix approximation these techniques exploit modern computational architectures more fully than classical methods and open the possibility of dealing with truly massive data sets this paper presents a modular framework for constructing randomized algorithms that compute partial matrix decompositions these methods use random sampling to identify a subspace that captures most of the action of a matrix the input matrix is then compressed either explicitly or implicitly to this subspace and the reduced matrix is manipulated deterministically to obtain the desired low rank factorization in many cases this approach beats its classical competitors in terms of accuracy robustness and or speed these claims are supported by extensive numerical experiments and a detailed error analysis the specific benefits of randomized techniques depend on the computational environment consider the model problem of finding the k dominant components of the singular value decomposition of an m times n matrix i for a dense input matrix randomized algorithms require bigo mn log k floating point operations flops in contrast to bigo mnk for classical algorithms ii for a sparse input matrix the flop count matches classical krylov subspace methods but the randomized approach is more robust and can easily be reorganized to exploit multiprocessor architectures iii for a matrix that is too large to fit in fast memory the randomized techniques require only a constant number of passes over the data as opposed to bigo k passes for classical algorithms in fact it is sometimes possible to perform matrix approximation with a single pass over the data

lower bounds for resonance counting functions for obstacle scattering in even dimensions in even dimensional euclidean scattering the resonances lie on the logarithmic cover of the complex plane this paper studies resonances for obstacle scattering in bbb r d with dirichlet neumann or admissible robin boundary conditions when d is even set n m r to be the number of resonances with norm at most r and argument between m pi and m 1 pi then lim sup r rightarrow infty log n m r over log r d if m in bbb z setminus 0

lower bounds on the number of scattering poles ii abstract we obtain lower bounds on the number of scattering poles for a class of abstract compactly supported perturbations of the laplacian in r n n odd they are applied to estimate the number of resonances for obstacle scattering and for hypoelliptic compactly supported perturbations of the laplacian the proof involves a development of lax phillips theory in a generalized setting a poisson formula for scattering poles and some simple tauberian arguments

oscillation criteria for delay and difference equations with non monotone arguments consider the first order linear delay differential equation 1 x t p t x t t 0 t t 0 where p email protected c t 0 r t t t 0 and lim t email protected t and the discrete analogue difference equation 1 dx n p n x t n 0 n 0 1 2 where d denotes the forward difference operator dx n x n 1 x n p n is a sequence of nonnegative real numbers and t n is a sequence of integers such that t n 0 and lim n email protected n the state of the art on the oscillation of all solutions to these equations are established especially in the case of non monotone arguments examples illustrating the results are given

numerical linear algebra and solvability of partial differential equations it was observed long ago that the obstruction to the accurate computation of eigenvalues of large non self adjoint matrices is inherent in the problem the basic idea is that the resolvent of a highly non normal operator can be very large far away from the spectrum this leads to an easily observable fact that algorithms for locating eigenvalues will typically find some false eigenvalues these false eigenvalues also explain one of the most surprising phenomena in linear pdes namely the fact discovered by hans lewy in 1957 in berkeley that one cannot always locally solve the pde p u f almost immediately after that discovery hormander provided an explanation of lewy s example showing that almost all operators with non constanct complex valued coefficients are not locally solvable in modern language that was done by considering the essentially dual problem of existence of non propagating singularities the purpose of this article is to review this work in the context of almost eigenvalues and from the point of view of semi classical analysis

preserving positivity in solutions of discretised stochastic differential equations we consider the euler discretisation of a scalar linear test equation with positive solutions and show for both strong and weak approximations that the probability of positivity over any finite interval of simulation tends to unity as the step size approaches zero although a s positivity in an approximation is impossible to achieve we develop for the strong maruyama approximation an asymptotic estimate of the number of mesh points required for positivity as our tolerance of non positive trajectories tends to zero and examine the effectiveness of this estimate in the context of practical numerical simulation we show how this analysis generalises to equations with a drift coefficient that may display a high level of nonlinearity but which must be linearly bounded from below i e when acting towards zero and a linearly bounded diffusion coefficient finally in the linear case we develop a refined asymptotic estimate that is more useful as an a priori guide to the number of mesh points required to produce positive approximations with a given probability

discrete retract principle for systems of discrete equations abstract a power tool for investigation of various problems of ordinary differential equations as well as delayed differential equations is a retraction method the development of this method is discussed here in the case of system discrete equations corresponding definition of a point of a type of strict egress for a given set with respect to the system of discrete equations u k f k u k is given in an example it is shown that this definition cannot exactly copy the analogous one known from the theory of ordinary differential equations conditions for existence of at least one solution with graph remaining in a given set are formulated the proof is based on idea of a retract principle in construction of a retract mapping the property of continuous dependence on their initial data is used illustrative applications are considered too

lieb thirring estimates for non self adjoint schrodinger operators for general nonsymmetric operators a we prove that the moment of order 1 of negative real parts of its eigenvalues is bounded by the moment of order of negative eigenvalues of its symmetric part h 12 a a as an application we obtain lieb thirring estimates for non self adjoint schrodinger operators in particular we recover recent results by frank et al lett math phys 77 309 2006 we also discuss moment of resonances of schrodinger self adjoint operators

simultaneous seismic data denoising and reconstruction via multichannel singular spectrum analysis we present a rank reduction algorithm that permits simultaneous reconstruction and random noise attenuation of seismic records we based our technique on multichannel singular spectrum analysis mssa the technique entails organizing spatial data at a given temporal frequency into a block hankel matrix that in ideal conditions is a matrix of rank k where k is the number of plane waves in the window of analysis additive noise and missing samples will increase the rank of the block hankel matrix of the data consequently rank reduction is proposed as a means to attenuate noise and recover missing traces we present an iterative algorithm that resembles seismic data reconstruction with the method of projection onto convex sets in addition we propose to adopt a randomized singular value decomposition to accelerate the rank reduction stage of the algorithm we apply mssa reconstruction to synthetic examples and a field data set synthetic examples were used to assess the performance of the method in two

positive and oscillating solutions of differential equations with delay in critical case abstract this article is devoted to the problem of existence of positive solution for common classes of nonlinear retarded functional differential equations some criteria of its existence are proved as well as some comparison results the obtained results are applied to the linear case moreover the existence of positive and oscillating solutions of a differential equation with delay x t a t x t in the critical case is considered some comparisons with known results are given

quasimodes and resonances sharp lower bounds we prove that asymptotically any cluster of quasimodes close to each other approximates at least the same number of resonances counting multiplicities as a consequence we get that the counting function of the number of resonances close to the real axis is bounded from below essentially by the counting function of the quasimodes

on the multilevel internal structure of the asymptotic distribution of resonances abstract we prove that the set of resonances h has a multilevel asymptotic structure for the following classes of hamiltonians h schrodinger operators with point interactions y j r 3 quantum graphs and 1 d photonic crystals in the case of n 2 point interactions h consists of a finite number of sequences with logarithmic asymptotics leading parameters n of these sequences are connected with the geometry of the family y y j j 1 n the minimal parameter min corresponds to sequences with more narrow and so more observable resonances the asymptotic density of narrow resonances is expressed via the multiplicity of min and is connected with symmetries of y in the cases of quantum graphs and 1 d photonic crystals the decomposition of h into asymptotic sequences is proved under commensurability conditions to address the general quantum graph we introduce asymptotic density functions for two classes of complex strips the obtained results and effects are compared with obstacle scattering

diophantine tori and weyl laws for non selfadjoint operators in dimension two we study the distribution of eigenvalues for non selfadjoint perturbations of selfadjoint semiclassical analytic pseudodifferential operators in dimension two assuming that the classical flow of the unperturbed part is completely integrable an asymptotic formula of weyl type for the number of eigenvalues in a spectral band bounded from above and from below by levels corresponding to diophantine invariant lagrangian tori is established the weyl law is given in terms of the long time averages of the leading non selfadjoint perturbation along the classical flow of the unperturbed part

asymptotic formulae for the solutions of a linear delay difference equation consider the linear delay difference equation xn 1 xn anxnk where an s are real numbers and k is a positive integer in this paper we establish asymptotic formulae for the solutions of the above equation assuming n 0 an p for some p 1 the asymptotic formulae are given by an appropriate recursive sequence for p 1 2 3 4 we present explicit formulae we also construct an example which shows that some comparison results in a recent paper of yan and qian are incorrect

efficient denoising algorithms for large experimental datasets and their applications in fourier transform ion cyclotron resonance mass spectrometry modern scientific research produces datasets of increasing size and complexity that require dedicated numerical methods to be processed in many cases the analysis of spectroscopic data involves the denoising of raw data before any further processing current efficient denoising algorithms require the singular value decomposition of a matrix with a size that scales up as the square of the data length preventing their use on very large datasets taking advantage of recent progress on random projection and probabilistic algorithms we developed a simple and efficient method for the denoising of very large datasets based on the qr decomposition of a matrix randomly sampled from the data this approach allows a gain of nearly three orders of magnitude in processing time compared with classical singular value decomposition denoising this procedure called urqrd uncoiled random qr denoising strongly reduces the computer memory footprint and allows the denoising algorithm to be applied to virtually unlimited data size the efficiency of these numerical tools is demonstrated on experimental data from high resolution broadband fourier transform ion cyclotron resonance mass spectrometry which has applications in proteomics and metabolomics we show that robust denoising is achieved in 2d spectra whose interpretation is severely impaired by scintillation noise these denoising procedures can be adapted to many other data analysis domains where the size and or the processing time are crucial

a fast reduced rank interpolation method for prestack seismic volumes that depend on four spatial dimensions rank reduction strategies can be employed to attenuate noise and for prestack seismic data regularization we present a fast version of cadzow reduced rank reconstruction method cadzow reconstruction is implemented by embedding 4d spatial data into a level four block toeplitz matrix rank reduction of this matrix via the lanczos bidiagonalization algorithm is used to recover missing observations and to attenuate random noise the computational cost of the lanczos bidiagonalization is dominated by the cost of multiplying a level four block toeplitz matrix by a vector this is efficiently implemented via the 4d fast fourier transform the proposed algorithm significantly decreases the computational cost of rank reduction methods for multidimensional seismic data denoising and reconstruction synthetic and field prestack data examples are used to examine the effectiveness of the proposed method

the asymptotic bounds of solutions of linear delay systems abstract our aim is to obtain conditions under which every solutionyof the vector functional differential equation formula can be estimated in terms of a solution of a scalar functional equation formula wherea x is a suitable real positive function andbis a suitable real negative constant moreover ifb x is diagonal matrix we derive more precise estimates ofyin terms of a solution of a vector functional equation formula where the elements ofa x are formed by the absolute values of the corresponding elements ofa x these asymptotic results enable us to obtain sharp estimates of solutions of certain linear delay systems

f xy eigenimage noise suppression f xy eigenimage filtering works by replacing constant frequency slices with the sum of their first few weighted eigenimages and can be used to remove random noise from stacked 3d seismic volumes it performs equally well on flat or dipping events and has no effect on noiseless data when the number of dips is no more than the number of summed eigenimages it is also independent of many x and y consistent effects such as reordering statics and filtering one consequence is that it tends to do a good job along the boundaries of the 3d grid execution time is comparable to f xy prediction filtering but can be greatly reduced using approximations based on lanczos bidiagonalization

fast dual domain reduced rank algorithm for 3d deblending via randomized qr decomposition abstractwe have developed a fast dual domain algorithm based on matrix rank reduction for separating simultaneous source seismic data our algorithm operates on 3d common receiver gathers or offset midpoint gathers at a given monochromatic frequency slice in the x y domain the spatial data of the ideal unblended common receiver or offset midpoint gather could be represented via a low rank matrix the interferences from the randomly and closely fired shots increased the rank of the aforementioned matrix therefore we could minimize the misfit between the blended observation and the predicted blended data subject to a low rank constraint that was applied to the data in the x y domain the low rank constraint could be implemented via the classic truncated singular value decomposition svd or via a randomized qr decomposition rqrd the rqrd yielded nearly one order of processing time improvement with respect to the truncated svd we have also discovered that the rqrd was less stringent on the select

scattering frequencies and gevrey 3 singularities on etudie les frequences de diffusion k pour le laplacien dans la region r n o exterieur a un obstacle analytique compact on suppose que n est impair et qu on impose des conditions de dirichlet sur

remark on positive solutions of discrete equation δu k n p k u k abstract a delayed discrete equation u k n p k u k with positive function p k is considered sufficient conditions are formulated to guarantee the existence of a positive solution if k

positive solutions of the equation ẋ t c t x t τ in the critical case abstract the equation t c t x t is considered in the critical case for it the asymptotic behavior of dominant and subdominant solutions is studied a generalization is made and connections with known results are discussed

sharp bounds on the number of scattering poles for perturbations of the laplacian sharp bounds on the numbern r of the scattering poles in the disc z r for a large class of compactly supported perturbations not necessarily selfadjoint of the laplacian in n n 3 odd are obtained in particular in the elliptic case the estimaten r cr n c is proved

simultaneous source separation using a robust radon transform abstractwe adopted the robust radon transform to eliminate erratic incoherent noise that arises in common receiver gathers when simultaneous source data are acquired the proposed robust radon transform was posed as an inverse problem using an l1 misfit that is not sensitive to erratic noise the latter permitted us to design radon algorithms that are capable of eliminating incoherent noise in common receiver gathers we also compared nonrobust and robust radon transforms that are implemented via a quadratic l2 or a sparse l1 penalty term in the cost function the results demonstrated the importance of incorporating a robust misfit functional in the radon transform to cope with simultaneous source interferences synthetic and real data examples proved that the robust radon transform produces more accurate data estimates than least squares and sparse radon transforms

separation of blended data by iterative estimation and subtraction of blending interference noise seismic acquisition is a trade off between economy and quality in conventional acquisition the time intervals between successive records are large enough to avoid interference in time to obtain an efficient survey the spatial source sampling is therefore often too large however in blending or simultaneous acquisition temporal overlap between shot records is allowed this additional degree of freedom in survey design significantly improves the quality or the economics or both deblending is the procedure of recovering the data as if they were acquired in the conventional unblended way a simple least squares procedure however does not remove the interference due to other sources or blending noise fortunately the character of this noise is different in different domains e g it is coherent in the common source domain but incoherent in the common receiver domain this property is used to obtain a considerable improvement we propose to estimate the blending noise and subtract it from the blended data the estimate does not need to be perfect because our procedure is iterative starting with the least squares deblended data the estimate of the blending noise is obtained via the following steps sort the data to a domain where the blending noise is incoherent apply a noise suppression filter apply a threshold to remove the remaining noise ending up with part of the signal compute an estimate of the blending noise from this signal at each iteration the threshold can be lowered and more of the signal is recovered promising results were obtained with a simple implementation of this method for both impulsive and vibratory sources undoubtedly in the future algorithms will be developed for the direct processing of blended data however currently a high quality deblending procedure is an important step allowing the application of contemporary processing flows