on polyhedral approximations of the positive semidefinite cone it is well known that state of the art linear programming solvers are more efficient than their semidefinite programming counterparts and can scale to much larger problem sizes this leads us to co

identifiability in blind deconvolution with subspace or sparsity constraints blind deconvolution bd the resolution of a signal and a filter given their convolution arises in many applications without further constraints bd is ill posed in practice subspace or sparsity constraints have been imposed to reduce the search space and have shown some empirical success however the existing theoretical analysis on uniqueness in bd is rather limited in an effort to address the still open question we derive sufficient conditions under which two vectors can be uniquely identified from their circular convolution subject to subspace or sparsity constraints these sufficient conditions provide the first algebraic sample complexities for bd we first derive a sufficient condition that applies to almost all bases or frames for bd of vectors in mathbb c n with two subspace constraints of dimensions m 1 and m 2 the required sample complexity is n geq m 1 m 2 then we impose a sub band structure on one basis and derive a sufficient condition that involves a relaxed sample complexity n geq m 1 m 2 1 which we show to be optimal we present the extensions of these results to bd with sparsity constraints or mixed constraints with the sparsity level replacing the subspace dimension the cost for the unknown support in this case is an extra factor of 2 in the sample complexity

prime phase retrieval via majorization minimization this paper considers the phase retrieval problem in which measurements consist of only the magnitude of several linear measurements of the unknown e g spectral components of a time sequence we develop low complexity algorithms with superior performance based on the majorization minimization mm framework the proposed algorithms are referred to as prime phase retrieval via the majorization minimization technique they are preferred to existing benchmark methods since at each iteration a simple surrogate problem is solved with a closed form solution that monotonically decreases the original objective function in total three algorithms are proposed using different majorization minimization techniques experimental results validate that our algorithms outperform existing methods in terms of successful recovery and mean square error under various settings

blind deconvolution using convex programming we consider the problem of recovering two unknown vectors w and x of length l from their circular convolution we make the structural assumption that the two vectors are members of known subspaces one with dimension n and the other with dimension k although the observed convolution is nonlinear in both w and x it is linear in the rank 1 matrix formed by their outer product wx this observation allows us to recast the deconvolution problem as low rank matrix recovery problem from linear measurements whose natural convex relaxation is a nuclear norm minimization program we prove the effectiveness of this relaxation by showing that for generic signals the program can deconvolve w and x exactly when the maximum of n and k is almost on the order of l that is we show that if x is drawn from a random subspace of dimension n and w is a vector in a subspace of dimension k whose basis vectors are spread out in the frequency domain then nuclear norm minimization recovers wx without error we discuss this result in the context of blind channel estimation in communications if we have a message of length n which we code using a random l x n coding matrix and the encoded message travels through an unknown linear time invariant channel of maximum length k then the receiver can recover both the channel response and the message when l n k to within constant and log factors

convex optimization approaches for blind sensor calibration using sparsity we investigate a compressive sensing framework in which the sensors introduce a distortion to the measurements in the form of unknown gains we focus on blind calibration using measures performed on multiple unknown but sparse signals and formulate the joint recovery of the gains and the sparse signals as a convex optimization problem we divide this problem in 3 subproblems with different conditions on the gains specifially i gains with different amplitude and the same phase ii gains with the same amplitude and different phase and iii gains with different amplitude and phase in order to solve the first case we propose an extension to the basis pursuit optimization which can estimate the unknown gains along with the unknown sparse signals for the second case we formulate a quadratic approach that eliminates the unknown phase shifts and retrieves the unknown sparse signals an alternative form of this approach is also formulated to reduce complexity and memory requirements and provide scalability with respect to the number of input signals finally for the third case we propose a formulation that combines the earlier two approaches to solve the problem the performance of the proposed algorithms is investigated extensively through numerical simulations which demonstrates that simultaneous signal recovery and calibration is possible with convex methods when sufficiently many unknown but sparse calibrating signals are provided

on lipschitz analysis and lipschitz synthesis for the phase retrieval problem in this paper we prove two results regarding reconstruction from magnitudes of frame coefficients the so called phase retrieval problem first we show that phase retrievability as an algebraic property implies that nonlinear maps are bi lipschitz with respect to appropriate metrics on the quotient space second we prove that reconstruction can be performed using lipschitz continuous maps specifically we show that when nonlinear analysis maps h r m are injective with x hx fki m 1 and x hx fki 2 m 1 where f1 fm is a frame for a hilbert space h and

fast unit modulus least squares with applications in beamforming unit modulus least squares uls problems arise in many applications including phase only beamforming sensor network localization synchronization phase retrieval and radar code design uls formulations can always be recast as unit modulus quadratic programs to which semidefinite relaxation sdr can be applied and is often the state of the art approach however sdr lifts the problem dimension i e the number of variables from n to n 2 which drastically increases the memory burden and computational cost when the problem size is already large e g when designing phase only beamformer weights for massive multiple input multiple output systems this paper focuses on scalable first order algorithms for the uls problem and some of its variants it advocates using simple gradient projection gp as a starting point for solving the uls problem establishes global convergence of gp to a karush kuhn tucker point for this np hard problem and bounds its iteration complexity then it proposes uls extensions tailored to reflect practical beamformer design objectives bringing in and exploiting new degrees of freedom to improve the beampattern designs simple variants of gp are proposed to deal with these extended uls problems simulations are used to showcase the effectiveness of the proposed algorithms in both the plain uls problem and in the context of phase only beamforming

on the rank minimization problem over a positive semidefinite linear matrix inequality we consider the problem of minimizing the rank of a positive semidefinite matrix subject to the constraint that an affine transformation of it is also positive semidefinite our method for solving this problem employs ideas from the ordered linear complementarity theory and the notion of the least element in a vector lattice this problem is of importance in many contexts for example in feedback synthesis problems and such an example is also provided

gauge optimization and duality gauge functions significantly generalize the notion of a norm and gauge optimization as defined by r m freund math programming 38 1987 pp 47 67 seeks the element of a convex set that is minimal with respect to a gauge function this conceptually simple problem can be used to model a remarkable array of useful problems including a special case of conic optimization and related problems that arise in machine learning and signal processing the gauge structure of these problems allows for a special kind of duality framework this paper explores the duality framework proposed by freund and proposes a particular form of the problem that exposes some useful properties of the gauge optimization framework such as the variational properties of its value function and yet maintains most of the generality of the abstract form of gauge optimization

the reconstruction of a multidimensional sequence from the phase or magnitude of its fourier transform this paper addresses two fundamental issues involved in the reconstruction of a multidimensional sequence from either the phase or magnitude of its fourier transform the first issue relates to the uniqueness of a multidimensional sequence in terms of its phase or magnitude although phase or magnitude information alone is not sufficient in general to uniquely specify a sequence a large class of sequences are shown to be recoverable from their phase or magnitude the second issue which is addressed in this paper concerns the actual reconstruction of a multidimensional sequence from its phase or magnitude for those sequences which are uniquely specified by their phase several practical algorithms are described which may be used to reconstruct a sequence from its phase several examples of phase only reconstruction are also presented unfortunately however even for those sequences which are uniquely defined by their magnitude it appears that a practical algorithm is yet to be developed for reconstructing a sequence from only its magnitude nevertheless an iterative procedure which has been proposed is briefly discussed and evaluated

constructing solutions of hamilton jacobi equations for 2d fields with one component by means of backlund transformations the hamilton jacobi formalism generalized to two dimensional field theories according to lepage s canonical framework is applied to several relativistic real scalar fields e g massless and massive klein gordon sinh and sine gordon liouville and 4 theories the relations between the euler lagrange and the hamilton jacobi equations are discussed in dedonder and weyl s and the corresponding wave fronts are calculated in caratheodory s formulation unlike mechanics one has to impose certain integrability conditions on the velocity fields to guarantee the transversality relations and especially the dynamical equivalence between hamilton jacobi wave fronts and fields of extremals embedded therein backlund transformations play a crucial role in solving the resulting system of coupled nonlinear pdes

identifiability in bilinear inverse problems with applications to subspace or sparsity constrained blind gain and phase calibration bilinear inverse problems bips the resolution of two vectors given their image under a bilinear mapping arise in many applications without further constraints bips are usually ill posed in practice the properties of natural signals are exploited to solve bips for example subspace constraints or sparsity constraints are imposed to reduce the search space these approaches have shown some success in practice however there are few results on uniqueness in bips for most bips the fundamental question of under what condition the problem admits a unique solution is yet to be answered for example blind gain and phase calibration bgpc is a structured bip which arises in many applications including inverse rendering in computational relighting albedo estimation with unknown lighting blind phase and gain calibration in sensor array processing and multichannel blind deconvolution mbd it is interesting to study the uniqueness of such problems in this paper we define identifiability of a bip up to a group of transformations we derive necessary and sufficient conditions for such identifiability i e the conditions under which the solutions can be uniquely determined up to the transformation group these conditions take the form of dividing the identifiability of the pair of unknown variables into the individual identifiability of each variable although verifying these individual conditions requires problem specific procedures this framework is universally applicable to all bips applying these results to bgpc we derive sufficient conditions for unique recovery under several scenarios including subspace joint sparsity and sparsity models for bgpc with joint sparsity or sparsity constraints we develop a procedure to compute the transformation groups corresponding to inherent ambiguities we also give necessary conditions in the form of tight lower bounds on sample complexities and demonstrate the tightness of these bounds by numerical experiments the results for bgpc not only demonstrate the application of the proposed general framework for identifiability analysis but are also of interest in their own right

phaseless recovery using the gauss newton method in this paper we propose a gauss newton algorithm to recover an n dimensional signal from its phaseless measurements the algorithm has two stages in the first stage the algorithm obtains a good initialization by calculating the eigenvector corresponding to the largest eigenvalue of a hermitian matrix in the second stage the algorithm solves an optimization problem iteratively using the gauss newton method our initialization method makes full use of all measurements and provides a good initial guess as long as the number of random measurements is o n for real valued signals we prove that a resampled version of gauss newton iterations converges to the global optimal solution quadratically with o n log n random measurements numerical experiments show that the gauss newton method has better empirical performance than other algorithms such as the wirtinger flow algorithm and altmin phase algorithm

phase retrievable operator valued frames and representations of quantum channels abstract we examine some connections among phase retrievable not necessarily self adjoint operator valued frames projective group representation frames and representations of quantum channels we first present some characterizations of phase retrievable frames for general operator systems acting on both finite and infinite dimensional hilbert spaces which generalize the known results for vector valued frames fusion frames and frames of hermitian matrices for an irreducible projective unitary representation of a finite group the image system is automatically phase retrievable and moreover it is a point wisely tight operator valued frames we generalize this notion to more general operator valued frames and prove that point wise tight operator valued frames are exactly the ones that are right equivalent to operator valued tight frames for an operator system that represent a quantum channel we show that phase retrievability of the system is independent of the choices of the representations of the quantum channel

a direct boundary integral method for the three dimensional lifting flow abstract a regularized integral formulation in velocity vector terms for three dimensional incompressible non lifting and lifting flow is developed in the case of lifting flow the circulatory flow around the body is generated by a discrete distribution of inner vortices and a vectorial equality in velocity terms is used as an equivalent of the equal pressure kutta condition the numerical comparison with the panel method in the case of non lifting flow shows the superiority of the present method also a two dimensional lifting flow is simulated using a high spanwise wing and comparison with the analytic solution shows a very good concordance even at high incidence

phase retrieval via wirtinger flow theory and algorithms we study the problem of recovering the phase from magnitude measurements specifically we wish to reconstruct a complex valued signal boldsymbol x in mathbb c n about which we have phaseless samples of the form y r left langle boldsymbol a r boldsymbol x rangle right 2 r 1 ldots m knowledge of the phase of these samples would yield a linear system this paper develops a nonconvex formulation of the phase retrieval problem as well as a concrete solution algorithm in a nutshell this algorithm starts with a careful initialization obtained by means of a spectral method and then refines this initial estimate by iteratively applying novel update rules which have low computational complexity much like in a gradient descent scheme the main contribution is that this algorithm is shown to rigorously allow the exact retrieval of phase information from a nearly minimal number of random measurements indeed the sequence of successive iterates provably converges to the solution at a geometric rate so that the proposed scheme is efficient both in terms of computational and data resources in theory a variation on this scheme leads to a near linear time algorithm for a physically realizable model based on coded diffraction patterns we illustrate the effectiveness of our methods with various experiments on image data underlying our analysis are insights for the analysis of nonconvex optimization schemes that may have implications for computational problems beyond phase retrieval

characterization of analytic wavelet transforms and a new phaseless reconstruction algorithm we obtain a characterization of all wavelets leading to analytic wavelet transforms wt the characterization is obtained as a byproduct of the theoretical foundations of a new method for wavelet phase reconstruction from magnitude only coefficients the cornerstone of our analysis is an expression of the partial derivatives of the continuous wt which results in phase magnitude relationships similar to the short time fourier transform setting and valid for the generalized family of cauchy wavelets we show that the existence of such relations is equivalent to analyticity of the wt up to a multiplicative weight and a scaling of the mother wavelet the implementation of the new phaseless reconstruction method is considered in detail and compared to previous methods it is shown that the proposed method provides significant performance gains and a great flexibility regarding accuracy versus complexity in addition we discuss the relation between scalogram reassignment operators and the wavelet transform phase gradient and present an observation on the phase around zeros of the wt

spectral matrix completion by cyclic projection and application to sound source reconstruction from non synchronous measurements a fundamental limitation of the inverse acoustic problem is determined by the size of the array and the microphone density a solution to achieve large array and or high microphone density is to scan the object of interest by moving sequentially a small prototype array which is referred to as sequential measurements in comparison to a large array and or high microphone density array that can acquire simultaneously all the information of the spectral matrix in particular all cross spectra sequential measurements can only acquire a block diagonal spectral matrix while the cross spectra between the sequential measurements remain unknown due to the missing phase relationships between consecutive positions nevertheless these unknown cross spectra are necessary for sound source reconstruction the objective of this paper is to propose an algorithm to recover the missing elements of the spectral matrix in the case where the acoustical field is highly coherent this issue is shown to boil down to a matrix completion problem subject to given constraint of hermitian symmetry measurements fitting reduced rank and spatial continuity of the sound field a cyclic projection cp algorithm is proposed in this work to find an optimal solution at the intersection between three predefined sets the proposed method is analyzed through numerical simulations of diverse setups and is also validated experimentally

on global convergence of gradient descent algorithms for generalized phase retrieval problem abstract in this paper we study the generalized phase retrieval problem to recover a signal x c n from the measurements y r a r x 2 r 1 2 m the problem can be reformulated as a least squares minimization problem although the cost function is nonconvex the global convergence of gradient descent algorithms from a random initialization is studied when m is large enough we improve the known result of the local convergence from a spectral initialization when the signal x is real valued we prove that the cost function is local convex near the solution x to accelerate the gradient descent we review and apply several efficient line search methods with exact line search stepsize we also perform a comparative numerical study of the line search methods and the alternative projection method numerical simulations demonstrate the superior ability of lbfgs algorithm than other algorithms

misspecified nonconvex statistical optimization for sparse phase retrieval existing nonconvex statistical optimization theory and methods crucially rely on the correct specification of the underlying true statistical models to address this issue we take a first step towards taming model misspecification by studying the high dimensional sparse phase retrieval problem with misspecified link functions in particular we propose a simple variant of the thresholded wirtinger flow algorithm that given a proper initialization linearly converges to an estimator with optimal statistical accuracy for a broad family of unknown link functions we further provide extensive numerical experiments to support our theoretical findings

sparse phase retrieval via truncated amplitude flow this paper develops a novel algorithm termed sparse truncated amplitude flow sparta to reconstruct a sparse signal from a small number of magnitude only measurements it deals with what is also known as sparse phase retrieval pr which is np hard in general and emerges in many science and engineering applications upon formulating sparse pr as an amplitude based nonconvex optimization task sparta works iteratively in two stages in stage one the support of the underlying sparse signal is recovered using an analytically well justified rule and subsequently a sparse orthogonality promoting initialization is obtained via power iterations restricted on the support and in the second stage the initialization is successively refined by means of hard thresholding based gradient type iterations sparta is a simple yet effective scalable and fast sparse pr solver on the theoretical side for any n dimensional k sparse k ll n signal boldsymbol x with minimum in modulus nonzero entries on the order of 1 sqrt k vert boldsymbol x vert 2 sparta recovers the signal exactly up to a global unimodular constant from about k 2 log n random gaussian measurements with high probability furthermore sparta incurs computational complexity on the order of k 2n log n with total runtime proportional to the time required to read the data which improves upon the state of the art by at least a factor of k finally sparta is robust against additive noise of bounded support extensive numerical tests corroborate markedly improved recovery performance and speedups of sparta relative to existing alternatives

on two dimensional hilbert integral equations generalized minimum phase signals and phase retrieval one dimensional 1 d causal signals admit hilbert integral relations between the real and imaginary parts of their fourier spectra for 1 d minimum phase signals the log magnitude and phase spectra also admit such hilbert relations in this paper we extend these results to 2 d signals we first establish the hilbert integral equations for 2 d first quadrant signals for continuous domain 2 d signals we present the partial hilbert transform relations between the real and imaginary parts of the spectrum for the discrete domain counterpart we show that the partial hilbert transform does not suffice which motivates us to introduce the composite hilbert transform considering the problem of phase retrieval we show that in both 1 d 2 d and continuous discrete domains there exists a generalized class of signals which subsumes the well known class of 1 d minimum phase signals with a rational transfer function for which hilbert integral relations exist between the log magnitude and phase spectra we introduce them under the nomenclature of generalized minimum phase signals for this class of signals phase retrieval is possible by using the hilbert transform without the requirement of a rational transfer function furthermore we show that a generalized minimum phase signal admits a stable convolutional inverse which also belongs to the same class simulation results demonstrating accurate reconstruction of 1 d and 2 d generalized minimum phase signals from their magnitude spectra are presented to support the theoretical developments

phase retrieval through regularization for seismic problems abstractthe most advanced data processing imaging algorithms aim at optimizing a suitably defined cost function consisting of a data misfit term posed in the data domain and a regularization term for strong phase errors in the data however such algorithms fail to achieve an acceptable solution because they force us to match the incorrect phase information to remedy this deficiency we replaced the time domain misfit term by a fourier magnitude fidelity term to recover the desired signal from just the amplitude spectrum of the observations such a problem is known as phase retrieval and it is a subject of interest for those who cannot measure the phase information about the system under study the proposed phase retrieval problem is solved by a fast iterative shrinkage thresholding algorithm and it is used to tackle two common phase problems arising in seismic processing data recovery in the presence of residual static shifts and deconvolution with a missing wavelet phase under sparsity constrain

phase retrieval via sparse wirtinger flow abstract phase retrieval pr problem is an inverse problem which arises in various applications based on the wirtinger flow method an algorithm utilizing the sparsity priority called swf sparse wirtinger flow is proposed in this paper to deal with the pr problem firstly the support of the signal is estimated besides the initialization is evaluated based on this support then the evaluation is updated by the hard thresholding method from this initialization we prove that for any k sparse signal with length n swf has a linear convergence with o k 2 log n phaseless gaussian random measurements to get e accuracy the computational complexity of swf is o k 3 n log n log 1 e numerical tests also demonstrate that swf has a higher recovery rate than other algorithms compared especially when we have no prior information about sparsity k moreover swf is robust to the noise

phase retrieval from incomplete magnitude information via total variation regularization the phase retrieval problem has drawn considerable attention as many optical detection devices can only measure magnitudes of the fourier transform of the underlying object signal or image this paper addresses the phase retrieval problem from incomplete data where only partial magnitudes of fourier transform are obtained in particular we consider structured illuminated patterns in holography and find that noninteger values used in designing such patterns often yield better reconstruction than the conventional integer valued ones furthermore we demonstrate theoretically and numerically that three diffracted sets of complete magnitude data are sufficient to recover the object to compensate for incomplete information we incorporate a total variation regularization a priori to guarantee that the reconstructed image satisfies some desirable properties the proposed model can be solved efficiently by an alternative directional multiplier method with provable convergence numerical experiments valid

phasecode fast and efficient compressive phase retrieval based on sparse graph codes we consider the problem of recovering a complex signal x in mathbb c n from m intensity measurements of the form a i mathrm h x 1 leq i leq m where a i mathrm h is the i th row of measurement matrix a in mathbb c m times n our main focus is on the case where the measurement vectors are unconstrained and where x is exactly k sparse or the so called general compressive phase retrieval problem we introduce phasecode a novel family of fast and efficient algorithms that are based on a sparse graph coding framework we show that in the noiseless case the phasecode algorithm can recover an arbitrarily close to one fraction of the k nonzero signal components using only slightly more than 4k measurements when the support of the signal is uniformly random with the order optimal time and memory complexity of theta k 1 it is known that the fundamental limit for the number of measurements in compressive phase retrieval problem is 4k o k for the more difficult problem of recovering the signal exactly and with no assumptions on its support distribution this shows that under mild relaxation of the conditions our algorithm is the first constructive capacity approaching compressive phase retrieval algorithm in fact our algorithm is also order optimal in complexity and memory furthermore we show that for any signal x phasecode can recover a random 1 p fraction of the nonzero components of x with high probability where p can be made arbitrarily close to zero with sample complexity m c p k where c p is a small constant depending on p that can be precisely calculated with optimal time and memory complexity as a result assuming that the nonzero components of x are lower bounded by theta 1 and upper bounded by theta k gamma for some positive constant gamma we are able to provide a strong ell 1 guarantee for the estimated signal hat x as follows hat x x 1 leq p x 1 1 o 1 where p can be made arbitrarily close to zero as one instance the phasecode algorithm can provably recover with high probability a random 1 10 7 fraction of the significant signal components using at most m 14k measurements next motivated by some important practical classes of optical systems we consider a fourier friendly constrained measurement setting and show that its performance matches that of the unconstrained setting when the signal is sparse in the fourier domain with uniform support in the fourier friendly setting that we consider the measurement matrix is constrained to be a cascade of fourier matrices corresponding to optical lenses and diagonal matrices corresponding to diffraction mask patterns finally we tackle the compressive phase retrieval problem in the presence of noise where measurements are in the form of y i a i mathrm h x 2 w i and w i is the additive noise to the i th measurement we assume that the signal is quantized and each nonzero component can take l m possible magnitudes and l p possible phases we consider the regime where k beta n delta delta in 0 1 we use the same architecture of phasecode for the noiseless case and robustify it using two schemes the almost linear scheme and the sublinear scheme we prove that with high probability the almost linear scheme recovers x with sample complexity theta k log n and computational complexity theta l m l p n log n and the sublinear scheme recovers x with sample complexity theta k log 3 n and computational complexity theta l m l p k log 3 n throughout we provide extensive simulation results that validate the practical power of our proposed algorithms for the sparse unconstrained and fourier friendly measurement settings for noiseless and noisy scenarios 1 here we define the notation mathcal o cdot theta cdot and omega cdot we have f mathcal o g if and only if there exists a constant c 1 0 such that left f g right f theta g if and only if there exist two constants c 1 c 2 0 such that c 1 and f omega g if and only if there exists a constant c 1 0 such that left f g right c 1

vectorial phase retrieval of 1 d signals reconstruction of signals from measurements of their spectral intensities also known as the phase retrieval problem is of fundamental importance in many scientific fields in this paper we present a novel framework denoted as vectorial phase retrieval for reconstruction of pairs of signals from spectral intensity measurements of the two signals and of their interference we show that this new framework can alleviate some of the theoretical and computational challenges associated with classical phase retrieval from a single signal first we prove that for compactly supported signals in the absence of measurement noise this new setup admits a unique solution next we present a statistical analysis of vectorial phase retrieval and derive a computationally efficient algorithm to solve it finally we illustrate via simulations that our algorithm can accurately reconstruct signals even at considerable noise levels

constructing a class of solutions for the hamilton jacobi equation in field theory a new approach leading to the formulation of the hamilton jacobi equation for field theories is investigated within the framework of jet bundles and multisymplectic manifolds an algorithm associating classes of solutions to given sets of boundary conditions of the field equations is provided the paper also puts into evidence the intrinsic limits of the hamilton jacobi method as an algorithm to determine families of solutions of the field equations showing how the choice of the boundary data is often limited by compatibility conditions

signal reconstruction from the magnitude of subspace components we consider signal reconstruction from the norms of subspace components generalizing standard phase retrieval problems in the deterministic setting a closed reconstruction formula is derived when the subspaces satisfy certain cubature conditions that require at least a quadratic number of subspaces moreover we address reconstruction under the erasure of a subset of the norms using the concepts of p fusion frames and list decoding we propose an algorithm that outputs a finite list of candidate signals one of which is the correct one in the random setting we show that a set of subspaces chosen at random and of cardinality scaling linearly in the ambient dimension allows for exact reconstruction with high probability by solving the feasibility problem of a semidefinite program

a boundary element approach to the 2d potential flow problem around airfoils with cusped trailing edge abstract the boundary integral method utilised in 1 for investigating the lifting flow past airfoils with non vanishing trailing edge angle is adjusted herein by imposing a different kutta joukovsky condition in order to study the incompressible inviscid flow past airfoils with cusped trailing edge using a linear interpolation for the normal derivative of the perturbation stream function we reduce by discretization the integral representation of the perturbation stream function to an algebraic system the solutions of this system are used in order to calculate the distribution of the tangential velocity and pressure coefficient over the airfoil comparison between calculated and analytical values of the pressure coefficient for a joukovsky and a von mises profile show very good agreement

statistical analysis of compressive low rank tomography with random measurements we consider the statistical problem of compressive estimation of low rank states r d with random basis measurements where r d are the rank and dimension of the state respectively we investigate whether for a fixed sample size n the estimation error associated with a compressive measurement setup is close to that of the setting where a large number of bases are measured we generalise and extend previous results and show that the mean square error mse associated with the frobenius norm attains the optimal rate rd n with only o rlogd random basis measurements for all states an important tool in the analysis is the concentration of the fisher information matrix fim we demonstrate that although a concentration of the mse follows from a concentration of the fim for most states the fim fails to concentrate for states with eigenvalues close to zero we analyse this phenomenon in the case of a single qubit and demonstrate a concentration of the mse about its optimal despite a lack of concentration of the fim for states close to the boundary of the bloch sphere we also consider the estimation error in terms of a different metric the quantum infidelity we show that a concentration in the mean infidelity minf does not exist uniformly over all states highlighting the importance of loss function choice specifically we show that for states that are nearly pure the minf scales as 1 n but the constant converges to zero as the number of settings is increased this demonstrates a lack of compressive recovery for nearly pure states in this metric

a boundary node method for airfoils based on the dirichlet condition abstract a nonsingular integral equation is developed for thin airfoils with the dirichlet boundary condition both singular and near singular kernels which accompany the original integral equation are flattened out after such regularizations the difficulty in the numerical quadrature is greatly reduced a program with an arbitrary order integral formulation becomes available in the present approach two kinds of airfoils the van de vooren and joukowski airfoils are analyzed for demonstration in this paper in comparison with the results from constant doublet methods the results from the proposed methods are shown to match the analytic solutions excellently

harmonic mean iteratively reweighted least squares for low rank matrix recovery we propose a new iteratively reweighted least squares irls algorithm for the recovery of a matrix x cd1 d2 of rank r min d1 d2 from incomplete linear observations solving a sequence of low complexity linear problems the easily implementable algorithm which we call harmonic mean iteratively reweighted least squares hm irls optimizes a non convex schatten p quasi norm penalization to promote low rankness and carries three major strengths in particular for the matrix completion setting first we observe a remarkable global convergence behavior of the algorithm s iterates to the low rank matrix for relevant interesting cases for which any other state of the art optimization approach fails the recovery secondly hm irls exhibits an empirical recovery probability close to 1 even for a number of measurements very close to the theoretical lower bound r d1 d2 r i e already for significantly fewer linear observations than any other tractable approach in the literature thirdly hm irls exhibits a locally superlinear rate of convergence of order 2 p if the linear observations fulfill a suitable null space property while for the first two properties we have so far only strong empirical evidence we prove the third property as our main theoretical result

sublinear time algorithms for compressive phase retrieval in the problem of compressed phase retrieval the goal is to reconstruct a sparse or approximately k sparse vector x in mathbb c n given access to y phi x where v denotes the vector obtained from taking the absolute value of v in mathbb c n coordinate wise in this paper we present sublinear time algorithms for a few for each variants of the compressive phase retrieval problem which are akin to the variants considered for the classical compressive sensing problem in theoretical computer science our algorithms use pure combinatorial techniques and near optimal number of measurements

performance of uniform and sparse non uniform samplers in presence of modeling errors a cramer rao bound based study this paper evaluates the performance of uniform and sparse nonuniform sampling techniques namely nested and coprime sampling for line spectrum estimation in presence of nonideal conditions such as perturbation in sampling instants and limited data for computing statistical averages coprime and nested sampling are well known deterministic sampling techniques that operate at rates significantly lower than nyquist and yet allow perfect reconstruction of the spectra of wide sense stationary signals however theoretical guarantees for these samplers assume ideal conditions such as synchronous sampling and ability to perfectly compute statistical expectations this paper studies the performance of coprime and nested samplers when these assumptions are violated using a general grid based signal model that applies to both spatial and temporal line spectrum estimation the effect of perturbations in sampling instants is evaluated by deriving fundamental cramer rao bounds crb for line spectrum estimation with perturbed samplers for the first time simplified expressions for the fisher information matrix for perturbed coprime and nested samplers are derived which explicitly highlight the role of coarray even in presence of perturbations it is possible to resolve o m 2 spectral lines under appropriate conditions on the size of the grid the effect of finite data on the crb is also studied and necessary and sufficient conditions are derived to ensure that the crb decreases monotonically to zero with the number of measurements even when there are more sources than sensors finally the theoretical results derived in this paper are supported by extensive numerical experiments

sparse phase retrieval uniqueness guarantees and recovery algorithms the problem of signal recovery from its fourier transform magnitude is of paramount importance in various fields of engineering and has been around for more than 100 years due to the absence of phase information some form of additional information is required in order to be able to uniquely identify the signal of interest in this paper we focus our attention on discrete time sparse signals of length n we first show that if the discrete fourier transform dimension is greater than or equal to 2n then almost all signals with aperiodic support can be uniquely identified by their fourier transform magnitude up to time shift conjugate flip and global phase then we develop an efficient two stage sparse phase retrieval algorithm tspr which involves identifying the support i e the locations of the nonzero components of the signal using a combinatorial algorithm and identifying the signal values in the support using a convex algorithm we show that tspr can provably recover most o n 1 2 epsilon sparse signals up to a time shift conjugate flip and global phase we also show that for most o n 1 4 epsilon sparse signals the recovery is robust in the presence of measurement noise these recovery guarantees are asymptotic in nature numerical experiments complement our theoretical analysis and verify the effectiveness of tspr

low rank phase retrieval we develop two iterative algorithms for solving the low rank phase retrieval lrpr problem lrpr refers to recovering a low rank matrix boldsymbol x from magnitude only phaseless measurements of random linear projections of its columns both methods consist of a spectral initialization step followed by an iterative algorithm to maximize the observed data likelihood we obtain sample complexity bounds for our proposed initialization approach to provide a good approximation of the true boldsymbol x when the rank is low enough these bounds are significantly lower than what existing single vector phase retrieval algorithms need via extensive experiments we show that the same is also true for the proposed complete algorithms

compressive multiplexing of correlated signals we present a general architecture for the acquisition of ensembles of correlated signals the signals are multiplexed onto a single line by mixing each one against a different code and then adding them together and the resulting signal is sampled at a high rate we show that if the m signals each band limited to w 2 hz can be approximated by a superposition of r m underlying signals then the ensemble can be recovered by sampling at a rate within a logarithmic factor of rw as compared with the cumulative nyquist rate of mw this sampling theorem shows that the correlation structure of the signal ensemble can be exploited in the acquisition process even though it is unknown a priori the reconstruction of the ensemble is recast as a low rank matrix recovery problem from linear measurements the architectures we are considering impose a certain type of structure on the linear operators although our results depend on the mixing forms being random this imposed structure results in a very different type of random projection than those analyzed in the low rank recovery literature to date

gradient descent with random initialization fast global convergence for nonconvex phase retrieval this paper considers the problem of solving systems of quadratic equations namely recovering an object of interest varvec x natural in mathbb r n from m quadratic equations samples y i varvec a i top varvec x natural 2 1 le i le m this problem also dubbed as phase retrieval spans multiple domains including physical sciences and machine learning we investigate the efficacy of gradient descent or wirtinger flow designed for the nonconvex least squares problem we prove that under gaussian designs gradient descent when randomly initialized yields an epsilon accurate solution in o big log n log 1 epsilon big iterations given nearly minimal samples thus achieving near optimal computational and sample complexities at once this provides the first global convergence guarantee concerning vanilla gradient descent for phase retrieval without the need of i carefully designed initialization ii sample splitting or iii sophisticated saddle point escaping schemes all of these are achieved by exploiting the statistical models in analyzing optimization algorithms via a leave one out approach that enables the decoupling of certain statistical dependency between the gradient descent iterates and the data

phase retrieval using alternating minimization phase retrieval problems involve solving linear equations but with missing sign or phase for complex numbers information more than four decades after it was first proposed the seminal error reduction algorithm of gerchberg and saxton and fienup is still the popular choice for solving many variants of this problem the algorithm is based on alternating minimization i e it alternates between estimating the missing phase information and the candidate solution despite its wide usage in practice no global convergence guarantees for this algorithm are known in this paper we show that a resampling variant of this approach converges geometrically to the solution of one such problem finding a vector bf x from bf y bf a where bf y vert bf a t bf x vert and vert bf z vert denotes a vector of element wise magnitudes of bf z under the assumption that bf a is gaussian empirically we demonstrate that alternating minimization performs similar to recently proposed convex techniques for this problem which are based on lifting to a convex matrix problem in sample complexity and robustness to noise however it is much more efficient and can scale to large problems analytically for a resampling version of alternating minimization we show geometric convergence to the solution and sample complexity that is off by log factors from obvious lower bounds we also establish close to optimal scaling for the case when the unknown vector is sparse our work represents the first theoretical guarantee for alternating minimization albeit with resampling for any variant of phase retrieval problems in the non convex setting

solving phaselift by low rank riemannian optimization methods for complex semidefinite constraints a framework phaselift was recently proposed to solve the phase retrieval problem in this framework the problem is solved by optimizing a cost function over the set of complex hermitian positive semidefinite matrices this approach to phase retrieval motivates a more general consideration of optimizing cost functions on semidefinite hermitian matrices where the desired minimizers are known to have low rank this paper considers an approach based on an alternative cost function defined on a union of appropriate manifolds it is related to the original cost function in a manner that preserves the ability to find a global minimizer and is significantly more efficient computationally a rank based optimality condition for stationary points is given and optimization algorithms based on state of the art riemannian optimization and dynamically reducing rank are proposed empirical evaluations are performed using the phaselift problem the new approach is shown to be an effective method of phase retrieval with

excitation retrieval of microwave linear arrays from phaseless far field data a methodology to recover the excitations of microwave linear arrays from the measurements of far field magnitude only is proposed the approach combines tools from convex optimization to solve the phase retrieval problem and a simple measurement procedure to mitigate the non uniqueness of the solution numerical simulations in various representative and realistic configurations of noise and measurement sampling are presented and discussed they show that it is possible to perfectly retrieve the array excitations from only the knowledge of far field magnitude by simply solving a convex optimization problem in addition the proposed approach that only calls for readily available routines is stable with respect to noise since the reconstruction performances degrades gracefully as the signal to noise ratio decreases

hypersingular boundary integral equations for exterior acoustic problems a bounded form of the hypersingular helmholtz integral equation for 3 d acoustic problems is developed in this paper all integrals with singular kernels are regularized within each regular surface element in the sense of kellogg with twice continuous differentiability the standard gaussian quadrature formula is applied without any special treatment for all the computational points the collocation points are chosen to be the gauss legendre nodes and no interpolation function is assumed for acoustic variables

exact semidefinite formulations for a class of random and non random nonconvex quadratic programs we study a class of quadratically constrained quadratic programs qcqps called diagonal qcqps which contain no off diagonal terms x j x k for j ne k and we provide a sufficient condition on the problem data guaranteeing that the basic shor semidefinite relaxation is exact our condition complements and refines those already present in the literature and can be checked in polynomial time we then extend our analysis from diagonal qcqps to general qcqps i e ones with no particular structure by reformulating a general qcqp into diagonal form we establish new polynomial time checkable sufficient conditions for the semidefinite relaxations of general qcqps to be exact finally these ideas are extended to show that a class of random general qcqps has exact semidefinite relaxations with high probability as long as the number of constraints grows no faster than a fixed polynomial in the number of variables to the best of our knowledge this is the first result establishing the exactness of the semidefinite relaxation for random general qcqps

undersampled sparse phase retrieval via majorization minimization in the undersampled phase retrieval problem the goal is to recover an n dimensional complex valued signal from only m intensity measurements without phase information this inverse system is not only nonconvex but also underdetermined in this paper we propose to exploit the sparsity in the original signal and develop two low complexity algorithms with superior performance based on the majorization minimization framework the proposed algorithms are preferred to existing benchmark methods since at each iteration a simple convex surrogate problem is solved with a closed form solution that monotonically decreases the objective function value when the unknown signal is sparse in the standard basis the first algorithm c prime can produce a stationary point of the corresponding nonconvex phase retrieval problem when the unknown signal is not sparse in the standard basis the second algorithm sc prime can find a coordinatewise stationary point of the more challenging phase retrieval problem through sparse coding experimental results validate that the proposed algorithms have higher successful recovery rate and less normalized mean square error than existing up to date methods under the same setting

reconstruction of signals from their autocorrelation and cross correlation vectors with applications to phase retrieval and blind channel estimation we consider the problem of reconstructing two signals from the autocorrelation and cross correlation measurements this inverse problem is a fundamental one in signal processing and arises in many applications including phase retrieval and blind channel estimation in a typical phase retrieval setup only the autocorrelation measurements are obtainable we show that when the measurements are obtained using three simple masks phase retrieval reduces to the aforementioned reconstruction problem the classic solution to this problem is based on finding common factors between the z transforms of the autocorrelation and cross correlation vectors this solution has enjoyed limited practical success mainly due to the fact that it is not sufficiently stable in the noisy setting in this paper inspired by the success of convex programming in provably and stably solving various quadratic constrained problems we develop a semidefinite programming based algorithm and provide theoretical guarantees in particular we show that almost all signals can be uniquely recovered by this algorithm up to a global phase comparative numerical studies demonstrate that the proposed method significantly outperforms the classic method in the noisy setting

statistical guarantees for the em algorithm from population to sample based analysis we develop a general framework for proving rigorous guarantees on the performance of the em algorithm and a variant known as gradient em our analysis is divided into two parts a treatment of these algorithms at the population level in the limit of infinite data followed by results that apply to updates based on a finite set of samples first we characterize the domain of attraction of any global maximizer of the population likelihood this characterization is based on a novel view of the em updates as a perturbed form of likelihood ascent or in parallel of the gradient em updates as a perturbed form of standard gradient ascent leveraging this characterization we then provide non asymptotic guarantees on the em and gradient em algorithms when applied to a finite set of samples we develop consequences of our general theory for three canonical examples of incompletedata problems mixture of gaussians mixture of regressions and linear regression with covariates missing completely at random in each case our theory guarantees that with a suitable initialization a relatively small number of em or gradient em steps will yield with high probability an estimate that is within statistical error of the mle we provide simulations to confirm this theoretically predicted behavior

identifiability and stability in blind deconvolution under minimal assumptions blind deconvolution bd arises in many applications without assumptions on the signal and the filter bd does not admit a unique solution in practice subspace or sparsity assumptions have shown the ability to reduce the search space and yield the unique solution however existing theoretical analysis on uniqueness in bd is rather limited in an earlier paper we provided the first algebraic sample complexities for bd that hold for lebesgue almost all bases or frames we showed that for bd of a pair of vectors in mathbb c n with subspace constraints of dimensions m 1 and m 2 respectively a sample complexity of n geq m 1 m 2 is sufficient this result is suboptimal since the number of degrees of freedom is merely m 1 m 2 1 we provided analogous results with similar suboptimality for bd with sparsity or mixed subspace and sparsity constraints in this paper taking advantage of the recent progress on the information theoretic limits of unique low rank matrix recovery we finally bridge this gap and derive an optimal sample complexity result for bd with generic bases or frames we show that for bd of an arbitrary pair respectively all pairs of vectors in mathbb c n with sparsity constraints of sparsity levels s 1 and s 2 a sample complexity of n s 1 s 2 respectively n 2 s 1 s 2 is sufficient we also present analogous results for bd with subspace constraints or mixed constraints with the subspace dimension replacing the sparsity level last but not least in all the above scenarios if the bases or frames follow a probabilistic distribution specified in this paper the recovery is not only unique but also stable against small perturbations in the measurements under the same sample complexities

explicit kutta condition for unsteady two dimensional high order potential boundary element method an explicit unsteady pressure kutta condition is discribed that was directly and efficiently implemented in a time domain high order potential panel method so as to ensure the pressure equality on the upper and lower surfaces at the trailing edge of the airfoil at each time step

existence of solutions for hamiltonian field theories by the hamilton jacobi technique the paper is devoted to prove the existence of a local solution of the hamilton jacobi equation in field theory whence the general solution of the field equations can be obtained the solution is adapted to the choice of the submanifold where the initial data of the field equations are assigned finally a technique to obtain the general solution of the field equations starting from the given initial manifold is deduced

effect of airfoil trailing edge thickness on the numerical solution of panel methods based on the dirichlet boundary condition the practical limit of airfoil thickness ratio for which acceptable engineering results are obtainable with the dirichlet boundary condition based numerical methods is investigated this is done by studying the effect of thickness on the calculated pressure distribution near the trailing edge and by comparing the aerodynamic coefficients with available exact solutions the first objective of this study owing to the wide use of such computational methods is to demonstrate the numerical symptoms that occur when the body or wing thickness approaches zero and to increase the awareness of potential users of these methods additionally an effort is made to obtain the practical limits of the trailing edge thickness where such problems will appear in the flow solution and to propose some possible cures for very thin airfoils or those with cusped trailing edges

a regularized boundary integral method in potential theory abstract a global method for solving the boundary integral equation in potential theory is presented with subtracting and adding back treatment several auxiliary functions are employed to reduce the singularities of boundary integrals after that all the integrals can be directly integrated by arbitrary numerical quadrature without any geometric approximation the advantage of this method is that high accuracy of numerical quadrature can be maintained with minimal computational nodes in boundary integral since collocation points can be arranged exactly as quadrature points the proposed auxiliary functions can also smooth the peaks of nearly singular integrals if they exist in such a case little extra work is needed to separate singular and nearly singular integrals into different elements after separation both kinds of integrals can be partially transformed into line integrals along their boundaries for the three dimensional case or several simple formulas for the two dimensional case

phase recovery maxcut and complex semidefinite programming phase retrieval seeks to recover a signal x in mathbb c p x c p from the amplitude a x a x of linear measurements ax in mathbb c n a x c n we cast the phase retrieval problem as a non convex quadratic program over a complex phase vector and formulate a tractable relaxation called phasecut similar to the classical maxcut semidefinite program we solve this problem using a provably convergent block coordinate descent algorithm whose structure is similar to that of the original greedy algorithm in gerchberg and saxton optik 35 237 246 1972 where each iteration is a matrix vector product numerical results show the performance of this approach over three different phase retrieval problems in comparison with greedy phase retrieval algorithms and matrix completion formulations

confidence region of singular subspaces for low rank matrix regression low rank matrix regression refers to the instances of recovering a low rank matrix based on specially designed measurements and the corresponding noisy outcomes numerous statistical methods have been developed over the recent decade for efficiently reconstructing the unknown low rank matrices it is often interesting in certain applications to estimate the unknown singular subspaces in this paper we revisit the low rank matrix regression model and introduce a two step procedure to construct confidence regions of the singular subspaces we investigate distributions of the joint projection distance between the empirical singular subspaces and the unknown true singular subspaces we prove asymptotical normality of the joint projection distance with data dependent centering and normalization when r 3 2 m 1 m 2 3 2 o n log n where m 1 m 2 denote the matrix row and column sizes r is the rank and n is the number of independent random measurements consequently data dependent confidence regions of the true singular subspaces are established which attain pre determined confidence levels asymptotically additionally non asymptotic convergence rates are also established numerical results are presented to show the merits of our methods

convex and nonconvex formulations for mixed regression with two components minimax optimal rates we consider the mixed regression problem with two components under adversarial and stochastic noise we give a convex optimization formulation that provably recovers the true solution as well as a nonconvex formulation that works under more general settings and remains tractable upper bounds are provided on the recovery errors for both arbitrary noise and stochastic noise models we also give matching minimax lower bounds up to log factors showing that our algorithm is information theoretical optimal in a precise sense our results represent the first tractable algorithm guaranteeing successful recovery with tight bounds on recovery errors and sample complexity moreover we pinpoint the statistical cost of mixtures our minimax optimal results indicate that the mixture poses a fundamentally more difficult problem in the low snr regime where the learning rate changes

low rank positive semidefinite matrix recovery from corrupted rank one measurements we study the problem of estimating a low rank positive semidefinite psd matrix from a set of rank one measurements using sensing vectors composed of i i d standard gaussian entries which are possibly corrupted by arbitrary outliers this problem arises from applications such as phase retrieval covariance sketching quantum space tomography and power spectrum estimation we first propose a convex optimization algorithm that seeks the psd matrix with the minimum ell 1 norm of the observation residual the advantage of our algorithm is that it is free of parameters therefore eliminating the need for tuning parameters and allowing easy implementations we establish that with high probability a low rank psd matrix can be exactly recovered as soon as the number of measurements is large enough even when a fraction of the measurements are corrupted by outliers with arbitrary magnitudes moreover the recovery is also stable against bounded noise with the additional information of an upper bound of the rank of the psd matrix we propose another nonconvex algorithm based on subgradient descent that demonstrates excellent empirical performance in terms of computational efficiency and accuracy

iterative hard thresholding for low rank recovery from rank one projections abstract a novel algorithm for the recovery of low rank matrices acquired via compressive linear measurements is proposed and analyzed the algorithm a variation on the iterative hard thresholding algorithm for low rank recovery is designed to succeed in situations where the standard rank restricted isometry property fails e g in case of subexponential unstructured measurements or of subgaussian rank one measurements the stability and robustness of the algorithm are established based on distinctive matrix analytic ingredients and its performance is substantiated numerically

non singular boundary integral formulations for plane interior potential problems in this article a non singular formulation of the boundary integral equation is developed to solve smooth and non smooth interior potential problems in two dimensions the subtracting and adding back technique is used toregularize the singularity of green s function and to simplify the calculation of the normal derivative of green s function after that a global numerical integration is directly applied at the boundary and those integration points are also taken as collocation points to simplify the algorithm of computation the result indicates that this simple method gives the convergence speed of order n 3 in the smooth boundary cases for both dirichlet and mix type problems for the non smooth cases the convergence speed drops at o n 1 2 for the dirichlet problems

phaselift exact and stable signal recovery from magnitude measurements via convex programming suppose we wish to recover a signal input amssym font abc cmmib10 def bi 1 hbox abc 1 bi x in bbb c n from m intensity measurements of the form that is from data in which phase information is missing we prove that if the vectors are sampled independently and uniformly at random on the unit sphere then the signal x can be recovered exactly up to a global phase factor by solving a convenient semidefinite program a trace norm minimization problem this holds with large probability provided that m is on the order of and without any assumption about the signal whatsoever this novel result demonstrates that in some instances the combinatorial phase retrieval problem can be solved by convex programming techniques finally we also prove that our methodology is robust vis a vis additive noise 2012 wiley periodicals inc

a boundary integral method for acoustic radiation and scattering in this article a computational method is proposed for computing the helmholtz integral equation for acoustic radiation and scattering problems with a three dimensional body shape unlike the previous boundary element method the integration of acoustic properties is based on the global body surface instead of local elements the approximation function of body geometry and the weighting function of numerical integration are independently chosen when the singular kernels of helmholtz equation are regularized as bounded discontinuous functions the order of integration polynomials for acoustic properties can be arbitrarily chosen therefore the accuracy and efficiency of computation can be increased