average consensus on general strongly connected digraphs we study the average consensus problem of multi agent systems for general network topologies with unidirectional information flow we propose two linear distributed algorithms deterministic and gossip respectively for the cases where the inter agent communication is synchronous and asynchronous in both cases the developed algorithms guarantee state averaging on arbitrary strongly connected digraphs in particular this graphical condition does not require that the network be balanced or symmetric thereby extending previous results in the literature the key novelty of our approach is to augment an additional variable for each agent called surplus whose function is to locally record individual state updates for convergence analysis we employ graph theoretic and nonnegative matrix tools plus the eigenvalue perturbation theory playing a crucial role

distributed subgradient methods for convex optimization over random networks we consider the problem of cooperatively minimizing the sum of convex functions where the functions represent local objective functions of the agents we assume that each agent has information about his local function and communicate with the other agents over a time varying network topology for this problem we propose a distributed subgradient method that uses averaging algorithms for locally sharing information among the agents in contrast to previous works on multi agent optimization that make worst case assumptions about the connectivity of the agents such as bounded communication intervals between nodes we assume that links fail according to a given stochastic process under the assumption that the link failures are independent and identically distributed over time possibly correlated across links we provide almost sure convergence results for our subgradient algorithm

exponential convergence of a distributed algorithm for solving linear algebraic equations abstract in a recent paper a distributed algorithm was proposed for solving linear algebraic equations of the form a x b assuming that the equation has at least one solution the equation is presumed to be solved by m agents assuming that each agent knows a subset of the rows of the matrix a b the current estimates of the equation s solution generated by each of its neighbors and nothing more neighbor relationships are represented by a time dependent directed graph n t whose vertices correspond to agents and whose arcs characterize neighbor relationships sufficient conditions on n t were derived under which the algorithm can cause all agents estimates to converge exponentially fast to the same solution to a x b these conditions were also shown to be necessary for exponential convergence provided the data about a b available to the agents is non redundant the aim of this paper is to relax this non redundant assumption this is accomplished by establishing exponential convergence under conditions which are the weakest possible for the problem at hand the conditions are based on a new notion of graph connectivity an improved bound on the convergence rate is also derived

distributed multi agent optimization with state dependent communication we study distributed algorithms for solving global optimization problems in which the objective function is the sum of local objective functions of agents and the constraint set is given by the intersection of local constraint sets of agents we assume that each agent knows only his own local objective function and constraint set and exchanges information with the other agents over a randomly varying network topology to update his information state we assume a state dependent communication model over this topology communication is markovian with respect to the states of the agents and the probability with which the links are available depends on the states of the agents we study a projected multi agent subgradient algorithm under state dependent communication the state dependence of the communication introduces significant challenges and couples the study of information exchange with the analysis of subgradient steps and projection errors we first show that the multi agent subgradient algorithm when used with a constant stepsize may result in the agent estimates to diverge with probability one under some assumptions on the stepsize sequence we provide convergence rate bounds on a disagreement metric between the agent estimates our bounds are time nonhomogeneous in the sense that they depend on the initial starting time despite this we show that agent estimates reach an almost sure consensus and converge to the same optimal solution of the global optimization problem with probability one under different assumptions on the local constraint sets and the stepsize sequence

approximate linearized inversion by optimal scaling of prestack depth migration linearized inversion provides one possible sense of image amplitude correctness an image or bandlimited model perturbation has correct amplitudes if it is an approximate inversion that is if linearized modeling demigration with the image as input reproduces the data approximately the theory of linearized acoustic inverse scattering with slowly varying background or macromodel shows that an approximate inversion may be recovered from the output of prestack depth migration by a combination of scaling and filtering the necessary filter is completely specified by the theory and the scale factor may be estimated via filtering linearized modeling a second migration and the solution of a small auxiliary inverse problem

interpolation with fourier radial adaptive thresholding many interpolation methods are effective with regularly sampled or randomly sampled data whereas the spatial sampling of seismic reflectivity data is typically neither regular nor random fourier radial adaptive thresholding frat is a sparsity promoting method in which the interpolated result is sparse in the frequency wavenumber domain and is coherent in a manner consistent with that of a collection of unaliased plane waves the sparsity and the desired pattern in the f k domain are promoted by iterative soft thresholding and adaptive weighting data in the f k domain are transformed to polar coordinates and then low pass filtered along the radial axis to generate the nonlinear weight frat interpolates data that are randomly sampled and aliased i e where the minimum distance between adjacent traces is greater than the nyquist sampling interval a conventional approach to solving this problem is to apply a cascade of two procedures first a sparsity based method such as projection onto convex sets

amplitude and kinematic corrections of migrated images for nonunitary imaging operators obtaining migrated images with meaningful amplitudes is a challenging problem when the migration operator is not unitary one possible solution to this problem is iterative inversion however inversion is an expensive process that can be rather difficult to apply especially with 3d data in this paper i propose estimating migrated images similar to the least squares inverse images by approximating the inverse hessian thus avoiding the need for iterative inversion the inverse hessian is approximated with a bank of nonstationary matching filters these filters are not exact impulse responses and are limited in their ability to mimic the full effects of least squares inversion tests on two data sets show that this filtering approach gives results similar to iterative least squares inversion at a lower cost this technique is flexible enough to be applied to images migrated from zero offset or angle domain common image point gathers

incorporating geologic information into reflection tomography in areas of complex geology prestack depth migration is often necessary if we are to produce an accurate image of the subsurface prestack depth migration requires an accurate interval velocity model with few exceptions the subsurface velocities are not known beforehand and should be estimated when the velocity structure is complex with significant lateral variations reflection tomography methods are often an effective tool for improving the velocity estimate unfortunately reflection tomography often converges slowly to a model that is geologically unreasonable or it does not converge at all the large null space of reflection tomography problems often forces us to add a sparse parameterization of the model and or regularization criteria to the estimation standard tomography schemes tend to create isotropic features in velocity models that are inconsistent with geology these isotropic features result in large part from using symmetric regularization operators or from choosing a poor model parameterization if we replace the symmetric operators with nonstationary operators that tend to spread information along structural dips the tomography will produce velocity models that are geologically more reasonable in addition by forming the operators in helical 1d space and performing polynomial division we apply the inverse of these space varying anisotropic operators the inverse operators can be used as a preconditioner to a standard tomography problem thereby significantly improving the speed of convergence compared with the typical regularized inversion problem results from 2d synthetic and 2d field data are shown in each case the velocity obtained improves the focusing of the migrated image

decentralized optimization over time varying directed graphs with row and column stochastic matrices in this article we provide a distributed optimization algorithm termed as tv mathcal ab that minimizes a sum of convex functions over time varying random directed graphs contrary to the existing work the algorithm we propose does not require eigenvector estimation to estimate the non mathbf 1 perron eigenvector of a stochastic matrix instead the proposed approach relies on a novel information mixing approach that exploits both row and column stochastic weights to achieve agreement toward the optimal solution when the underlying graph is directed we show that tv mathcal ab converges linearly to the optimal solution when the global objective is smooth and strongly convex and the underlying time varying graphs exhibit bounded connectivity i e a union of every c consecutive graphs is strongly connected we derive the convergence results based on the stability analysis of a linear system of inequalities along with a matrix perturbation argument simulations confirm the findings in this article

improving explicit seismic depth migration with a stabilizing wiener filter and spatial resampling we present a new approach to the design and implementation of explicit wavefield extrapolation for seismic depth migration in the space frequency domain instability of the wavefield extrapolation operator is addressed by splitting the operator into two parts one to control phase accuracy and a second to improve stability the first partial operator is simply a windowed version of the exact operator for a half step the second partial operator is designed using the wiener filter method as a band limited least squares inverse of the first the final wavefield extrapolation operator for a full step is formed as a convolution of the first partial operator with the complex conjugate of the second this resulting wavefield extrapolation operator can be designed to have any desired length and is generally more stable and more accurate than a simple windowed operator of similar length additional stability is gained by reducing the amount of evanescent filtering and by spatially downsampling the lower tempor

a fast proximal gradient algorithm for decentralized composite optimization over directed networks abstract this paper proposes a fast decentralized algorithm for solving a consensus optimization problem defined in a directed networked multi agent system where the local objective functions have the smooth nonsmooth composite form and are possibly nonconvex examples of such problems include decentralized compressed sensing and constrained quadratic programming problems as well as many decentralized regularization problems we extend the existing algorithms pg extra and extrapush to a new algorithm pg extrapush for composite consensus optimization over a directed network this algorithm takes advantage of the proximity operator like in pg extra to deal with the nonsmooth term and employs the push sum protocol like in extrapush to tackle the bias introduced by the directed network with a proper step size we show that pg extrapush converges to an optimal solution at a linear rate under some regular assumptions we conduct a series of numerical experiments to show the effectiveness of the proposed algorithm specifically with a proper step size pg extrapush performs linear rates in most of cases even in some nonconvex cases and is significantly faster than subgradient push even if the latter uses a hand optimized step size the established theoretical results are also verified by the numerical results

stochastic proximal gradient consensus over random networks we consider solving a convex optimization problem with possibly stochastic gradient and over a randomly time varying multiagent network each agent has access to some local objective function and it only has unbiased estimates of the gradients of the smooth component we develop a dynamic stochastic proximal gradient consensus algorithm with the following key features 1 it works for both the static and certain randomly time varying networks 2 it allows the agents to utilize either the exact or stochastic gradient information 3 it is convergent with provable rate in particular the proposed algorithm converges to a global optimal solution with a rate of mathcal o 1 r resp mathcal o 1 sqrt r when the exact resp stochastic gradient is available where r is the iteration counter interestingly the developed algorithm establishes a close connection among a number of seemingly unrelated distributed algorithms such as the extra the pg extra the ic idc admm the dlm and the classical distributed subgradient method

decentralized quasi newton methods we introduce the decentralized broyden fletcher goldfarb shanno d bfgs method as a variation of the bfgs quasi newton method for solving decentralized optimization problems decentralized quasi newton methods are of interest in problems that are not well conditioned making first order decentralized methods ineffective and in which second order information is not readily available making second order decentralized methods impossible d bfgs is a fully distributed algorithm in which nodes approximate curvature information of themselves and their neighbors through the satisfaction of a secant condition we additionally provide a formulation of the algorithm in asynchronous settings convergence of d bfgs is established formally in both the synchronous and asynchronous settings and strong performance advantages relative to existing methods are shown numerically

convergence rates of distributed nesterov like gradient methods on random networks we consider distributed optimization in random networks where n nodes cooperatively minimize the sum i 1n fi x of their individual convex costs existing literature proposes distributed gradient like methods that are computationally cheap and resilient to link failures but have slow convergence rates in this paper we propose accelerated distributed gradient methods that 1 are resilient to link failures 2 computationally cheap and 3 improve convergence rates over other gradient methods we model the network by a sequence of independent identically distributed random matrices w k drawn from the set of symmetric stochastic matrices with positive diagonals the network is connected on average and the cost functions are convex differentiable with lipschitz continuous and bounded gradients we design two distributed nesterov like gradient methods that modify the d ng and d nc methods that we proposed for static networks we prove their convergence rates in terms of the expected optimality gap at the cost function let k and k be the number of per node gradient evaluations and per node communications respectively then the modified d ng achieves rates o logk k and o logk k and the modified d nc rates o 1 k2 and o 1 k2 where 0 is arbitrarily small for comparison the standard distributed gradient method cannot do better than 1 k2 3 and 1 k2 3 on the same class of cost functions even for static networks simulation examples illustrate our analytical findings

regularizing seismic inverse problems by model reparameterization using plane wave construction we define plane wave construction pwc an operator for generating data aligned along predefined locally variable slopes as the inverse of plane wave destruction an operator used for measuring the slopes pwc can be applied for efficient regularization of seismic estimation problems using simple examples we demonstrate how pwc enhances the coherency of seismic images improves velocity estimation methods and separates primaries and multiples with a pattern based approach

dextra a fast algorithm for optimization over directed graphs this paper develops a fast distributed algorithm termed dextra to solve the optimization problem when n agents reach agreement and collaboratively minimize the sum of their local objective functions over the network where the communication between the agents is described by a directed graph existing algorithms solve the problem restricted to directed graphs with convergence rates of o ln k sqrt k for general convex objective functions and o ln k k when the objective functions are strongly convex where k is the number of iterations we show that with the appropriate step size dextra converges at a linear rate o tau k for 0 given that the objective functions are restricted strongly convex the implementation of dextra requires each agent to know its local out degree simulation examples further illustrate our findings

adams operations and integral hermitian galois representations we give an algebraic interpretation of the action of the j th adams operator j on hermitian galois structure classes for tame covers of schemes of dimension 1 this generalizes work of erez and of erez taylor concerning rings of algebraic integers when j 2

multidimensional recursive filters via a helix wind a wire onto a cylinder to create a helix i show that a filter on the 1 d space of the wire mimics a 2 d filter on the cylindrical surface thus 2 d convolution can be done with a 1 d convolution program i show some examples of 2 d recursive filtering also called 2 d deconvolution or 2 d polynomial division in 2 d as in 1 d the computational advantage of recursive filters is the speed with which they propagate information over long distances we can estimate 2 d prediction error filters pefs that are assured of being stable for 2 d recursion such 2 d and 3 d recursions are general purpose preconditioners that vastly speed the solution of a wide class of geophysical estimation problems the helix transformation also enables use of the partial differential equation of wave extrapolation as though it were an ordinary differential equation

target oriented wave equation inversion a target oriented strategy can be applied to estimate a wave equation least squares inverse lsi image by explicitly computing the wave equation hessian the lsi image is obtained as the solution of a nonstationary least squares inverse filtering problem the rows of the hessian are the nonstationary filters containing information about the acquisition geometry the velocity model and the band limited characteristics of the seismic data by exploiting the sparsity and the structure of the hessian matrix a large number of iterations necessary to achieve convergence can be computed cheaply the results on a structurally complex model show the improvements of the lsi image versus the migrated image

on the convergence of decentralized gradient descent consider the consensus problem of minimizing f x sum i 1 n f i x where x in mathbb r p and each f i is only known to the individual agent i in a connected network of n agents to solve this problem and obtain the solution all the agents collaborate with their neighbors through information exchange this type of decentralized computation does not need a fusion center offers better network load balance and improves data privacy this paper studies the decentralized gradient descent method a nedic and a ozdaglar ieee trans automat control 54 2009 pp 48 61 in which each agent i updates its local variable x i in mathbb r n by combining the average of its neighbors with a local negative gradient step alpha nabla f i x i the method is described by the iteration x i k 1 gets sum j 1 n w ij x j k alpha nabla f i x i k text for each agent i where w ij is nonzero only if i and j are neighbors or i j and the matrix

convex programming design of linear and nonlinear phase wideband blocking filters blocking filters are commonly used in array processing to excise targets from data when suitable target free training data is not available a drawback is the reduction of the effective size of the array by the order of the filter motivating a search for the smallest filter that meets the design specifications common linear phase approaches lead to convex optimization and efficient global solutions but can be inefficient when the phase of the blocking filter is unimportant although direct magnitude response optimization is nonconvex in general for narrowband linear arrays optimal magnitude blocking filters can be found using constrained optimization followed by spectral factorization this approach cannot be directly applied to wideband arrays as finite support spectral factors do not generally exist in higher dimensions instead a procedure is presented in which an approximate multidimensional spectral factor of an optimized spectral density is used as a target response in a second optimization stage linear and nonlinear phase solutions are found and compared for both narrowband and wideband notch filters with the nonlinear phase outperforming the linear phase in both notch width and passband error the various optimizations are performed using second order cone programming an efficient class of convex optimization

achieving geometric convergence for distributed optimization over time varying graphs this paper considers the problem of distributed optimization over time varying graphs for the case of undirected graphs we introduce a distributed algorithm referred to as diging based on a combination of a distributed inexact gradient method and a gradient tracking technique the diging algorithm uses doubly stochastic mixing matrices and employs fixed step sizes and yet drives all the agents iterates to a global and consensual minimizer when the graphs are directed in which case the implementation of doubly stochastic mixing matrices is unrealistic we construct an algorithm that incorporates the push sum protocol into the diging structure thus obtaining the push diging algorithm push diging uses column stochastic matrices and fixed step sizes but it still converges to a global and consensual minimizer under the strong convexity assumption we prove that the algorithms converge at r linear geometric rates as long as the step sizes do not exceed some upper bounds we establish explicit est

extra an exact first order algorithm for decentralized consensus optimization recently there has been growing interest in solving consensus optimization problems in a multiagent network in this paper we develop a decentralized algorithm for the consensus optimization problem mathrm minimize x in mathbb r p bar f x frac 1 n sum i 1 n f i x which is defined over a connected network of n agents where each function f i is held privately by agent i and encodes the agent s data and objective all the agents shall collaboratively find the minimizer while each agent can only communicate with its neighbors such a computation scheme avoids a data fusion center or long distance communication and offers better load balance to the network this paper proposes a novel decentralized exact first order algorithm abbreviated as extra to solve the consensus optimization problem exact means that it can converge to the exact solution extra uses a fixed large step size which can be determined independently of the network size or topology the local variable of every a

hermitian modules in galois extensions of number fields and adams operations 0 introduction and statement of results 1 some definitions and notation 2 the module m n k 2 1 the group algebra fa 2 2 determinants 2 3 resolvends and resolvents 2 4 adams operations 2 5 valuation results the integers v 8 3 semilocal isometries 3 1 local tamely ramified galois extensions 3 2 local isometries 4 double coset description of li and minus determinants 4 1 minus determinants the invariant d 5 strong approximation 6 proof of the main theorem 7 the abelian case appendix references

distributed heavy ball a generalization and acceleration of first order methods with gradient tracking we study distributed optimization to minimize a sum of smooth and strongly convex functions recent work on this problem uses gradient tracking to achieve linear convergence to the exact global minimizer however a connection among different approaches has been unclear in this paper we first show that many of the existing first order algorithms are related with a simple state transformation at the heart of which lies a recently introduced algorithm known as mathcal ab we then present distributed heavy ball denoted as mathcal ab m that combines mathcal ab with a momentum term and uses nonidentical local step sizes by simultaneously implementing both row and column stochastic weights mathcal ab m removes the conservatism in the related work due to doubly stochastic weights or eigenvector estimation mathcal ab m thus naturally leads to optimization and average consensus over both undirected and directed graphs we show that mathcal ab m has a global r linear rate when the largest step size and momentum parameter are positive and sufficiently small we numerically show that mathcal ab m achieves acceleration particularly when the objective functions are ill conditioned

on frohlich s conjecture for rings of integers of tame extensions where is the complex conjugate of z the constant w z is called the artin root number of in m l j martinet showed that om is not always a free zf module when n q and when f is a quaternion group of order 8 subsequently in f5 a frtihlich showed that the class of u in this special case is determined by the sign of the artin root number of the irreducible sympletic character of 2 we now return to the general case and let ci tif be the class group of locally free zf modules given a locally free 2 f module x we denote its class by x philippe cassou nogu6s defined a class t w in ci f which is defined solely in terms of the values of artin root numbers of sympletic characters this class will be defined in w after numerous computations by fr6hlich et al cf f6 c t3 for instance fr6hlich conjectured the result given below which we prove in this paper

convergence of asynchronous distributed gradient methods over stochastic networks we consider distributed optimization problems in which a number of agents are to seek the global optimum of a sum of cost functions through only local information sharing in this paper we are particularly interested in scenarios where agents are operating asynchronously over stochastic networks subject to random failures most existing algorithms require coordinated and decaying stepsizes to ensure zero gap between the estimated value of each agent and the exact optimum restricting it from asynchronous implementation and resulting in slower convergence results to deal with this issue we develop a new asynchronous distributed gradient method asyndgm based on consensus theory the proposed algorithm not only allows for asynchronous implementation in a completely distributed manner but also most importantly is able to seek the exact optimum even with constant stepsizes we will show that the assumption of boundedness of gradients which is widely used in the literature can be dropped by instead imposing the standard lipschitz continuity condition on gradients moreover we derive an upper bound of stepsize within which the proposed asyndgm can achieve a linear convergence rate for strongly convex functions with lipschitz gradients a canonical example of sensor fusion problems is provided to illustrate the effectiveness of the proposed algorithm