simultaneous multifrequency inversion of full waveform seismic data we present a simultaneous multifrequency inversion approach for seismic data interpretation this algorithm inverts all frequency data components simultaneously a data weighting scheme balances the contributions from different frequency data components so the inversion process does not become dominated by high frequency data components which produce a velocity image with many artifacts a gauss newton minimization approach achieves a high convergence rate and an accurate reconstructed velocity image by introducing a modified adjoint formulation we can calculate the jacobian matrix efficiently allowing the material properties in the perfectly matched layers pmls to be updated automatically during the inversion process this feature ensures the correct behavior of the inversion and implies that the algorithm is appropriate for realistic applications where a priori information of the background medium is unavailable two different regularization schemes an l2 norm and a weighted l2 norm function a

error bounds for the methods of glimm godunov and leveque the expected error in l 1 r at time t for glimm s scheme when applied to a scalar conservation law is bounded by left h frac 2 sqrt 3 left frac h delta t right 1 2 right left u 0 right bv bf r where h is the mesh size and delta t is the time step we show that the error in godunov s scheme is bounded by the same expression and use this result to analyze a variant of a large time step numerical method of leveque that may be viewed as a combination of godunov s scheme and a method of dafermos

kraft inequality and zero error source coding with decoder side information for certain source coding problems it is well known that the kraft inequality provides a simple sufficient condition for the existence of codes with given codeword lengths motivated by this fact a sufficient condition based on the kraft inequality can also be sought for the problem of zero error instantaneous coding with decoder side information more specifically it can be envisioned that a sufficient condition for the existence of such codes with codeword lengths l x is that for some 0 alphales 1 sigmaxepsivfscr 2 l x lesalpha for each clique fscr in the characteristic graph g of the source side information pair in this correspondence it is shown that 1 if the above is indeed a sufficient condition for a class g of graphs then it is possible to come as close as 1 log2 alpha bits to the asymptotic limits for each graph in g 2 there exist graph classes of interest for which such a can indeed be found and finally 3 no such n can be found for the class of all graphs

source coding and graph entropies a sender wants to accurately convey information to a receiver who has some possibly related data we study the expected number of bits the sender must transmit for one and for multiple instances in two communication scenarios and relate this number to the chromatic and korner 1973 entropies of a naturally defined graph

three dimensional visco acoustic modeling using a renormalized integral equation iterative solver we present a frequency domain renormalized integral equation formulation for solving a three dimensional visco acoustic medium using an iterative solver upon applying this special renormalization the resulting integral equation operator can be proven to have a contraction property hence solving the linear system of equations using a krylov optimization method will result in a good convergence rate furthermore since the matrix vector multiplication can be done using a fast fourier transform fft technique its operation is of the order of o n log n where n is the size of the discretization grid this technique also allows us to use matrix free implementation hence the memory usage is about o n numerical tests show that the computational time and memory usage of this renormalized integral equation approach can be quite competitive with the frequency domain finite difference iterative solver further the numerical examples demonstrate that it is possible to solve a problem with over 100 million unknowns using an integral equation approach

efficient reconstruction of sequences we introduce and solve some new problems of efficient reconstruction of an unknown sequence from its versions distorted by errors of a certain type these erroneous versions are considered as outputs of repeated transmissions over a channel either a combinatorial channel defined by the maximum number of permissible errors of a given type or a discrete memoryless channel we are interested in the smallest n such that n erroneous versions always suffice to reconstruct a sequence of length n either exactly or with a preset accuracy and or with a given probability we are also interested in simple reconstruction algorithms complete solutions for combinatorial channels with some types of errors of interest in coding theory namely substitutions transpositions deletions and insertions of symbols are given for these cases simple reconstruction algorithms based on majority and threshold principles and their nontrivial combination are found in general for combinatorial channels the considered problem is reduced to a new problem of reconstructing a vertex of an arbitrary graph with the help of the minimum number of vertices in its metrical ball of a given radius a certain sufficient condition for solution of this problem is presented for a discrete memoryless channel the asymptotic behavior of the minimum number of repeated transmissions which are sufficient to reconstruct any sequence of length n within hamming distance d with error probability spl epsiv is found when d n and spl epsiv tend to 0 as n spl rarr spl infin a similar result for the continuous channel with discrete time and additive gaussian noise is also obtained

solutions a variations bornees pour certains systemes hyperboliques de lois de conservation abstract blake temple trans amer math soc 280 1983 781 795 has described the hyperbolic systems of two conservation laws whose shock and rarefaction curves coincide in this note we prove the global existence of weak solutions for such systems with any bounded variation initial condition the proof is based upon standard numerical schemes as well as upon parabolic regularization the key is that the total variation of the riemann invariants is decreasing in time at least in the case of the initial condition with compact support we prove by using the glimm scheme that the system is decoupling in two conservation laws in one unknown in finite time

quasi linear approximation in 3 d electromagnetic modeling the born approximation in electromagnetic em numerical modeling has limited application for solving 3 d electromagnetic induction problems because in structures with high conductivity contrasts and at high frequencies this approximation is inaccurate in this paper we develop a new and relatively simple approximation for the em field called a quasi linear approximation which is based on the evaluation of the anomalous field ea by a linear transformation of the normal primary field ea en where is called the electrical reflectivity tensor the reflectivity tensor inside inhomogeneities can be approximated by a slowly varying function that can be determined numerically by a simple optimization technique the new approximation gives an accurate estimate of the em response for conductivity contrasts of more than one hundred to one and for a wide range of frequencies it also opens the possibility for fast 3 d electromagnetic inversion

quasi analytical approximations and series in electromagnetic modeling the quasi linear approximation for electromagnetic forward modeling is based on the assumption that the anomalous electrical field within an inhomogeneous domain is linearly proportional to the background normal field through an electrical reflectivity tensor in the original formulation of the quasi linear approximation was determined by solving a minimization problem based on an integral equation for the scattering currents this approach is much less time consuming than the full integral equation method however it still requires solution of the corresponding system of linear equations in this paper we present a new approach to the approximate solution of the integral equation using through construction of quasi analytical expressions for the anomalous electromagnetic field for 3 d and 2 d models quasi analytical solutions reduce dramatically the computational effort related to forward electromagnetic modeling of inhomogeneous geoelectrical structures in the last sections of this paper

approximate solutions of acoustic 3d integral equation and their application to seismic modeling and full waveform inversion abstract over the recent decades a number of fast approximate solutions of lippmann schwinger equation which are more accurate than classic born and rytov approximations were proposed in the field of electromagnetic modeling those developments could be naturally extended to acoustic and elastic fields however until recently they were almost unknown in seismology this paper presents several solutions of this kind applied to acoustic modeling for both lossy and lossless media we evaluated the numerical merits of those methods and provide an estimation of their numerical complexity in our numerical realization we use the matrix free implementation of the corresponding integral operator we study the accuracy of those approximate solutions and demonstrate that the quasi analytical approximation is more accurate than the born approximation further we apply the quasi analytical approximation to the solution of the inverse problem it is demonstrated that this approach improves the estimation of the data gradient comparing to the born approximation the developed inversion algorithm is based on the conjugate gradient type optimization numerical model study demonstrates that the quasi analytical solution significantly reduces computation time of the seismic full waveform inversion we also show how the quasi analytical approximation can be extended to the case of elastic wavefield

an information spectrum approach to weak variable length source coding with side information this paper studies variable length vl source coding of general sources with side information novel one shot coding bounds for slepian wolf sw coding which give nonasymptotic tradeoff between the error probability and the codeword length of vl sw coding are established one shot results are applied to asymptotic analysis and a general formula for the optimal coding rate achievable by weakly lossless vl sw coding i e vl sw coding with vanishing error probability is derived our general formula reveals how the encoder side information and or vl coding improve the optimal coding rate in the general setting in addition it is shown that if the encoder side information is useless in weakly lossless vl coding then it is also useless even in the case where the error probability may be positive asymptotically

on network functional compression in this thesis we consider different aspects of the functional compression problem in functional compression the computation of a function or some functions of sources is desired at the receiver s the rate region of this problem has been considered in the literature under certain restrictive assumptions in chapter 2 of this thesis we consider this problem for an arbitrary tree network and asymptotically lossless computations in particular for one stage tree networks we compute a rate region and for an arbitrary tree network we derive a rate lower bound based on the graph entropy we introduce a new condition on colorings of source random variables characteristic graphs called the coloring connectivity condition c c c we show that unlike the condition mentioned in doshi et al this condition is necessary and sufficient for any achievable coding scheme based on colorings we also show that unlike entropy graph entropy does not satisfy the chain rule for one stage trees with correlated sources and general trees with independent sources we propose a modularized coding scheme based on graph colorings to perform arbitrarily closely to the derived rate lower bound we show that in a general tree network case with independent sources to achieve the rate lower bound intermediate nodes should perform some computations however for a family of functions and random variables called chain rule proper sets it is sufficient to have intermediate nodes act like relays to perform arbitrarily closely to the rate lower bound in chapter 3 of this thesis we consider a multi functional version of this problem with side information where the receiver wants to compute several functions with different side information random variables and zero distortion our results are applicable to the case with several receivers computing different desired functions we define a new concept named multi functional graph entropy which is an extension of graph entropy defined by k6rner we show that the minimum achievable rate for this problem is equal to conditional multi functional graph entropy of the source random variable given the side information we also propose a coding scheme based on graph colorings to achieve this rate in these proposed coding schemes one needs to compute the minimum entropy coloring a coloring random variable which minimizes the entropy of a characteristic graph in general finding this coloring is an np hard problem however in chapter 4 we show that depending on the characteristic graph s structure there are some interesting cases where finding the minimum entropy coloring is not np hard but tractable and practical in one of these cases we show that by having a non zero joint probability condition on random variables distributions for any desired function finding the minimum entropy coloring can be solved in polynomial time in another case we show that if the desired function is a quantization function this problem is also tractable we also consider this problem in a general

error bounds for fractional step methods for conservation laws with source terms fractional step methods have been used to approximate solutions of scalar conservation laws with source terms in this paper the stability and accuracy of the basic fractional step algorithms are analyzed when these algorithms are used to compute discontinuous solutions of nonhomogeneous scalar conservation laws the authors show that time splitting methods for conservation laws with source terms always converge to the unique weak solution satisfying the entropy condition in particular it is proved that the l 1 errors in the splitting methods are bounded by o sqrt delta t where delta t is the splitting time step the l 1 convergence rate of a class of fully discrete splitting methods is also investigated

zero error instantaneous coding of correlated sources with length constraints is np complete it is well known that the kraft inequality gives a necessary and sufficient condition on the codeword lengths of a zero error instantaneous code for a single source however generalization for two correlated sources is nontrivial we show that in the slepian wolf configuration even if one source is known at the decoder designing a zero error instantaneous code with given codeword lengths for the other source is np complete

three dimensional born inversion with an arbitrary reference recent work of g beylkin helped set the stage for very general seismic inversions we have combined these broad concepts for inversion with classical high frequency asymptotics and perturbation methods to bring them closer to practically implementable algorithms applications include inversion schemes for both stacked and unstacked seismic data basic assumptions are that the data have relative true amplitude and that a reasonably accurate background velocity c x y z is available the perturbation from this background is then sought since high frequency approximations are used throughout the resulting algorithms essentially locate discontinuities in velocity an expression for a full 3 d velocity inversion can be derived for a general data surface in this degree of generality the formula does not represent a computationally feasible algorithm primarily because a key jacobian determinant is not expressed in practical terms in several important cases however this shortcoming can be overcome and

an l 1 error bound for a semi implicit difference scheme applied to a stiff system of conservation laws a straightforward semi implicit finite difference method approximating a system of conservation laws including a stiff relaxation term is analyzed we show that the error measured in lone is bounded by o sqrt dt independent of the stiffness where the time step dt represents the mesh size as a simple corollary we obtain that solutions of the stiff system converge toward the solution of an equilibrium model at a rate of o delta 1 3 in lone as the relaxation time delta tends to zero

an error estimate for a finite difference scheme approximating a hyperbolic system of conservation laws a rigorous proof of an error estimate for a finite difference scheme applied to a 2 times 2 nonstrictly hyperbolic system of conservation laws is presented it is established that the error in the l 1 norm is of order delta x 1 2 where delta x denotes the mesh size this convergence estimate is optimal since for special data the system degenerates to a scalar equation for which a similar estimate is known to be optimal

amortized communication complexity in this work we study the direct sum problem with respect to communication complexity consider a relation f defined over 0 1 n times 0 1 n can the communication complexity of simultaneously computing f on cal l instances x 1 y 1 ldots x cal l y cal l be smaller than the communication complexity of computing f on the cal l instances separately let the amortized communication complexity of f be the communication complexity of simultaneously computing f on cal l instances divided by cal l we study the properties of the amortized communication complexity we show that the amortized communication complexity of a relation can be smaller than its communication complexity more precisely we present a partial function whose deterministic communication complexity is theta log n and its amortized deterministic communication complexity is o 1 similarly for randomized protocols we present a function whose randomized communication complexity is theta log n and its amortized randomized communication complexity is o 1 we also give a general lower bound on the amortized communication complexity of any function f in terms of its communication complexity c f for every function f the amortized communication complexity of f is omega left sqrt c f log n right

data verification and reconciliation with generalized error control codes we consider the problem of data reconciliation which we model as two separate multisets of data that must be reconciled with minimum communication under this model we show that the problem of reconciliation is equivalent to a variant of the graph coloring problem and provide consequent upper and lower bounds on the communication complexity of reconciliation further we show by means of an explicit construction that the problem of reconciliation is under certain general conditions equivalent to the problem of finding error correcting codes for a general class of errors under this equivalence reconciling with little communication is linked to codes with large size and vice versa we show analogous results for the problem of multiset verification in which we wish to determine whether two multisets are equal using minimum communication as a result a wide body of literature in coding theory may be applied to the problems of reconciliation and verification

on seismic deghosting using integral representation for the wave equation use of green s functions with neumann or dirichlet boundary conditions abstractthe recent interest in broadband seismic technology has spurred research into new and improved seismic deghosting solutions one starting point for deriving deghosting methods is the representation theorem which is an integral representation for the wave equation recent research results show that by using green s functions with dirichlet boundary conditions in the representation theorem source side deghosting of already receiver side deghosted wavefields can be achieved we found that the choice of green s functions with neumann boundary conditions on the sea surface and the plane that contains the sources leads to an identical but simpler solution with fewer processing steps in addition we found that pressure data can be receiver side deghosted by introducing green s functions with dirichlet boundary conditions on the sea surface and the plane containing the receivers into a modified representation theorem the deghosting methods derived from the representation theorem are wave theoretic alg

forward problem of nonlinear fredholm integral equation in reference medium via velocity weighted wavefield function a nonlinear fredholm integral equation of the first kind for the perturbation potential can be derived by interpreting the acoustic velocity as a perturbation of a reference velocity we present an accurate and efficient method to formulate and numerically solve this equation with no restriction on the size of the perturbation by introducing a velocity weighted wavefield function i e the scalar product of the potential function with the perturbed velocity function inside a scatterer which is an intermediate function associating the observed wavefield with the perturbed velocity function we start with a singularity analysis of the fredholm integral equation when the observation point coincides with the scattering point to establish a linear singular integral equation with respect to the velocity weighted wavefield function then we formulate the relation between the velocity weighted wavefield and scattered field in the f k domain a numerical scheme is developed to solve the forward problem for the