a projection regularized newton method for nonlinear ill posed problems and its application to parameter identification problems with finite element discretization this paper is concerned with nonlinear ill posed operator equations f a y e g parameter identification problems and their approximate solution by a newton type method that is regularized by projecting the linearized equation in each newton step onto a finite dimensional space e g a finite element space and by stopping the newton iteration at an appropriate index we prove convergence as the iteration index n goes to infinity in the noise free case and convergence as the data noise level delta goes to zero in the case of noisy data as well as convergence rates under certain additional conditions

efficient preconditioners for boundary element matrices based on grey box algebraic multigrid methods this paper is concerned with the iterative solution of the boundary element equations arising from standard galerkin boundary clement discretizations of first kind boundary integral operators of positive and negative order we construct efficient preconditioners on the basis of so called grey box algebraic multigrid methods that are well adapted to the treatment of boundary element matrices in particular the coarsening is based on an auxiliary matrix that represents the underlying topology in a certain sense this auxiliary matrix is additionally used for the construction of the smoothers and the transfer operators finally we present the results of some numerical studies that show the efficiency of the proposed algebraic multigrid preconditioners

simultaneous multitrace impedance inversion with transform domain sparsity promotion abstractthe impedance inversion technique plays a crucial role in seismic reservoir properties prediction however most existing impedance inversion methods often suffer from spatial discontinuities and instability because each vertical profile is processed independently in the inversion we tested a transform domain sparsity promotion simultaneous multitrace impedance inversion method to address this issue the approach was implemented through minimizing a data misfit term and a transform domain sparsity constraint term that incorporates the 2d or 3d structural information into the inversion processing a 2d synthetic data example was applied to mainly explain the roles of the transform domain sparsity constraint we determined that the transform domain sparsity constraint can help in stabilizing the inversion reducing the influence of high wavenumber noise on the inverted result and exploring spatial continuities of structures furthermore a 3d field data example was used to examine the effectiven

seismic sparse layer reflectivity inversion using basis pursuit decomposition a basis pursuit inversion of seismic reflection data for reflection coefficients is introduced as an alternative method of incorporating a priori information in the seismic inversion process the inversion is accomplished by building a dictionary of functions representing reflectivity patterns and constituting the seismic trace as a superposition of these patterns basis pursuit decomposition finds a sparse number of reflection responses that sum to form the seismic trace when the dictionary of functions is chosen to be a wedge model of reflection coefficient pairs convolved with the seismic wavelet the resulting reflectivity inversion is a sparse layer inversion rather than a sparse spike inversion synthetic tests suggest that a sparse layer inversion using basis pursuit can better resolve thin beds than a comparable sparse spike inversion application to field data indicates that sparse layer inversion results in the potentially improved detectability and resolution of some thin layers and reveals a

the analysis of multigrid algorithms for pseudodifferential operators of order minus one multigrid algorithms are developed to solve the discrete systems approximating the solutions of operator equations involving pseudodifferential operators of order minus one classical multigrid theory deals with the case of differential operators of positive order the pseudodifferential operator gives rise to a coercive form on h sup minus 1 2 omega effective multigrid algorithms are developed for this problem these algorithms are novel in that they use the inner product on h sup minus 1 omega as a base inner product for the multigrid development the authors show that the resulting rate of iterative convergence can at worst depend linearly on the number of levels in these novel multigrid algorithms in addition it is shown that the convergence rate is independent of the number of levels and unknowns in the case of a pseudodifferential operator defined by a single layer potential finally the results of numerical experiments illustrating the theory are presented 19 refs 1 fig 2 tabs

new convergence estimates for multigrid algorithms in this paper new convergence estimates are proved for both symmetric and nonsymmetric multigrid algorithms applied to symmetric positive definite problems our theory relates the convergence of multigrid algorithms to a regularity and approximation parameter cap alpha epsilon 0 1 and the number of relaxations m we show that for the symmetric and nonsymmetric nu cycles the multigrid iteration converges for any positive m at a rate which deteriorates no worse than 1 cj sup 1 sup cap alpha sup sup cap alpha where j is the number of grid levels we then define a generalized nu cycle algorithm which involves exponentially increasing for example doubling the number of smoothings on successively coarser grids we show that the resulting symmetric and nonsymmetric multigrid iterations converge for any cap alpha with rates that are independent of the mesh size the theory is presented in an abstract setting which can be applied to finite element multigrid and finite difference multigrid methods

multichannel matching pursuit for seismic trace decomposition the technique of matching pursuit can adaptively decompose a seismic trace into a series of wavelets however the solution is not unique and is also strongly affected by data noise multichannel matching pursuit mcmp exploiting lateral coherence as a constraint might improve the uniqueness of the solution it extracts a constituent wavelet that has an optimal correlation coefficient to neighboring traces instead of to a single trace only according to linearity theory a wavelet shared by neighboring traces is the best match to the average of multiple traces and therefore it might effectively suppress the data noise and stabilize the performance it is found that the mcmp scheme greatly improves spatial continuity in decomposition and can generate a plausible time frequency spectrum with high resolution for reservoir detection

convergence estimates for multigrid algorithms without regularity assumptions a new technique for proving rate of convergence estimates of multi grid algorithms for symmetric positive definite problems will be given in this paper the standard multigrid theory requires a regularity and approxima tion assumption in contrast the new theory requires only an easily verified approximation assumption this leads to convergence results for multigrid re finement applications problems with irregular coefficients and problems whose coefficients have large jumps in addition the new theory shows why it suffices to smooth only in the regions where new nodes are being added in multigrid refinement applications

on the stability of the cut locus abstract in r d we consider a riemannian metric g and an open bounded subset we study the stability of the cut locus associated with and g w r t perturbations both of the set and of the metric g in order to have the stability of the cut locus we assume c 2 regularity of the data the metrics and the sets in the case of sets with c 1 1 boundaries the cut locus may be unstable we prove that to c 2 perturbations both of the set and of the metric correspond small changes of the cut locus w r t the hausdorff distance i e the cut locus is stable in the c 2 category we give some examples showing that c 1 perturbations may lead to large variations of the cut locus

on the extension of the solutions of hamilton jacobi equations abstract we consider the viscosity solution of a homogeneous dirichlet problem for the eikonal equation in a bounded set we suppose that the hamiltonian h x p a x p p 1 is strictly convex w r t the variables p and of class c 1 1 w r t the variables x then the solution of the dirichlet problem admits an extension to a neighbourhood of u such that u is still a viscosity solution of the eikonal equation if and only if satisfies an exterior sphere condition the above result in particular provides a characterization of the boundary singularities and a regularity theorem up to the boundary for the solution of the eikonal equation

recovery of the acoustic impedance from reflection seismograms this paper examines the problem of recovering the acoustic impedance from a band limited normal incidence reflection seismogram the convolutional model for the seismogram is adopted at the outset and it is therefore required that initial processing has removed multiples and recovered true amplitudes as well as possible in the first portion of the paper we investigate the effect of substituting the deconvolved seismic trace that is the band limited version of the reflectivity function into the standard recursion formula for the acoustic impedance the formalism of linear inverse theory is used to show that the logarithm of the normalized acoustic impedance estimated from the deconvolved seismogram is approximately an average of the true logarithm of the impedance moreover the averaging function is identical to that used in deconvolving the initial seismogram the advantage of these averages is that they are unique their disadvantage is that low frequency information which is crucial to making a g

distance to ck hypersurfaces abstract it is shown by elementary means that a ck hypersurface m of positive reach in r n 1 has the property that the signed distance function to it is ck k 1 this extends and complements work of federer gilbarg and trudinger and serrin

the ciarlet raviart method for biharmonic problems on general polygonal domains mapping properties and preconditioning for biharmonic boundary value problems the ciarlet raviart mixed method is considered on polygonal domains without additional convexity assumptions mapping properties of the involved operators on the continuous as well as on the discrete level is studied based on this efficient preconditioners are constructed and numerical experiments are shown

on the convergence of the multigrid method for a hypersingular integral equation of the first kind we present a multigrid method to solve linear systems arising from galerkin schemes for a hypersingular boundary integral equation governing three dimensional neumann problems for the laplacian our algorithm uses damped jacobi iteration gauss seidel iteration or sor as preand post smoothers if the integral equation holds on a closed lipschitz surface we prove convergence ofv andw cycles with any number of smoothing steps if the integral equation holds on an open lipschitz surface covering crack problems we show convergence of thew cycle numerical experiments are given which underline the theoretical results

additive schwarz method for the p version of the boundary element method for the single layer potential operator on a plane screen we analyze an additive schwarz preconditioner for the p version of the boundary element method for the single layer potential operator on a plane screen in the three dimensional euclidean space we decompose the ansatz space which consists of piecewise polynomials of degree p on a mesh of size h by introducing a coarse mesh of size h ge h after subtraction of the coarse subspace of piecewise constant functions on the coarse mesh this results in local subspaces of piecewise polynomials living only on elements of size h this decomposition yields a preconditioner which bounds the spectral condition number of the stiffness matrix by o log frac hhp 2 numerical results supporting the theory are presented

the analysis of smoothers for multigrid algorithms the purpose of this paper is to provide a general technique for defining and analyzing smoothing operators for use in multigrid algorithms the smoothing operators considered are based on subspace decomposition and include point line and block versions of jacobi and gauss seidel iteration as well as generalizations we shall show that these smoothers will be effective in multigrid algorithms provided that the subspace decomposition satisfies two simple conditions in many applications these conditions are trivial to verify

fast parallel solvers for symmetric boundary element domain decomposition equations the boundary element method bem is of advantage in many applications including far field computations in magnetostatics and solid mechanics as well as accurate computations of singularities since the numerical approximation is essentially reduced to the boundary of the domain under consideration the mesh generation and handling is simpler than for example in a finite element discretization of the domain in this paper we discuss fast solution techniques for the linear systems of equations obtained by the bem be equations utilizing the non overlapping domain decomposition dd we study parallel algorithms for solving large scale galerkin be equations approximating linear potential problems in plane bounded domains with piecewise homogeneous material properties we give an elementary spectral equivalence analysis of the bem schur complement that provides the tool for constructing and analysing appropriate preconditioners finally we present numerical results obtained on a massively parallel machine using up to 128 processors and we sketch further applications to elasticity problems and to the coupling of the finite element method fem with the boundary element method as shown theoretically and confirmed by the numerical experiments the methods are of o h 2 algebraic complexity and of high parallel efficiency where h denotes the usual discretization parameter

seismic trace interpolation in the f x domain interpolation of seismic traces is an effective means of improving migration when the data set exhibits spatial aliasing a major difficulty of standard interpolation methods is that they depend on the degree of reliability with which the various geological events can be separated in this respect a multichannel interpolation method is described which requires neither a priori knowledge of the directions of lateral coherence of the events nor estimation of these directions the method is based on the fact that linear events present in a section made of equally spaced traces may be interpolated exactly regardless of the original spatial interval without any attempt to determine their true dips the predictability of linear events in the f x domain allows the missing traces to be expressed as the output of a linear system the input of which consists of the recorded traces the interpolation operator is obtained by solving a set of linear equations whose coefficients depend only on the spectrum of the spatial prediction filter defined by the recorded traces synthetic examples show that this method is insensitive to random noise and that it correctly handles curvatures and lateral amplitude variations assessment of the method with a real data set shows that the interpolation yields an improved migrated section

some estimates for a weighted l 2 projection this paper is devoted to the error estimates for some weighted l2 projections nearly optimal estimates are obtained these estimates can be applied to the analysis of the usual multigrid method multilevel preconditioner and domain decomposition method for solving elliptic boundary problems whose coefficients have large jump discontinuities

rectifiability results for singular and conjugate points of optimal exit time problems we consider the value function v of optimal control problems with exit time under suitable assumptions through the study of the conjugate points we prove that the closure of the singular set of v is rectifiable moreover a sharper hausdorff estimate is given on the set of the conjugate nonsingular points

the application of coupled be fe formulations in technical magnetic field computations abstract we present a coupled finite element fe boundary element be formulation based on domain decomposition dd ideas leading to a discrete system which can be solved naturally in parallel on mimd parallel computers the main components of the dd algorithm are discussed finally the advantage of coupled be fe discretizations is demonstrated for problems arising in technical magnetic field computations

multichannel blind deconvolution of seismic signals a new algorithm for simultaneous wavelet estimation and deconvolution of seismic reflection signals is given to remove the inherent ambiguity in this blind deconvolution problem we introduce relevant a priori information our major assumption is sparseness of the reflectivity which corresponds to a layered earth model this allows nonminimum phase wavelets to be recovered reliably and closely spaced reflectors to be resolved to combine a priori knowledge and data we use a bayesian framework and derive a maximum a posteriori estimate computing this estimate is a difficult optimization problem solved by a suboptimal iterative procedure the procedure alternates steps of wavelet estimation and reflectivity estimation the first step only requires a simple least squares fit while the second step is solved by the iterated window maximization algorithm proposed by kaaresen this enables better efficiency and optimality than established alternatives the resulting optimization method can easily handle multichannel models with only a moderate increase of the computational load lateral continuity of the reflectors is achieved by modeling local dependencies between neighboring traces major improvements in both wavelet and reflectivity estimates are obtained by taking the wavelet to be invariant across several traces the practicality of the algorithm is demonstrated on synthetic and real seismic data an application to multivariate well log segmentation is also given

stochastic sparse spike deconvolution sparse spike deconvolution can be viewed as an inverse problem where the locations and amplitudes of a number of spikes reflectivity are estimated from noisy data seismic traces the main objective is to find the least number of spikes that when convolved with the available band limited seismicwaveletestimate fitthedatawithinagiventolerance errormisfit thedetectionofthespikes timelagsisahighly nonlinearoptimizationproblemthatcanbesolvedusingvery fastsimulatedannealingsa amplitudesareeasilyestimated using linear least squares at each sa iteration at this stage quadratic regularization is used to stabilize the solution to reduce its nonuniqueness and to provide meaningful reflectivitysequences thusavoidingtheneedtoconstrainthe spikes time lags and or amplitudes to force valid solutions impedanceconstraintsalsocanbeincludedatthisstage providing the low frequencies required to recover the acoustic impedance oneadvantageoftheproposedmethodoverother sparse spike deconvolution techniques is that the uncertaintyoftheobtainedsolutionscanbeestimatedstochastically further errorsinthephaseofthewaveletestimatearetolerated foranoptimumconstant phaseshiftisobtainedtocalibratetheeffectivewaveletthatispresentinthedata results using synthetic data including simulated data for the marmousi2 modelandfield 3d data show that physically meaningful high resolution sparse spike sections can be derived fromband limitednoisydata evenwhentheavailablewaveletestimateisinaccurate

the regularity of the distance function propagates along minimizing geodesics abstract we consider the distance function from the boundary of an open bounded set r n associated to a riemannian metric with c 1 1 coefficients we show that the c 1 1 regularity propagates towards the boundary along the distance minimizing geodesics hence we show that the cut locus is invariant with respect to the generalized gradient flow associated to the distance function and that it has the same homotopy type as

v cycle convergence of some multigrid methods for ill posed problems for ill posed linear operator equations we consider some v cycle multigrid approaches that in the framework of bramble pasciak wang and xu 1991 we prove to yield level independent contraction factor estimates consequently we can incorporate these multigrid operators in a full multigrid method that together with a discrepancy principle is shown to act as an iterative regularization method for the underlying infinite dimensional ill posed problem numerical experiments illustrate the theoretical results

new estimates for multilevel algorithms including the v cycle the purpose of this paper is to provide new estimates for certain multilevel algorithms in particular we are concerned with the simple additive multilevel algorithm discussed recently together with j xu and the standard v cycle algorithm with one smoothing step per grid we shall prove that these algorithms have a uniform reduction per iteration independent of the mesh sizes and number of levels even on nonconvex domains which do not provide full elliptic regularity for example the theory applies to the standard multigrid vcycle on the l shaped domain or a domain with a crack and yields a uniform convergence rate we also prove uniform convergence rates for the multigrid v cycle for problems with nonuniformly refined meshes finally we give a new multigrid approach for problems on domains with curved boundaries and prove a uniform rate of convergence for the corresponding multigrid v cycle algorithms

on the local semiconcavity of the solutions of the eikonal equation abstract we show that if the hamiltonian is locally semiconvex with respect to the state variables and strictly convex with respect to the gradient then every viscosity solution of the eikonal equation is locally semiconcave furthermore in the 1d case we show that every viscosity solution of the eikonal equation is semiconcave if and only if the hamiltonian is lipschitz continuous with respect to the state variable

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

a least squares coupling method with finite elements and boundary elements for transmission problems we analyze a least squares formulation for the numerical solution of second order lineartransmission problems in two and three dimensions which allow jumps on the interface in a bounded domain the second order partial differential equation is rewritten as a first order system the part of the transmission problem which corresponds to the unbounded exterior domain is reformulated by means of boundary integral equations on the interface the least squares functional is given in terms of sobolev norms of order 1 and of order 1 2 these norms are computed by approximating the corresponding inner products using multilevel preconditioners for a second order elliptic problem in a bounded domain and for the weakly singular integral operator of the single layer potential on its boundary email protected as preconditioners we use both multigrid and bpx algorithms and the preconditioned system has bounded or mildly growing condition number numerical experiments confirm our theoretical results

on the cut locus of closed sets abstract let c be a closed subset of a smooth manifold of dimension n m and let m be endowed with a riemannian metric of class c 2 we study the cut locus of c cut c first we show that cut c is a set of measure zero then we assume that c is the boundary of an open bounded set m in particular this assumption implies that cut c 0 we deduce that cut c is invariant w r t the generalized gradient flow associated with the distance function from the set c as a consequence of the invariance we have that cut c has the same homotopy type as the set furthermore if m is a compact manifold then cut c has the same homotopy type as m c finally we show that the closure of the cut locus stays away from c if and only if c is a manifold of class c 1 1

a new convergence proof for the multigrid method including the v cycle for a positive definite finite element equation we describe a multigrid iteration and prove convergence under natural assumptions on the discretization and the elliptic problem hitherto existing convergence proofs require a sufficiently large number of smoothing iterations and exclude the v cycle the presented proof applies to procedures with any number of smoothing iterations and to the v cycle

an optimal order process for solving finite element equations a k level iterative procedure for solving the algebraic equations which arise from the finite element approximation of elliptic boundary value problems is presented and analyzed the work estimate for this procedure is proportional to the number of unknowns an optimal order result general geometry is permitted for the underlying domain but the shape plays a role in the rate of convergence through elliptic regularity finally a short discussion of the use of this method for parabolic problems is presented

some properties of viscosity solutions of hamilton jacobi equations abstract recently m g crandall and p l lions introduced the notion of viscosity solutions of scalar nonlinear first order partial differential equations viscosity solutions need not be differentiable anywhere and thus are not sensitive to the classical problem of the crossing of characteristics the value of this concept is established by the fact that very general existence uniqueness and continuous dependence results hold for viscosity solutions of many problems arising in fields of application the notion of a viscosity solution admits several equivalent formulations here we look more closely at two of these equivalent criteria and exhibit their virtues by both proving several new facts and reproving various known results in a simpler manner moreover by forsaking technical generality we hereby provide a more congenial introduction to this subject than the original paper author

multilevel matrix multiplication and fast solution of integral equations abstract a fast multigrid approach is described for the task of calculating k x y u y dy for each x r d discretizing by an equidistant grid with n points and meshsize h and approximating the integrations to o h2s accuracy it is shown that the complexity of this calculation can be reduced from o n2 to o sn provided the kernel k is sufficiently smooth for potential type kernels the complexity is reduced to o sn log n corresponding integral equations can be solved to a similar accuracy in basically the same amount of work using a special kind of distributed relaxation in a multigrid algorithm one and two dimensional numerical tests and theoretical derivations of optimal strategies are reported the method is applicable to the task of multiplying by any matrix with appropriate smoothness properties including most types of many body interactions