postprocessing the galerkin method the finite element case a postprocessing technique developed earlier for spectral methods is extended here to galerkin finite element methods for dissipative evolution partial differential equations the postprocessing amounts to solving a linear elliptic problem on a finer grid or higher order space once the time integration on the coarser mesh is completed this technique increases the convergence rate of the finite element method to which it is applied and this is done at almost no additional computational cost the numerical experiments presented here show that the resulting postprocessed method is computationally more efficient than the method to which it is applied say quadratic finite elements as well as standard methods of similar order of convergence as the postprocessed one say cubic finite elements the error analysis of the new method is performed in l2 and in l infty norms

entropic approach to interior point solution of linear programs we present a maxentropic approach for obtaining interior suboptimal solutions to a linear programming problem not necessarily in karmarkar s canonical form each suboptimal solution lies in the relative interior set of feasible solutions we present an application to the problem of obtaining reconstructions of minimal weighted l 1 norm of a function from the knowledge of a few of its fourier coefficients

optimal l error estimates for nonconforming and mixed finite element methods of lowest order for second order linear elliptic problems it is proved that thep1 nonconforming finite element method has the samel asymptotic accuracy as thep1 conforming one this result is applied to derive optimall error estimates for both the displacement and the stress fields of the lowest order raviart thomas mixed finite element method and a superconvergence result at the barycenter of each element

pointwise localized error estimates for parabolic finite element equations we derive pointwise weighted error estimates for a semidiscrete finite element method applied to parabolic equations the results extend those obtained by a h schatz for stationary elliptic problems in particular they show that the error is more localized for higher order elements

feasibility of approximate broadband estimation of acoustic impedance profile from first principles and band limited reflection data abstractbecause seismic reflection data are band limited acoustic impedance profiles derived from them are nonunique the conventional inversion methods counter the nonuniqueness either by stabilizing the answer with respect to an initial model or by imposing mathematical constraints such as sparsity of the reflection coefficients by making a nominal assumption of an earth model locally consisting of a stack of homogeneous and horizontal layers we have formulated a set of linear equations in which the reflection coefficients are the unknowns and the recursively integrated seismic trace constitute the data drawing only on first principles the zoeppritz equation in this case the approach makes a frontal assault on the problem of reconstructing reflection coefficients from band limited data the local layer cake assumption and the strategy of seeking a singular value decomposition solution of the linear equations counter the nonuniqueness provided that the objective is to reconstruct a smooth version

error estimates for the numerical approximation of time dependent flow of bingham fluid in cylindrical pipes by the regularization method the flow of a bingham fluid in a cylindrical pipe can give rise to free boundary problems the fluid behaves like a viscous fluid if the shear stress expressed as a linear function of the shear rate exceeds a yield value and like a rigid body otherwise the surfaces dividing fluid and rigid zones are the free boundaries therefore the solution for such highly nonlinear problems can in general only be obtained by numerical methods considerable progress has been made in the development of numerical algorithms for bingham fluids 2 20 27 29 30 32 however very little research can be found in the literature regarding the rate of convergence of the numerical solution to the true continuous solution that is the error estimate of these numerical methods error estimates are a critically important issue because they tells us how to control the error by appropriately choosing the grid sizes and other related parameters this paper concerns the error estimates of a unsteady bingham fluid modeled as a variational inequality due to duvaut lions 16 and glowinski 30 the difficulty both in the analytical and numerical treatment of the mathematical model is due to the fact that it contains a nondifferentiable term a common technique called the regularization method is to replace the non differentiable term by a perturbed differentiable term which depends on a small regularization parameter the regularization method effectively reduces the variational inequality to an equation a regularized problem which is much easier to cope with this paper has achieved the following 1 error estimates are derived for a continuous time galerkin method in suitable norms 2 we give an estimate of the difference between the true solution and the regularized solution in terms of 3 some regularity properties for both regularized solution and the true solution are proved 4 the error estimates for full discretization of the regularized problem using piecewise linear finite elements in space and backward differencing in time are established for the first time by coupling the regularization parameter and the discretization parameters h and t 5 we are able to improve our estimates in the one dimensional case or under stronger regularity assumptions on the true solution the estimates for the one dimensional case are optimal and confirmed by numerical experiment the estimates from 4 and 5 provide very important information on the measure of the error and give us a powerful mechanism to properly choose the parameters h t and in order to control the error see corollary 4 4 the above estimates extend the error bounds derived in glowinski lions and tremolieres 32 chapter 5 pp 348 404 for the stationary bingham fluid to the time dependent one which is the main contribution of this paper

anl2 error estimate for an approximation of the solution of a parabolic variational inequality we estimate the order of the difference between the numerical approximation and the solution of a parabolic variational inequality the numerical approximation is obtained using a finite element discretization in space and a finite difference discretization in time which is more general than is used in the literature we obtain better error estimates than those given in the literature the error estimates are compared with numerical experiments

maximum norm estimates for resolvents of elliptic finite element operators let be a convex domain with smooth boundary in rd it has been shown recently that the semigroup generated by the discrete laplacian for quasi uniform families of piecewise linear finite element spaces on is analytic with respect to the maximum norm uniformly in the mesh width this implies a resolvent estimate of standard form in the maximum norm outside some sector in the right halfplane and conversely here we show directly that such a resolvent estimate holds outside any sector around the positive real axis with arbitrarily small angle this is useful in the study of fully discrete approximations based on a stable rational functions with small

adaptive iterative algorithms for spiky deconvolution considering the joint detection estimation character that spiky deconvolution problems have an adaptively contracted projection selection operator is introduced to detect the nonzero values of the solution which can be combined with iterative algorithms to offer very efficient schemes for solving these problems a number of gradient type algorithms based on this principle are described and their performance is illustrated through simulation examples the approach is based on the idea of reducing the noise and defining the signal in an iterative form another possibility is to define the signal and reduce the overall energy outside its domain also in an iterative form this algorithm is more expensive from a computational point of view however simulations indicate that it has somewhat different properties it seems to be slightly less robust against the noise while it offers better resolution and even more accurate amplitude estimates

uniform error estimates in the finite element method for a singularly perturbed reaction diffusion problem consider the problem e 2 u u f with homogeneous neumann boundary condition in a bounded smooth domain in r n the whole range 0 e 1 is treated the galerkin finite element method is used on a globally quasi uniform mesh of size h the mesh is fixed and independent of e a precise analysis of how the error at each point depends on h and e is is presented as an application first order error estimates in h which are uniform with respect to e are given

simultaneous deconvolution and wavelet inversion as a global optimization estimates of the source wavelet and band limited earth reflectivity are obtained simultaneously from an optimization of deconvolution outputs similar to minimum entropy deconvolution med the only inputs required beyond the observed seismogram are wavelet length and an inversion parameter cooling rate the objective function to be minimized is a measure of the spikiness of the deconvolved seismogram i assume that the wavelet whose deconvolution from the data results in the most spike like trace is the best wavelet estimate because this is a highly nonlinear problem simulated annealing is used to solve it the procedure yields excellent results on synthetic data and disparate field data sets is robust in the presence of noise and is fast enough to operate in a desktop computer environment

multichannel spatially correlated reflectivity inversion using block sparse bayesian learning abstractthe reflectivity inversion approach based on a variety of regularization terms was extensively developed and applied to image subsurface structure in recent years in addition multichannel reflectivity inversion or deconvolution considering the lateral continuity of reflection interfaces or reflectivity in adjacent channels has been developed however these processing operations seldom adaptively judge the stratal continuity or automatically alter the parameters of the corresponding algorithm to use the special correlation of the reflection information contained in the seismic data a multichannel spatially correlated reflectivity inversion using block sparse bayesian learning bsbl is introduced the method adopts a covariance matrix that describes the spatial relationship of reflectivity and simultaneously controls the temporal sparsity with an expectation maximization algorithm we can obtain the parameters of the multichannel reflectivity model including the mean i e the estimated mul

shape calculus and finite element method in smooth domains the use of finite elements in smooth domains leads naturally to polyhedral or piecewise polynomial approximations of the boundary hence the approximation error consists of two parts the geometric part and the finite element part we propose to exploit this decomposition in the error analysis by introducing an auxiliary problem defined in a polygonal domain approximating the original smooth domain the finite element part of the error can be treated in the standard way to estimate the geometric part of the error we need quantitative estimates related to perturbation of the geometry we derive such estimates using the techniques developed for shape sensitivity analysis

an adaptive newton algorithm based on numerical inversion regularization as postconditioner we examine a class of approximate inversion processes satisfying estimates similar to those defined by finite element or truncated spectral approximations these are to be used as approximate right inverses for newton iteration methods when viewed at the operator level these approximations introduce a defect or loss of derivatives of order one or more regularization is introduced as a form of defect correction a superlinearly convergent approximate newton iteration is thereby obtained by using the numerical inversion adaptively i e with spectral or grid parameters correlated to the magnitude of the current residual in an intermediate norm defined by the defect this adaptive choice makes possible ascribing an order to the convergent process and this is identified as essentially optimal for elliptic problems relative to complexity the design of the algorithm involves multi parameter selection thereby opening up interesting avenues for elliptic problems relative to complexity this applies also to the regularization which may be carried out in the fourier transform space and is band limited in the language of whittaker shannon sampling theory the norms employed in the analysis are of holder space type the iteration is an adaptation of nash moser interation and the complexity studies use vituskin s theory of information processing computational experience is described in the final section

projected gabor deconvolution abstractgabor deconvolution consists of approximating a nonstationary problem as several stationary subproblems via the gabor frame solving each subproblem independently and then recombining projecting the subsolutions into an approximate solution to the original nonstationary problem the approximations however cause inherent instability due to systematic errors that prevent the algorithm from converging to the true solution even for noise free cases furthermore the method will be time consuming when a nonlinear optimization is used for shaping the reflectivity structure an alternative projected gabor deconvolution is considered which is based on the gabor expansion however in contrast to the conventional form in the new method first the problem is projected into the time domain and then an inversion is performed to obtain the final reflectivity compared with the gabor deconvolution the projected alternative 1 exhibits an improved convergence property 2 is more efficient for sparse de

an l error estimate for finite element solution of nonlinear elliptic problem with a source term the l error estimate for the finite element solution of nonlinear elliptic problem with a source term is discussed the lumped mass discretization method is used in the discretization and the error estimate o h 2 logh in l norm is obtained

a linear programming approach to the estimation of the power spectra of harmonic processes a new approach to the estimation of high resolution power spectra of harmonic processes is presented this approach uses linear programming lip to exploit the spiky character of the l 1 norm results for the lip power spectral estimate are contrasted with those using various formulations of the maximum entropy algorithm

some optimal error estimates for piecewise linear finite element approximations it is shown that the ritz projection onto spaces of piecewise linear finite elements is bounded in the sobolev space wl for 2 p oc this implies that for functions in wf n w2 the error in approximation behaves like 0 h in wp for 2 p oo and like 0 h2 in lp for 2 p oo in all these cases the additional logarithmic factor previously included in error estimates for linear finite elements does not occur

fully discrete convergence estimates for vortex methods in bounded domains in this paper the authors study vortex method for 2 dimensionsal euler equations of incompressible flow in bounded domains to approximate the initial vortex field by a sum of vortex blobs with arbitrary high accuracy this field is extended smoothly to a small neighborhood of the boundary and the computation is carried out in the extended domain to construct the velocity field from vorticity a second order isoparametric finite element method is applied and to solve the characteristic equations the explicit euler s scheme is considered optimal error bounds for this fully discrete scheme are obtained

convergence past singularities for a fully discrete approximation of curvature drive interfaces consider a closed surface in rn of codimension 1 which propagates in the normal direction with velocity proportional to its mean curvature plus a forcing term this geometric problem is first approximated by a singularly perturbed parabolic double obstacle problem with small parameter eps 0 conforming piecewise linear finite elements over a quasi uniform and strongly acute mesh of size h are further used for space discretization and combined with backward differences for time discretization with uniform time step tau it is shown that the zero level set of the fully discrete solution converges past singularities to the true interface provided tau h 2 approx o eps 3 and no fattening occurs if the more stringent relations tau h 2 approx o eps 4 are enforced then a linear rate of convergence o eps for interfaces is derived in the vicinity of regular points namely those for which the underlying viscosity solution is nondegenerate singularities and their smearing effect are also studied the analysis is based on constructing discrete barriers via a parabolic projection lipschitz dependence of viscosity solutions with respect to perturbations of data and discrete nondegeneracy these issues are proven along with quasi optimality in two dimensions of the parabolic projection in l infty with respect to both order and regularity requirements for functions in w 2 1 p

sharpl error estimates for semilinear elliptic problems with free boundaries a numerical scheme to approximate a semilinear pde involving a singular maximal monotone graph is analyzed inl a preliminary regularization is combined with piecewise linear finite elements defined on a triangulation which is not assumed to be acute the discrete maximum principle is thus avoided sharp pointwise error estimates are derived for both the smoothing and the discretization procedures an optimal choice of the regularization parameter as a function of the mesh size leads to a sharp global rate of convergence these error estimates for solutions in conjunction with nondegeneracy properties of continuous problems provide sharp interface error estimates two model examples are discussed the obstacle problem and a combustion equation

combined effect of explicit time stepping and quadrature for curvature driven flows the flow of a closed surface of codimension 1 in bbb r n driven by curvature is first approximated by a singularly perturbed parabolic double obstacle problem with small parameter varepsilon 0 conforming piecewise linear finite elements with mass lumping over a quasi uniform and weakly acute mesh of size h are further used for space discretization and combined with forward differences for time discretization with uniform time step tau the resulting explicit schemes are the basis for an efficient algorithm the so called dynamic mesh algorithm and exhibit finite speed of propagation and discrete ondegeneracy no iteration is required not even to handle the obstacle constraints the zero level set of the fully discrete solution is shown to converge past singularities to the true interface provided tau h 2 approx o varepsilon 4 and no fattening occurs if the more stringent relations tau h 2 approx o varepsilon 6 are enforced then an interface rate of convergence o varepsilon is derived in the vicinity of regular points along with a companion o varepsilon 1 2 for type i singularities for smooth flows an interface rate of convergence of o varepsilonq is proven provided tau h 2 approx o varepsilon 5 and exact integration is used for the potential term the analysis is based on constructing fully discrete barriers via an explicit parabolic projection with quadrature which bears some intrinsic interest lipschitz properties of viscosity solutions of the level set approach and discrete nondegeneracy these basic ingredients are also discussed

cosine methods for nonlinear second order hyperbolic equations construction et analyse des methodes efficaces avec une precision d ordre eleve pour l approximation des solutions regulieres d une classe d equations hyperboliques non lineaires du second ordre application des discretisations de type galerkin et des schemas a deux pas estimations d erreurs d ordre optimal

interior maximum norm estimates for finite element methods part ii we consider bilinear forms a connected with second order elliptic problems and assume that for u h in a finite element space s h we have a u u h x f x for x in s h with local compact support we give local estimates for u u h in l and w 1 of the type local best approximation plus weak outside influences plus the local size of f

3d magnetic sparse inversion using an interior point method rock susceptibility measurements are sometimes taken on outcrop and borehole rocks and provide valuable information for constraining magnetic data inversion we present two approaches for 3d magnetic sparse inversion that effectively take advantage of the rock susceptibility information both approaches minimize a total objective function subject to bound constraints using an interior point method the first approach directly minimizes an l1 norm of the susceptibility model by keeping the bounds positive in which case the objective function is differentiable in the feasible region the second approach minimizes a more generalized lp like norm 0 p 1 of the susceptibility model by approximating the lp like norm inversion as an iteratively reweighed least squares problem moreover this approach allows the model values to be either positive or negative we also demonstrate the equivalence of the proposed approaches and the binary inversion the recovered models of both approaches are characterized by

analysis of fully discrete approximations to parabolic problems with rough or distribution valued initial data we consider fully discrete approximations to a parabolic initial boundary value problem with rough or distribution valued initial data in two space dimensions for discretization in time and space we apply single step methods and the standard galerkin method with piecewise linear test functions respectively for spatial discretization of the initial condition we are however forced to use more involved constructions our main result is stability and error estimates of the discrete solutions

an adaptive finite element method for two phase stefan problems in two space dimensions i stability and error estimates a simple and efficient adaptive local mesh refinement algorithm is devised and analyzed for two phase stefan problems in 2d a typical triangula tion is coarse away from the discrete interface where discretization parameters satisfy a parabolic relation whereas it is locally refined in the vicinity of the dis crete interface so that the relation becomes hyperbolic several numerical tests are performed on the computed temperature to extract information about its first and second derivatives as well as to predict discrete free boundary locations mesh selection is based upon equidistributing pointwise interpolation errors be tween consecutive meshes and imposing that discrete interfaces belong to the so called refined region consecutive meshes are not compatible in that they are not produced by enrichment or coarsening procedures but rather regenerated a general theory for interpolation between noncompatible meshes is set up in lp based norms the resulting scheme is stable in various sobolev norms and necessitates fewer spatial degrees of freedom than previous practical methods 3 2 2 on quasi uniform meshes namely 0 r as opposed to 0 x to achieve the same global asymptotic accuracy here r 0 is the uniform time step 112 a rate of convergence of essentially 0 x is derived in the natural energy spaces provided the total number of mesh changes is restricted to 0 x which in turn is compatible with the mesh selection procedure an auxiliary quasi optimal pointwise error estimate for the laplace operator is proved as well numerical results illustrate the scheme s efficiency in approximating both solutions and interfaces

postprocessing the linear finite element method we extend the idea of the postprocessing galerkin methods for dissipative evolution equations to the case of the linear finite element method the postprocessing technique has been developed earlier for spectral methods and for higher order finite element methods the analysis shows that this procedure improves the order of convergence of the piecewise linear galerkin finite element approximation in the h1 norm we show by means of numerical experiments that there is no improvement in the order of convergence in the l2 norm

a technique for stabilizing interval velocities from the dix equation interval velocities the velocities at which sounds travel in the earth can be computed from stacking or root mean square rms velocities by applying the dix equation dix 1955 ij j 1 rj 12tj 1 rj2tj tj 1 tj 1 2 eq 1 where rj are the stacking velocity picks tj are the associated times and ij j 1 is the average interval velocity between tj and tj 1

3d magnetic data space inversion with sparseness constraints i have developed an inversion approach that determines a 3d susceptibility distribution that produces a given magnetic anomaly the subsurface model consists of a 3d equally spaced array of dipoles the inversion incorporates a model norm that enforces sparseness and depth weighting of the solution sparseness is imposed by using the cauchy norm on model parameters the inverse problem is posed in the data space leading to a linear system of equations with dimensions based on the number of data n this contrasts with the standard least squares solution derived through operations within the m dimensional model space m being the number of model parameters hence the data space method combined with a conjugate gradient algorithm leads to computational efficiency by dealing with an n n system versus an m m one where n m tests on synthetic data show that sparse inversion produces a much more focused solution compared with a standard model space least squares inversion the inversion of aeromagnet

sparse reflectivity inversion for nonstationary seismic data abstractconventional reflectivity inversion methods are based on a stationary convolution model and theoretically require stationary seismic traces as input i e those free of attenuation and dispersion effects reflectivity inversion for nonstationary data which is typical for field surveys requires us to first compensate for the earth s q filtering effects by inverse q filtering however the attenuation compensation for inverse q filtering is inherently unstable and offers no perfect solution thus we presented a sparse reflectivity inversion method for nonstationary seismic data we referred to this method as nonstationary sparse reflectivity inversion nsri it makes the novel contribution of avoiding intrinsic instability associated with inverse q filtering by integrating the earth s q filtering operator into the stationary convolution model nsri also avoids time variant wavelets that are typically required in time variant deconvolution although nsri is initially designed for nonstationary

a new numerical method for backward parabolic problems in the maximum norm setting a new method for solving numerically backward parabolic problems is proposed as usual for this kind of ill posed problems it is assumed that an a priori bound for the solution is available the algorithm consists of two basic steps first a standard forward integration is performed in order to approximate the solution at suitable future time levels second a holomorphic recovery procedure is carried out providing the required approximations for the preceding times the analysis is valid in the maximum norm setting and rigorous estimates are derived among other advantages the method can also be applied to nonlinear problems and it produces a continuous output some numerical illustrations are presented

l error estimate for an approximation of a parabolic variational inequality almost optimall convergence of an approximation of a variational inequality of parabolic type is proved under regularity assumptions which are met by the solution of a one phase stefan problem the discretization employs piecewise linear finite elements in space and the backward euler scheme in time by means of a maximum principle the problem is reduced to an error estimate for an auxiliary parabolic equation the latter bound is obtained by using the smoothing property of the galerkin method

postprocessing the finite element method for semilinear parabolic problems in this paper we consider postprocessing of the finite element method for semilinear parabolic problems the postprocessing amounts to solving a linear elliptic problem on a finer grid or higher order space once the time integration on the coarser mesh is completed the convergence rate is increased at almost no additional computational cost this procedure was introduced and analyzed in garcia archilla and titi siam j numer anal 37 2000 pp 470 499 we extend the analysis to the fully discrete case and prove error estimates for both space and time discretization the analysis is based on error estimates for the approximation of time derivatives by difference quotients

the relativistic linear singular oscillator exactly solvable model of the linear singular oscillator in the relativistic configurational space is considered we have found wavefunctions and energy spectrum for the model under study it is shown that they have the correct non relativistic limits

fixed point continuation for ell_1 minimization methodology and convergence we present a framework for solving the large scale ell 1 regularized convex minimization problem min x 1 mu f x our approach is based on two powerful algorithmic ideas operator splitting and continuation operator splitting results in a fixed point algorithm for any given scalar mu continuation refers to approximately following the path traced by the optimal value of x as mu increases in this paper we study the structure of optimal solution sets prove finite convergence for important quantities and establish q linear convergence rates for the fixed point algorithm applied to problems with f x convex but not necessarily strictly convex the continuation framework motivated by our convergence results is demonstrated to facilitate the construction of practical algorithms

improving the rate of convergence of high order finite elements on polygons and domains with cusps let u and uv v be the solution and respectively the discrete solution of the non homogeneous dirichlet problem u f on u 0 for any m and any bounded polygonal domain we provide a construction of a new sequence of finite dimensional subspaces vn such that where f hm 1 is arbitrary and c is a constant that depends only on and not on n we do not assume u hm 1 the existence of such a sequence of subspaces was first proved in a ground breaking paper by babuska 8 our method is different it is based on the homogeneity properties of sobolev spaces with weights and the well posedness of non homogeneous dirichlet problem in suitable sobolev spaces with weights for which we provide a new proof and which is a substitute of the usual shift theorems for boundary value problems in domains with smooth boundary our results extended right away to domains whose boundaries have conical points we also indicate some of the changes necessary to deal with domains with cusps our numerical computation are in agreement with our theoretical results

conforming and discontinuous galerkin fem in space for solving parabolic obstacle problem abstract in this article we propose and analyze conforming and discontinuous galerkin dg finite element methods for numerical approximation of the solution of the parabolic variational inequality associated with a general obstacle in r d d 2 3 for fully discrete conforming method we use globally continuous and piecewise linear finite element space whereas for the fully discrete dg scheme we employ piecewise linear finite element space for spatial discretization the time discretization has been done by using the implicit backward euler method we present the error analysis for the conforming and the dg fully discrete schemes and derive an error estimate of optimal order o h t in a certain energy norm defined precisely in the article the analysis is performed without any assumptions on the speed of propagation of the free boundary but only assumes the pragmatic regularity that u t l 2 0 t l 2 the obstacle constraints are incorporated at the lagrange nodes of the triangular mesh and the analysis exploits the lagrange interpolation we present some numerical experiment to illustrate the performance of the proposed methods

reconstruction of a sparse spike train from a portion of its spectrum and application to high resolution deconvolution an algorithm is proposed for the reconstruction of a sparse spike train from an incomplete set of its fourier components it is shown that as little as 20 25 percent of the fourier spectrum is sufficient in practice for a high quality reconstruction the method employs linear programming to minimize the l1 norm of the output because minimization of this norm favors solutions with isolated spikes given a wavelet this technique can be used to perform deconvolution of noisy seismograms when the desired output is a sparse spike series relative reliability of the data is assessed in the frequency domain and only the reliable spectral data are included in the calculation of the spike series equations for the unknown spike amplitudes are solved to an accuracy compatible with the uncertainties in the reliable data in examples with 10 percent random noise the output is superior to that obtained using conventional least squares techniques

error estimates for spatially discrete approximations of semilinear parabolic equations with nonsmooth initial data on considere les approximations spatialement discretes continues par rapport au temps des problemes mixtes par la methode des elements finis de galerkin pour les equations paraboliques semi lineaires avec des donnees initiales irregulieres ou incompatibles on etablit des estimations d erreur

optimal l error estimates for piecewise linear finite elements in 1 n 3 for 1 n 3 in this paper for two second order elliptic boundary value problems uniform convergence error estimates are derived using piecewise linear conforming and non conforming finite elements the estimates are valid for polyhedral convex regions rn with 1 n 3 the proof of the estimates is mainly based on sobolev s inequality by which the uniform estimates are reduced to estimates in the mean in this way it is possible to obtain the desired uniformo h2 estimate in the plane as a special case thus confirming theoretically natterer s numerical results

superconvergence of least squares methods for a coupled system of elliptic equations abstract we present a numerical method for solving a coupled system of elliptic partial differential equations pdes our method is based on the least squares ls approach we develop ellipticity estimates and error bounds for the method the main idea of the error estimates is the establishment of supercloseness of the ls solutions and solutions of the mixed finite element methods and ritz projections using the supercloseness property we obtain l 2 norm error estimates and the error estimates for each quantity of interest show different convergence behaviors depending on the choice of the approximation spaces moreover we present maximum norm error estimates and construct asymptotically exact a posteriori error estimators under mild conditions application to optimal control problems is briefly considered

a simple difference realization of the heisenberg q algebra a realization of the heisenberg q algebra whose generators are first order difference operators on the full real line is discussed herein the eigenfunctions of the corresponding q oscillator hamiltonian are given explicitly in terms of the q 1 hermite polynomials the nonuniqueness of the measure for these q oscillator states is also studied

optimal rates of convergence for degenerate parabolic problems in two dimensions fully discrete implicit schemes for the solution of degenerate parabolic equations in two dimensions are shown to satisfy error estimates of the form o tau ln 1 h alpha h where tau is the time step size and h is the mesh size these results apply to both the stefan problem and the porous medium equation

suq 1 1 and the relativistic oscillator it is shown that the generalization of the quantum harmonic oscillator to the case of the relativistic configurational space is a q oscillator the corresponding group of dynamical symmetry is the quantum group suq 1 1 the deformation parameter being q e omega omega 0 where omega 0 4mc2 and omega is a frequency of the oscillator the deformed creation and annihilation operators are finite difference ones the corresponding deformation of the heisenberg weyl group and new coherent states are also considered

the covariant linear oscillator and generalized realization of the dynamical su 1 1 symmetry algebra an exactly solvable problem for the finite difference schrodinger equation in the relativistic configurational space is considered the appropriate finite difference generalization of the factorization method is developed the theory of new special functions the relativistic hermite polynomials in which the solutions are expressed is constructed

recovery of a sparse spike time series by l sub 1 norm deconvolution an l sub 1 norm minimization scheme is applied to the determination of the impulse response vector h of flaws detected in practical examples of ultrasonic nondestructive evaluation in candu nuclear reactors for each problem parametric programming is applied to find the optimum value of the damping parameter that will yield the best estimate of h according to a quantified performance factor this performance factor is based on a quantified analysis of the transitions in estimates of h as the damping parameter is varied over a wide range of possible values it is shown that for the examined cases in which the true impulse response is a sparsely filled spike strain the l sub 1 norm provides significantly better results than the more commonly used l sub 2 norm minimization schemes these results are shown to be consistent with theoretical predictions

uq covariant oscillators and vertex operators uq sl 2 covariant bose and fermi oscillators are studied systematically and applied to constructing vertex operators and uq invariant blocks for qp 1 p 2 3 a p2 dimensional unitary fock space representation of the bose oscillator algebra a q is constructed this representation does not allow however for the construction of all uq invariant blocks associated with an indecomposable representation of uq an extension of a is needed in this case which involves operators creating states of q spin projection p 2 from the fock vacuum the extended algebra only has infinite dimensional indefinite metric space representations an algebraic notion of trace is introduced provided q2 1 in the bose case this allows the introduction of a family of in general mixed uq invariant temperature states such that the vacuum expectation value appears in the zero temperature limit

pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids part i global estimates this part contains new pointwise error estimates for the finite element method for second order elliptic boundary value problems on smooth bounded domains in r n in a sense to be discussed below these sharpen known quasi optimal l and w 1 estimates for the error on irregular quasi uniform meshes in that they indicate a more local dependence of the error at a point on the derivatives of the solution u we note that in general the higher order finite element spaces exhibit more local behavior than lower order spaces as a consequence of these estimates new types of error expansions will be derived which are in the form of inequalities these expansion inequalities are valid for large classes of finite elements defined on irregular grids in r n and have applications to superconvergence and extrapolation and a posteriori estimates part ii of this series will contain local estimates applicable to non smooth problems

on the discretization in time of semilinear parabolic equations with nonsmooth initial data single step discretization methods are considered for equations of the form u au f t u where a is a linear positive definite operator in a hubert space h it is shown that if the method is consistent with the differential equation then the convergence is essentially of first order in the stepsize even if the initial data v are only in h but also that in contrast to the situation in the linear homogeneous case higher order convergence is not possible in general without further assumptions on v

the algebra of two dimensional generalized chebyshev koornwinder oscillator in the previous works of borzov and damaskinsky chebyshev koornwinder oscillator theor math phys 175 3 765 772 2013 and ladder operators for chebyshev koornwinder oscillator in proceedings of the days on diffraction 2013 the authors have defined the oscillator like system that is associated with the two variable chebyshev koornwinder polynomials we call this system the generalized chebyshev koornwinder oscillator in this paper we study the properties of infinite dimensional lie algebra that is analogous to the heisenberg algebra for the chebyshev koornwinder oscillator we construct the exact irreducible representation of this algebra in a hilbert space of functions that are defined on a region which is bounded by the steiner hypocycloid the functions are square integrable with respect to the orthogonality measure for the chebyshev koornwinder polynomials and these polynomials form an orthonormalized basis in the space the generalized oscillator which is studied in the work can be considered as the simplest nontrivial example of multiboson quantum system that is composed of three interacting oscillators h h

high resolution three term avo inversion by means of a trivariate cauchy probability distribution three term avo inversion can be used to estimate p wave velocity s wave velocity and density perturbations from reflection seismic data the density term however exhibits little sensitivity to amplitudes and therefore its inversion is unstable one way to stabilize the density term is by including a scale matrix that provides correlation information between the three unknown avo parameters we investigate a bayesian procedure to include sparsity and a scale matrix in the three term avo inversion problem to this end we model the prior distribution of the avo parameters via a trivariate cauchy distribution we found an iterative algorithm to solve the bayesian inversion and in addition comparisons are provided with the classical inversion approach that uses a multivariate gaussian prior it is important to point out that the multivariate gaussian prior allows us to include the correlation of the avo parameters in the solution of the inverse problem the trivariate cauchy prior not only permits us

numerical solution of the obstacle problem by the penalty method part ii time dependent problems in a previous paper we considered the possibility of using the penalty method to get approximations of the solution of elliptic variational inequalities in this note we show that the same idea can be used for the treatment of parabolic problems by coupling the penalty parameter e and the discretization parametersh and t quasi optimal error estimates are derived in suitable norms

applications of discrete maximal l p regularity for finite element operators in this paper we present applications of discrete maximal lp regularity for finite element operators more precisely we show error estimates of order h2 for linear and certain semilinear problems in various lp norms discrete maximal regularity allows us to prove error estimates in a very easy and efficient way moreover we also develop interpolation theory for fractional powers of finite element operators and extend the results on discrete maximal lp regularity formerly proved by the author

stability and accuracy of adapted finite element methods for singularly perturbed problems the stability and accuracy of a standard finite element method fem and a new streamline diffusion finite element method sdfem are studied in this paper for a one dimensional singularly perturbed connvection diffusion problem discretized on arbitrary grids both schemes are proven to produce stable and accurate approximations provided that the underlying grid is properly adapted to capture the singularity often in the form of boundary layers of the solution surprisingly the accuracy of the standard fem is shown to depend crucially on the uniformity of the grid away from the singularity in other words the accuracy of the adapted approximation is very sensitive to the perturbation of grid points in the region where the solution is smooth but in contrast it is robust with respect to perturbation of properly adapted grid inside the boundary layer motivated by this discovery a new sdfem is developed based on a special choice of the stabilization bubble function the new method is shown to have an optimal maximum norm stability and approximation property in the sense that u u n infty leq c inf v n in v n u v n infty where un is the sdfem approximation in linear finite element space vn of the exact solution u finally several optimal convergence results for the standard fem and the new sdfem are obtained and an open question about the optimal choice of the monitor function for the moving grid method is answered

a sparsity regularization approach to the electromagnetic inverse scattering problem we investigate solving the electromagnetic inverse scattering problem using the distorted born iterative method dbim in conjunction with a variable selection approach known as the elastic net the elastic net applies both l 1 and l 2 penalties to regularize the system of linear equations that result at each iteration of the dbim the elastic net thus incorporates both the stabilizing effect of the l 2 penalty with the sparsity encouraging effect of the l 1 penalty the dbim with the elastic net outperforms the commonly used l 2 regularizer when the unknown distribution of dielectric properties is sparse in a known set of basis functions we consider two very different 3 d examples to demonstrate the efficacy and applicability of our approach for both examples we use a scalar approximation in the inverse solution in the first example the actual distribution of dielectric properties is exactly sparse in a set of 3 d wavelets the performances of the elastic net and l 2 approaches are compared to the ideal case where it is known a priori which wavelets are involved in the true solution the second example comes from the area of microwave imaging for breast cancer detection for a given set of 3 d gaussian basis functions we show that the elastic net approach can produce a more accurate estimate of the distribution of dielectric properties in particular the effective conductivity within an anatomically realistic 3 d numerical breast phantom in contrast the dbim with an l 2 penalty produces an estimate which suffers from multiple artifacts

finite element approximation of a semilinear elliptic problem with a singular nonlinearity given nu in bbb r we consider the following problem find u 0 such that where omega subset bbb r d d 1 2 or 3 and c 0 in bar omega we prove h 1 and l infty error bounds for the standard continuous piecewise linear galerkin finite element approximation with a weakly acute triangulation our bounds are nearly optimal in addition for d 1 and 2 and c in bbb r we analyze a more practical scheme involving numerical integration on the nonlinear term we obtain nearly optimal h 1 and l infty error bounds for d 1 for this case we also present some numerical results

spiking band limited traces with a relative entropy algorithm a novel method for the inversion of band limited seismic traces to full bandwidth reflectivity traces is based on a probabilistic spiky model of the reflectivity trace in which position indicators and amplitudes of the spikes occur as random variables and relies on relative entropy inference from information theory first an a priori model for general reflectivity traces in the prospect is derived from nearby wells second the a priori distribution is updated into an a posteriori distribution for the specific trace being studied by the addition of the fourier data of the seismic trace within a passband uncertainty about the fourier coefficients can be accounted for by specification of a noise variance which implicitly is infinite outside the passband the update with relative entropy inference is justified because of its relationship with bayesian inference application of maximum a posteriori map estimation to the a posteriori distribution results in the most likely spiky reflectivity trace of f

the clebsch gordan coefficients of the three dimensional lorentz algebra in the parabolic basis starting from the oscillator representation of the three dimensional lorentz algebra so 2 1 we build a lie algebra of second order differential operators which realizes all series of self adjoint irreducible representations the choice of the common self adjoint extention over a two chart function space determines whether they lead to single or multivalued representations over the corresponding lie group the diagonal operator defining the basis is the parabolic subgroup generator the direct product of two such algebras allows for the calculation of all clebsch gordan coefficients explicitly as solutions of schrodinger equations for poschl teller potentials over one d d two d c or three c c charts all coefficients are given in terms of up to two 2f1 hypergeometric functions

pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids part ii interior estimates this part contains new interior pointwise error estimates for the finite element method for second order elliptic problems in mathbb r n global estimates were considered in part i in the sense to be discussed below these sharpen known interior quasi optimal l infty and w 1 infty estimates in that they indicate a more local dependence of the error at a point on the derivatives of the solution near the point the higher the order of the finite element the more local the behavior of the finite element approximation as a consequence of these estimates new types of local error expansions will be derived which are in the form of inequalities these expansion inequalities are valid for large classes of finite elements defined on irregular grids in mathbb r n and have applications to superconvergence extrapolation and a posteriori estimates for both smooth and nonsmooth problems

linearly and nonlinearly transforming fields on homogeneous spaces of the 4 1 de sitter group scalar functions on the homogeneous spaces hr of the de sitter group g so 4 1 are studied where the spaces hr are of the form g k with k being a subgroup of the lorentz group h so 3 1 contained in so 4 1 the spaces hr can be regarded as fiber bundles er er g h h k with the base v 4 g h being a space of constant negative curvature characterized by a fundamental length parameter r 4 1 de sitter space and the fiber s h k being a homogeneous space of the lorentz group the action of g on the spaces er is a linear action on v4 and a nonlinear action on s where the latter action is defined by a generalized wigner rotation a gauge theory based on the 4 1 de sitter group is investigated with matter represented in terms of a generalized wave function x y with x u4 riemann cartan space time v 4 and y s which is defined as a map from a cross section on the bundle e e u4 f er g so 4 1 over space time u4 with fiber f er g k and structural group g so 4 1 into the complex numbers th

on the rogers szego polynomials an orthogonality relation on the full real line for the rogers szego polynomials is discussed it is argued that fourier transformation with the standard exponential kernel exp ixy relates the rogers szego and stieltjes wigert functions

towards a mathematical theory of super resolution this paper develops a mathematical theory of super resolution broadly speaking super resolution is the problem of recovering the fine details of an object the high end of its spectrum from coarse scale information only from samples at the low end of the spectrum suppose we have many point sources at unknown locations in 0 1 and with unknown complex valued amplitudes we only observe fourier samples of this object up to a frequency cutoff fc we show that one can super resolve these point sources with infinite precision i e recover the exact locations and amplitudes by solving a simple convex optimization problem which can essentially be reformulated as a semidefinite program this holds provided that the distance between sources is at least 2 fc this result extends to higher dimensions and other models in one dimension for instance it is possible to recover a piecewise smooth function by resolving the discontinuity points with infinite precision as well we also show that the theory and methods are robust to noise in particular in the discrete setting we develop some theoretical results explaining how the accuracy of the super resolved signal is expected to degrade when both the noise level and the super resolution factor vary 2014 wiley periodicals inc

effect of numerical integration for elliptic obstacle problems an elliptic obstacle problem is approximated by piecewise linear finite elements with numerical integration on the penalty and forcing terms this leads to diagonal nonlinearities and thereby to a practical scheme optimal error estimates in the maximum norm are derived the proof is based on constructing suitable super and subsolutions that exploit the special structure of the penalization and using quite precise pointwise error estimates for an associated linear elliptic problem with quadrature via the discrete maximum principle

automatic phase correction of common midpoint stacked data the residual wavelet on a processed seismic section is often not zero phase despite all efforts to make it so in this paper we adopt the convolutional model for the processed seismogram assume that the residual phase shift can be approximated by a frequency independent constant and use the varimax norm to generate an algorithm to estimate the residual phase directly application of our algorithm to reflectivities from well logs suggests that it should work in the majority of cases so long as the reflectivity is non gaussian an application of our algorithm to stacked data enhances the interpretability of the seismic section and leads to an improved match between the recovered relative acoustic impedance and a measured velocity log

linear finite elements may be only first order pointwise accurate on anisotropic triangulations we give a counterexample of an anisotropic triangulation on which the exact solution has a second order error of linear interpolation while the computed solution obtained using linear finite elements is only first order pointwise accurate our example is given in the context of a singularly perturbed reaction diffusion equation whose exact solution exhibits a sharp boundary layer furthermore we give a theoretical justification of the observed numerical phenomena using a finite difference representation of the considered finite element methods both standard and lumped mass cases are addressed

minimum entropy deconvolution with frequency domain constraints a method for reconstructing the reflectivity spectrum using the minimum entropy criterion is presented the algorithm fmed described is compared with the classical minimum entropy deconvolution med as well as with the linear programming lp and autoregressive ar approaches the med is performed by maximizing an entropy norm with respect to the coefficients of a linear operator that deconvolves the seismic trace by comparison the approach presented here maximizes the norm with respect to the missing frequencies of the reflectivity series spectrum this procedure reduces to a nonlinear algorithm that is able to carry out the deconvolution of band limited data avoiding the inherent limitations of linear operators the proposed method is illustrated under a variety of synthetic examples field data are also used to test the algorithm the results show that the proposed method is an effective way to process band limited data the fmed and the lp arise from similar conceptions both methods seek an extremum of a particular norm subjected to frequency constraints in the lp approach the linear programming problem is solved using an adaptation of the simplex method which is a very expensive procedure the fmed uses only two fast fourier transforms ffts per iteration hence the computational cost of the inversion is reduced

finite element approximations for the stokes equations on curved domains and their errors we construct finite element spaces for the stokes equations on curved domains which satisfy the email protected ka condition moreover we estimate the errors of the finite element approximations for the stokes equations

convergence analysis of finite element method for a parabolic obstacle problem abstract a conforming finite element method is proposed and analyzed for numerical approximation of the solution of a parabolic variational inequality of the obstacle problem the model problem which is originally proposed using a general obstacle is reframed as a model problem with zero obstacle but with non homogeneous dirichlet boundary conditions subsequently the discrete problem is reframed and the error analysis proving the convergence of the method is performed the analysis requires a positive preserving interpolation with non homogeneous dirichlet boundary condition and a post processed solution that satisfies the boundary conditions sharply the results in the article extend the results of johnson sinum 1976 for a zero obstacle function to a more general obstacle function

from finite differences to finite elements a short history of numerical analysis of partial differential equations abstract this is an account of the history of numerical analysis of partial differential equations starting with the 1928 paper of courant friedrichs and lewy and proceeding with the development of first finite difference and then finite element methods the emphasis is on mathematical aspects such as stability and convergence analysis