fully discrete finite element approximations of the forced fisher equation abstract we consider fully discrete finite element approximations of the forced fisher equation that models the dynamics of gene selection migration for a diploid population with two available alleles in a multidimensional habitat and in the presence of an artificially introduced genotype finite element methods are used to effect spatial discretization and a nonstandard backward euler method is used for the time discretization error estimates for the fully discrete approximations are derived by applying the brezzi rappaz raviart theory for the approximation of a class of nonlinear problems the approximation schemes and error estimates are applicable under weaker regularity hypotheses than those that are typically assumed in the literature the algorithms and analyses although presented in the concrete setting of the forced fisher equation also apply to a wide class of semilinear parabolic partial differential equations

remarks on the kuramoto sivashinsky equation we report here a joint work in progress on the kuramoto sivashinsky equation the question we address is the analytical study of a fourth order nonlinear evolution equation this equation has been obtained by sivashinsky in the context of combustion and independently by kuramoto in the context of reaction diffusion systems both were motivated by nonlinear stability of travelling waves numerical calculations have been done on this equation all the results seem to indicate a chaotic behavior of the solution therefore the analytical study is of interest in analogy with the burger s and navier stokes equations here we give some existence and uniqueness results for the equation in space dimension one and we also study a fractional step method of numerical resolution in a forthcoming joint paper with r temam we will study the asymptotic behavior as t approaches infinity of the solution of 0 1 and give an estimate on the number of determining modes

arbitrary high order c0 tensor product galerkin finite element methods for the electromagnetic scattering from a large cavity the paper is concerned with the electromagnetic scattering from a large cavity embedded in an infinite ground plane the electromagnetic cavity problem is described by the helmholtz equation with a nonlocal boundary condition on the aperture of the cavity and dirichlet or neumann boundary conditions on the walls of the cavity a tensor product galerkin finite element method fem is proposed in which spaces of c 0 piecewise polynomials of degree k 1 are employed by the fast fourier transform and the toeplitz type structure of the approximation to the nonlocal operator in the nonlocal boundary condition a fast algorithm is designed for solving the linear system arising from the cavity problem with vertically layered media which requires o n 2logn operations on an nxn uniform partition numerical results for model problems illustrate the efficiency of the fast algorithm and exhibit the expected optimal global convergence rates of the finite element methods moreover our numerical results also show that the high order approximations are especially effective for problems with large wave numbers

full waveform inversion and the truncated newton method full waveform inversion fwi is a powerful method for reconstructing subsurface parameters from local measurements of the seismic wavefield this method consists in minimizing the distance between

legendre spectral galerkin method for electromagnetic scattering from large cavities the paper is concerned with the electromagnetic scattering from a large cavity embedded in an infinite ground plane which is governed by a helmholtz type equation with nonlocal hypersingular transparent boundary condition on the aperture we first present some stability estimates with the explicit dependency of wavenumber for the helmholtz type cavity problem then a legendre spectral galerkin method is proposed in which the legendre gauss interpolatory approximation is applicable to the hypersingular integral and a legendre galerkin scheme is used for the approximation to the helmholtz equation the existence and the uniqueness of the approximation solution are established for large wavenumbers the stability and the spectral convergence of the numerical method are then proved illustrative numerical results presented confirm our theoretical estimates and show that the proposed spectral method compared with low order finite difference methods is especially effective for problems with large wavenumbers

multiple emitter location and signal parameter estimation processing the signals received on an array of sensors for the location of the emitter is of great enough interest to have been treated under many special case assumptions the general problem considers sensors with arbitrary locations and arbitrary directional characteristics gain phase polarization in a noise interference environment of arbitrary covariance matrix this report is concerned first with the multiple emitter aspect of this problem and second with the generality of solution a description is given of the multiple signal classification music algorithm which provides asymptotically unbiased estimates of 1 number of incident wavefronts present 2 directions of arrival doa or emitter locations 3 strengths and cross correlations among the incident waveforms 4 noise interference strength examples and comparisons with methods based on maximum likelihood ml and maximum entropy me as well as conventional beamforming are included an example of its use as a multiple frequency estimator operating on time series is included

convergence of the peaceman rachford approximation for reaction diffusion systems consider a reaction diffusion system of the form u t m u f u 0 where m is a m m matrix whose spectrum is included in z 0 we approximate it by the peaceman rachford approximation defined by p t 1 tf 2 1 1 tm 2 1 tm 2 1 1 tf 2 we prove convergence of this scheme and show that it is of order two

an efficient and long time accurate third order algorithm for the stokes darcy system a third order in time numerical imex type algorithm for the stokes darcy system for flows in fluid saturated karst aquifers is proposed and analyzed a novel third order adams moulton scheme is used for the discretization of the dissipative term whereas a third order explicit adams bashforth scheme is used for the time discretization of the interface term that couples the stokes and darcy components the scheme is efficient in the sense that one needs to solve at each time step decoupled stokes and darcy problems therefore legacy stokes and darcy solvers can be applied in parallel the scheme is also unconditionally stable and with a mild time step restriction long time accurate in the sense that the error is bounded uniformly in time numerical experiments are used to illustrate the theoretical results to the authors knowledge the novel algorithm is the first third order accurate numerical scheme for the stokes darcy system possessing its favorable efficiency stability and accuracy properties

semi implicit time discretization schemes for the bidomain model the bidomain model is a system of partial differential equations used to model the propagation of electrical potential waves in the myocardium it is composed of coupled parabolic and elliptic partial differential equations as well as at least one ordinary differential equation to model the ion activity through the cardiac cell membranes the purpose of this paper is to propose and analyze several implicit semi implicit and explicit time stepping methods to solve that model in particular to avoid the expensive resolution of a nonlinear system through the newton raphson method we identify necessary stability conditions on the time step delta t for the proposed methods through a theoretical analysis based on energy estimates we next compare the methods for one and two dimensional test cases in terms of both stability and accuracy of the numerical solutions the theoretical stability conditions are seen to be consistent with those observed in practice our analysis allows us to recommend using either the crank nicolson adams bashforth method or the second order semi implicit backward differention method these semi implicit methods produce a good numerical solution unlike the explicit methods their stability does not depend on the spatial grid size and unlike the implicit methods they do not require the resolution of a system of nonlinear equations

numerical resolution of an electromagnetic inverse medium problem at fixed frequency the aim of this paper is to solve numerically the inverse problem of determining the complex refractive index of an electromagnetic medium from partial boundary field measurements at a fixed frequency the governing equations are the time harmonic maxwell equations formulated in electric field in a two dimensional bounded domain we express the inverse problem as the minimization of a cost function representing the difference between the measured and predicted fields our numerical reconstruction algorithm combines the bfgs method and an iterative process called the adaptive eigenspace inversion the unknown complex coefficient is expanded in terms of eigenfunctions of an elliptic operator both the eigenspace and the mesh are iteratively adapted during the minimization procedure numerical experiments illustrate the performance of the reconstruction for various configurations

discontinuous galerkin methods for the helmholtz equation with large wave number this paper develops and analyzes some interior penalty discontinuous galerkin ipdg methods using piecewise linear polynomials for the helmholtz equation with the first order absorbing boundary condition in two and three dimensions it is proved that the proposed discontinuous galerkin methods are stable hence well posed without any mesh constraint for each fixed wave number k optimal order with respect to h error estimate in the broken h 1 norm and suboptimal order estimate in the l 2 norm are derived without any mesh constraint the latter estimate improves to optimal order when the mesh size h is restricted to the preasymptotic regime i e k 2 h gtrsim 1 numerical experiments are also presented to gauge the theoretical result and to numerically examine the pollution effect with respect to k in the error bounds the novelties of the proposed ipdg methods include the following first the methods penalize not only the jumps of the function values across the element edges but also the jumps of the normal and tangential derivatives second the penalty parameters are taken as complex numbers of positive imaginary parts so essentially and practically no constraint is imposed on the penalty parameters since the helmholtz problem is a non hermitian and indefinite linear problem as expected the crucial and the most difficult part of the whole analysis is to establish the stability estimates i e a priori estimates for the numerical solutions to this end the cruxes of our analysis are to establish and to make use of a local version of the rellich identity for the laplacian and to mimic the stability analysis for the pde solutions given in p cummings analysis of finite element based numerical methods for acoustic waves elastic waves and fluid solid interactions in the frequency domain ph d thesis the university of tennessee knoxville tn 2001 p cummings and x feng math models methods appl sci 16 2006 pp 139 160 u hetmaniuk commun math sci 5 2007 pp 665 678

hierarchical generalized linear models we consider hierarchical generalized linear models which allow extra error components in the linear predictors of generalized linear models the distribution of these components is not restricted to be normal this allows a broader class of models which includes generalized linear mixed models we use a generalization of henderson s joint likelihood called a hierarchical or h likelihood for inferences from hierarchical generalized linear models this avoids the integration that is necessary when marginal likelihood is used under appropriate conditions maximizing the h likelihood gives fixed effect estimators that are asymptotically equivalent to those obtained from the use of marginal likelihood at the same time we obtain the random effect estimates that are asymptotically best unbiased predictors an adjusted profile h likelihood is shown to give the required generalization of restricted maximum likelihood for the estimation of dispersion components a scaled deviance test for the goodness of fit a model selection criterion for choosing between various dispersion models and a graphical method for checking the distributional assumption of random effects are proposed the ideas of quasi likelihood and extended quasi likelihood are generalized to the new class we give examples of the poisson gamma binomial beta and gamma inverse gamma hierarchical generalized linear models a resolution is proposed for the apparent difference between population averaged and subject specific models a unified framework is provided for viewing and extending many existing methods

a numerical study on the stability of a class of helmholtz problems this paper concerns the stability of a class of helmholtz problems in rectangular domains a well known application is the electromagnetic scattering from a rectangular cavity embedded in an infinite ground plane error analysis of numerical methods for cavity problems relies heavily on the stability estimates however it is extremely difficult to derive an optimal stability bound with the explicit dependency on wave numbers in this paper a high order finite element approximation is proposed for calculating the stability bound numerical experiments show that the stability depends strongly on wave numbers in extreme case and it is almost independent on the wave numbers in an average sense our numerical results also help to understand the stability of the multi frequency inverse problems

inversion of seismic reflection data in the acoustic approximation the nonlinear inverse problem for seismic reflection data is solved in the acoustic approximation the method is based on the generalized least squares criterion and it can handle errors in the data set and a priori information on the model multiply reflected energy is naturally taken into account as well as refracted energy or surface waves the inverse problem can be solved using an iterative algorithm which gives at each iteration updated values of bulk modulus density and time source function each step of the iterative algorithm essentially consists of a forward propagation of the actual sources in the current model and a forward propagation backward in time of the data residuals the correlation at each point of the space of the two fields thus obtained yields the corrections of the bulk modulus and density models this shows in particular that the general solution of the inverse problem can be attained by methods strongly related to the methods of migration of unstacked data and commerc

wavenumber explicit convergence analysis for galerkin discretizations of the helmholtz equation we develop a stability and convergence theory for a class of highly indefinite elliptic boundary value problems bvps by considering the helmholtz equation at high wavenumber k as our model problem the key element in this theory is a novel k explicit regularity theory for helmholtz bvps that is based on decomposing the solution into two parts the first part has the sobolev regularity properties expected of second order elliptic pdes but features k independent regularity constants the second part is an analytic function for which k explicit bounds for all derivatives are given this decomposition is worked out in detail for several types of bvps namely the helmholtz equation in bounded smooth domains or convex polygonal domains with robin boundary conditions and in exterior domains with dirichlet boundary conditions we present an error analysis for the classical hp version of the finite element method hp fem where the dependence on the mesh width h the approximation order p and the wavenumber k is given explicitly in particular under the assumption that the solution operator for helmholtz problems is polynomially bounded in k it is shown that quasi optimality is obtained under the conditions that kh p is sufficiently small and the polynomial degree p is at least o log k

frequency independent approach to calculate physical optics radiations with the quadratic concave phase variations in this work we develop the numerical steepest descent path nsdp method to calculate the physical optics po radiations with the quadratic concave phase variations with the surface integral equation method the physical optics po scattered fields are formulated and further reduced to the surface integrals the high frequency physical critical points contributions including the stationary phase points the boundary resonance points and the vertex points are comprehensively studied via the proposed nsdp method the key contributions of this work are twofold one is that together with the po integrals taking the quadratic parabolic and hyperbolic phase terms this work makes the nsdp theory be complete for treating the po integrals with quadratic phase variations another is that in order to illustrate the transition effect of the high frequency physical critical points in this work we consider and further extend the nsdp method to calculate the po integrals with the coalescence of the high frequency critical points numerical results for the highly oscillatory po integral with the coalescence of the critical points are given to verify the efficiency of the proposed nsdp method the nsdp method could achieve the frequency independent computational workload and error controllable accuracy in all the numerical experiments especially for the case of the coalescence of the high frequency critical points

a posteriori error estimates for the crank nicolson method for parabolic equations we derive optimal order a posteriori error estimates for time discretizations by both the crank nicolson and the crank nicolson galerkin methods for linear and nonlinear parabolic equations we examine both smooth and rough initial data our basic tool for deriving a posteriori estimates are second order crank nicolson reconstructions of the piecewise linear approximate solutions these functions satisfy two fundamental properties i they are explicitly computable and thus their difference to the numerical solution is controlled a posteriori and ii they lead to optimal order residuals as well as to appropriate pointwise representations of the error equation of the same form as the underlying evolution equation the resulting estimators are shown to be of optimal order by deriving upper and lower bounds for them depending only on the discretization parameters and the data of our problem as a consequence we provide alternative proofs for known a priori rates of convergence for the crank nicolson method

a θ stable approximations of abstract cauchy problems we study the approximation of linear parabolic cauchy problems by means of galerkin methods in space anda stable multistep schemes of arbitrary order in time the error is evaluated in the norm ofl t 2 h x 1 l t l x 2

convergence analysis for finite element discretizations of the helmholtz equation with dirichlet to neumann boundary conditions a rigorous convergence theory for galerkin methods for a model helmholtz problem in bbb r d d in 1 2 3 is presented general conditions on the approximation properties of the approximation space are stated that ensure quasi optimality of the method as an application of the general theory a full error analysis of the classical hp version of the finite element method is presented for the model problem where the dependence on the mesh width h the approximation order p and the wave number k is given explicitly in particular it is shown that quasi optimality is obtained under the conditions that kh p is sufficiently small and the polynomial degree p is at least o log k

assessing differential gene expression with small sample sizes in oligonucleotide arrays using a mean variance model the identification of the genes that are differentially expressed in two sample microarray experiments remains a difficult problem when the number of arrays is very small we discuss the implications of using ordinary t statistics and examine other commonly used variants for oligonucleotide arrays with multiple probes per gene we introduce a simple model relating the mean and variance of expression possibly with gene specific random effects parameter estimates from the model have natural shrinkage properties that guard against inappropriately small variance estimates and the model is used to obtain a differential expression statistic a limiting value to the positive false discovery rate pfdr for ordinary t tests provides motivation for our use of the data structure to improve variance estimates our approach performs well compared to other proposed approaches in terms of the false discovery rate

empirical bayes analysis of a microarray experiment microarrays are a novel technology that facilitates the simultaneous measurement of thousands of gene expression levels a typical microarray experiment can produce millions of data points raising serious problems of data reduction and simultaneous inference we consider one such experiment in which oligonucleotide arrays were employed to assess the genetic effects of ionizing radiation on seven thousand human genes a simple nonparametric empirical bayes model is introduced which is used to guide the efficient reduction of the data to a single summary statistic per gene and also to make simultaneous inferences concerning which genes were affected by the radiation although our focus is on one specific experiment the proposed methods can be applied quite generally the empirical bayes inferences are closely related to the frequentist false discovery rate fdr criterion

stability estimates and reconstructions in inverse acoustic scattering using singular sources abstract the problem of stability for the reconstruction of the scattered field us in the exterior of a scatterer d from far field data u and for the reconstruction of a sound soft or sound hard scatterer d from the far field pattern u x d x d for all incident plane waves is investigated in particular we show how stability estimates are linked to the behaviour of the scattered field s for incident point sources and to special minimum norm solutions for the herglotz wave operator the results can also be used to formulate a method of singular sources for the reconstruction of d which is new in inverse scattering the reconstruction method is described and numerical examples are provided

near field imaging of infinite rough surfaces this paper is concerned with an inverse rough surface scattering problem in near field optical imaging which is to reconstruct the scattering surface with a resolution beyond the diffraction limit the surface is assumed to be a small and smooth deformation of a plane surface based on a transformed field expansion the boundary value problem with complex scattering surface is converted into a successive sequence of two point boundary value problems in the frequency domain where an analytic solution for the direct scattering problem is derived from the method of integrated solution by neglecting the high order terms in the power series expansion the nonlinear inverse problem is linearized and an explicit inversion formula is obtained a spectral cut off regularization is adopted to suppress the exponential growth of the noise in the evanescent wave components which carry high spatial frequency of the scattering surface and contribute to the superresolution in the near field regime the method works f

new resolution strategy for multiscale reaction waves using time operator splitting space adaptive multiresolution and dedicated high order implicit explicit time integrators we tackle the numerical simulation of reaction diffusion equations modeling multi scale reaction waves this type of problem induces peculiar difficulties and potentially large stiffness which stem from the broad spectrum of temporal scales in the nonlinear chemical source term as well as from the presence of steep spatial gradients in the reaction fronts spatially very localized in this paper we introduce a new resolution strategy based on time operator splitting and space adaptive multiresolution in the context of very localized and stiff reaction fronts the paper considers a high order implicit time integration of the reaction and an explicit one for the diffusion term in order to build a time operator splitting scheme that exploits efficiently the special features of each problem based on recent theoretical studies of numerical analysis such a strategy leads to a splitting time step which is restricted by neither the fastest scales in the source term nor by stability constraints of the diffusive steps but only by the physics of the phenomenon we aim thus at solving complete models including all time and space scales within a prescribed accuracy considering large simulation domains with conventional computing resources the efficiency is evaluated through the numerical simulation of configurations which were so far out of reach of standard methods in the field of nonlinear chemical dynamics for two dimensional spiral waves and three dimensional scroll waves as an illustration future extensions of the proposed strategy to more complex configurations involving other physical phenomena as well as optimization capability on new computer architectures are discussed

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

finite element analysis of electromagnetic scattering from a cavity a finite element method fem is implemented to compute the radar cross section of a two dimensional 2d cavity embedded in an infinite ground plane the method is based on the variational formulation which uses the fourier transform to couple the fields outside the cavity and those inside the cavity hence the scattering problem can be reduced to a bounded domain the convergence of the discrete finite element problem is analyzed numerical results are presented and compared with those obtained by the standard finite element green function method and by the 2d integral equation method

an incomplete assembly with thresholding algorithm for systems of reaction diffusion equations in three space dimensions iat for reaction diffusion systems solving systems of reaction diffusion equations in three space dimensions can be prohibitively expensive both in terms of storage and cpu time herein i present a new incomplete assembly procedure that is designed to reduce storage requirements incomplete assembly is analogous to incomplete factorization in that only a fixed number of nonzero entries are stored per row and a drop tolerance is used to discard small values the algorithm is incorporated in a finite element method of lines code and tested on a set of reaction diffusion systems the effect of incomplete assembly on cpu time and storage and on the performance of the temporal integrator daspk algebraic solver gmres and preconditioner ilut is studied

phase transitions and generalized motion by mean curvature we study the limiting behavior of solutions to appropriately rescaled versions of the allen cahn equation a simplified model for dynamic phase transitions we rigorously establish the existence in the limit of a phase antiphase interface evolving according to mean curvature motion this assertion is valid for all positive time the motion interpreted in the generalized sense of evans spruck and chen giga goto after the onset of geometric singularities

a multi frequency inverse source problem this paper is concerned with an inverse source problem that determines the source from measurements of the radiated fields away at multiple frequencies rigorous stability estimates are established when the background medium is homogeneous it is shown that the ill posedness of the inverse problem decreases as the frequency increases under some regularity assumptions on the source function it is further proven that by increasing the frequency the logarithmic stability converts to a linear one for the inverse source problem

linearly implicit methods for nonlinear parabolic equations we construct and analyze combinations of rational implicit and explicit multistep methods for nonlinear parabolic equations the resulting schemes are linearly implicit and include as particular cases implicit explicit multistep schemes as well as the combination of implicit runge kutta schemes and extrapolation an optimal condition for the stability constant is derived under which the schemes are locally stable we establish optimal order error estimates

a composite preconditioner for the electromagnetic scattering from a large cavity in this paper we propose a composite preconditioner for the electromagnetic scattering from a large cavity the electromagnetic cavity problem is described by the helmholtz equation with a nonlocal boundary condition on the aperture of the cavity and dirichlet or neumann boundary conditions on the walls of the cavity the preconditioner proposed here combines the optimal sine transform based approximation with a layered medium model using fast fourier transforms the computational cost of every iteration is o n 2logn on an nxn uniform partition of the unit square numerical results for a model problem show that the new preconditioner is more efficient than those recently considered in the literature for the cavity with a small portion of non layered media we propose a sparse preconditioned conjugate orthogonal conjugate gradient solver combined with the new preconditioner numerical results for a model problem are reported to demonstrate the efficiency of the sparse solver

unconditional superconvergence analysis of a new mixed finite element method for nonlinear sobolev equation in this paper a new mixed finite element scheme is proposed for the nonlinear sobolev equation by employing the finite element pair q11 q01 q10 based on the combination of interpolation and projection skill as well as the mean value technique the independent superclose results of the original variable u in h1 norm and the variable q a u u t b u u in l2 norm are deduced for the semi discrete and linearized fully discrete systems is the temporal partition parameter what s more the new interpolated postprocessing operators are put forward and the corresponding global superconvergence results are obtained unconditionally while previous literature always require certain time step conditions finally some numerical results are provided to confirm our theoretical analysis and show the efficiency of the method

computational study of the dispersively modified kuramoto sivashinsky equation we analyze and implement fully discrete schemes for periodic initial value problems for a general class of dispersively modified kuramoto sivashinsky equations time discretizations are constructed using linearly implicit schemes and spectral methods are used for the spatial discretization the general case analyzed covers several physical applications arising in multiphase hydrodynamics and the emerging dynamics arise from a competition of long wave instability negative diffusion short wave damping fourth order stabilization nonlinear saturation burgers nonlinearity and dispersive effects the solutions of such systems typically converge to compact absorbing sets of finite dimension i e global attractors and are characterized by chaotic behavior our objective is to employ schemes which capture faithfully these chaotic dynamics in the general case the dispersive term is taken to be a pseudodifferential operator which is allowed to have higher order than the familiar fourth order stabilizing term in the kuramoto sivashinsky equation in such instances we show that first and second order time stepping schemes are appropriate and provide convergence proofs for the schemes in physical situations when the dispersion is of lower order than the fourth order stabilization term for example a hybrid kuramoto sivashinsky korteweg devries equation also known as the kawahara equation in hydrodynamics higher order time stepping schemes can be used and we analyze and implement schemes of order six or less we derive optimal order error estimates throughout and utilize the schemes to compute the long time dynamics and to characterize the attractors various numerical diagnostic tools are implemented such as the projection of the infinite dimensional dynamics to one dimensional return maps that enable us to probe the geometry of the attractors quantitatively such results are possible only if computations are carried out for very long times we provide examples where integrations are carried out for 10 8 time units and it is shown that the schemes used here are very well suited for such tasks for illustration computations are carried out for third order dispersion the kawahara equation as well as fifth order dispersion the benney lin equation but the methods developed here are applicable for rather general dispersive terms with similar accuracy characteristics

imaging of local surface displacement on an infinite ground plane the multiple frequency case this paper is concerned with the inverse electromagnetic scattering from a perfectly conducting local surface displacement on an infinite ground plane by using multiple frequency data to image the profile of the surface displacement a stable and accurate reconstruction method is presented and investigated with numerical simulations an integral equation approach is also introduced for the solution of the forward scattering problem by recovering the main features at low frequencies and capturing the fine details at high frequencies the proposed method yields a stable and convergent reconstruction of the general profile in particular for a multiple scale profile the method resolves the fine scales with sufficiently high frequency information numerical examples are provided to illustrate the effectiveness of our approach

seismic waveform inversion in the frequency domain part 1 theory and verification in a physical scale model seismic waveforms contain much information that is ignored under standard processing schemes seismic waveform inversion seeks to use the full information content of the recorded wavefield in this paper i present apply and evaluate a frequency space domain approach to waveform inversion the method is a local descent algorithm that proceeds from a starting model to refine the model in order to reduce the waveform misfit between observed and model data the model data are computed using a full wave equation viscoacoustic frequency domain finite difference method ray asymptotics are avoided and higher order effects such as diffractions and multiple scattering are accounted for automatically the theory of frequency domain waveform wavefield inversion can be expressed compactly using a matrix formalism that uses finite difference finite element frequency domain modeling equations expressions for fast local descent inversion using back propagation techniques then follow naturally implementation of

bayesian models for gene expression with dna microarray data two of the critical issues that arise when examining dna microarray data are 1 determination of which genes best discriminate among the different types of tissue and 2 characterization of expression patterns in tumor tissues for 1 there are many genes that characterize dna expression and it is of critical importance to try and identify a small set of genes that best discriminate between normal and tumor tissues for 2 it is critical to be able to characterize the dna expression of the normal and tumor tissue samples and develop suitable models that explain patterns of dna expression for these types of tissues toward this goal we propose a novel bayesian model for analyzing dna microarray data and propose a model selection methodology for identifying subsets of genes that show different expression levels between normal and cancer tissues in addition we propose a novel class of hierarchical priors for the parameters that allow us to borrow strength across genes for making inference the pro

inexact interior point method for pde constrained nonlinear optimization starting from the inexact interior point framework from curtis schenk and wachter siam j sci comput 32 2012 pp 3447 3475 we propose an effective schur complement slack control preconditioner for the full lagrangian hessian matrix needed at each newton iteration together they yield a scalable robust and highly parallel method for the numerical solution of large scale nonconvex pde constrained optimization problems with inequality constraints because it uses the full hessian matrix modifying it whenever needed the method not only is globally convergent but also converges fast locally our preconditioner is not tailored to any particular class of pdes or constraints but instead judiciously exploits the sparsity structure of the hessian numerical examples from pde constrained optimal control parameter estimation and full waveform inversion demonstrate the robustness and efficiency of the method even in the presence of active inequality constraints

analysis of a spectral galerkin approximation to the helmholtz equation in exterior domains an error analysis is presented for the spectral galerkin method to the helmholtz equation in 2 and 3 dimensional exterior domains the problem in unbounded domains is first reduced to a problem on a bounded domain via the dirichlet to neumann operator and then a spectral galerkin method is employed to approximate the reduced problem the error analysis is based on exploring delicate asymptotic behaviors of the hankel functions and on deriving a priori estimates with explicit dependence on the wave number for both the continuous and the discrete problems explicit error bounds with respect to the wave number are derived and some illustrative numerical examples are also presented

a three level finite element approximation of a pattern formation model in developmental biology this paper concerns a second order three level piecewise linear finite element scheme 2 sbdf ruuth in j math biol 34 148 176 1995 for approximating the stationary turing patterns of a well known experimental substrate inhibition reaction diffusion thomas system thomas in analysis and control of immobilized enzyme systems pp 115 150 1975 a numerical analysis of the semi discrete in time approximations leads to semi discrete a priori bounds and an optimal error estimate the analysis highlights the technical challenges in undertaking the numerical analysis of multi level ge 3 3 schemes we illustrate the effectiveness of the numerical method by repeating an important classical experiment in mathematical biology namely to approximate the turing patterns of the thomas system over a schematic mammal skin domain with fixed geometry at various scales we also make some comments on the correct procedure for simulating turing patterns in general reaction diffusion systems

unconditional convergence and optimal error estimates of a galerkin mixed fem for incompressible miscible flow in porous media in this paper we study the unconditional convergence and error estimates of a galerkin mixed fem with the linearized semi implicit euler scheme for the equations of incompressible miscible flow in porous media we prove that the optimal l 2 error estimates hold without any time step convergence conditions while all previous works require certain time step restrictions our theoretical results provide a new understanding on commonly used linearized schemes the proof is based on a splitting of the error into two parts the error from the time discretization of the pdes and the error from the finite element discretization of corresponding time discrete pdes the approach used in this paper can be applied to more general nonlinear parabolic systems and many other linearized semi implicit time discretizations

implicit explicit multistep methods for nonlinear parabolic equations implicit explicit multistep methods for nonlinear parabolic equations were recently analyzed in 2 3 1 in these papers the linear operator of the equation is assumed time independent self adjoint and positive definite then the linear part is discretized implicitly and the remaining part explicitly here we slightly relax the hypotheses on the linear operator by allowing part of it to be time dependent or nonself adjoint we establish optimal order a priori error estimates

the superconvergence of newton cotes rules for the hadamard finite part integral on an interval we study the general composite newton cotes rules for the computation of hadamard finite part integral with the second order singularity and focus on their pointwise superconvergence phenomenon i e when the singular point coincides with some a priori known point the convergence rate is higher than what is globally possible we show that the superconvergence rate of the composite newton cotes rules occurs at the zeros of a special function and prove the existence of the superconvergence points several numerical examples are provided to validate the theoretical analysis

second order implicit explicit scheme for the gray scott model a second order scheme for the gray scott gs model used to describe the pattern formation is studied the linear part of the gs equation for the time derivative and the viscous terms is discretized implicitly while the other or nonlinear part of the gs equation explicitly galerkin finite element approximation methods are presented and analyzed as well as methods for solving the resulting system of equations the optimal l 2 norm error estimates are derived numerical experiments are presented

spectral approximation of the helmholtz equation with high wave numbers a complete error analysis is performed for the spectral galerkin approximation of a model helmholtz equation with high wave numbers the analysis presented in this paper does not rely on the explicit knowledge of continuous discrete green s functions and does not require any mesh condition to be satisfied furthermore new error estimates are also established for multi dimensional radial and spherical symmetric domains illustrative numerical results in agreement with the theoretical analysis are presented

optimal shape design of a cavity for radar cross section reduction this paper focuses on the reduction of radar cross section rcs for a cavity embedded in an infinite ground plane by introducing a transparent boundary condition the unbounded scattering problem is reduced to a bounded domain problem rcs reduction for the cavity is formulated as a shape optimization problem involving the helmholtz equation the existence of the minimizer is proved under an appropriate constraint descent directions of the objective function with respect to the boundary may be found via the domain derivative it is used in a gradient based optimization scheme to find the optimal shape of the cavity numerical examples show that the rcs is effectively reduced at different incident frequencies

numerical solutions of the generalized kuramoto sivashinsky equation by chebyshev spectral collocation methods chebyshev spectral collocation methods known as el gendi method s e el gendi chebyshev solution of differential integral and integro differential equations comput j 12 1969 282 287 b mihaila i mihaila numerical approximation using chebyshev polynomial expansions el gendi s method revisited j phys a 35 2002 731 746 are extended to deal with the generalized kuramoto sivashinsky equation the problem is reduced to a system of ordinary differential equations that are solved by combinations of backward differential formula and appropriate explicit schemes implicit explicit bdf methods g akrivis y s smyrlis implicit explicit bdf methods for the kuramoto sivashinsky equation appl numer math 51 2004 151 169 good numerical results have been obtained and compared with the exact solutions

stiffly accurate runge kutta methods for nonlinear evolution problems governed by a monotone operator stiy accurate implicit runge kutta methods are studied for the time discretisation of nonlinear rst order evolution equations the equation is supposed to be governed by a time dependent hemicontinuous operator that is up to a shift monotone and coercive and fullls a certain growth condi tion it is proven that the piecewise constant as well as the piecewise linear interpolant of the time discrete solution converge towards the exact weak solu tion provided the runge kutta method is consistent and satises a stability criterion that implies algebraic stability examples are the radau iia and lo batto iiic methods the convergence analysis is also extended to problems involving a strongly continuous perturbation of the monotone main part

adaptive eigenspace for multi parameter inverse scattering problems abstract a nonlinear optimization method is proposed for inverse scattering problems in the frequency domain when the unknown medium is characterized by one or several spatially varying parameters the time harmonic inverse medium problem is formulated as a pde constrained optimization problem and solved by an inexact truncated newton type method combined with frequency stepping instead of a grid based discrete representation each parameter is projected to a separate finite dimensional subspace which is iteratively adapted during the optimization each subspace is spanned by the first few eigenfunctions of a linearized regularization penalty functional chosen a priori the small and slowly increasing finite number of eigenfunctions effectively introduces regularization into the inversion and thus avoids the need for standard tikhonov type regularization numerical results illustrate the accuracy and efficiency of the resulting adaptive eigenspace regularization for single and multi parameter problems including the well known marmousi model from geosciences

efficient and long time accurate second order methods for the stokes darcy system we propose and study two second order in time implicit explicit methods for the coupled stokes darcy system that governs flows in karst aquifers and other subsurface flow systems the first method is a combination of a second order backward differentiation formula and the second order gear s extrapolation approach the second is a combination of the second order adams moulton and second order adams bashforth methods both algorithms only require the solution of decoupled stokes and darcy problems at each time step hence these schemes are very efficient and can be easily implemented using legacy codes we establish the unconditional and uniform in time stability for both schemes the uniform in time stability leads to uniform in time control of the error which is highly desirable for modeling physical processes e g contaminant sequestration and release that occur over very long time scales error estimates for fully discretized schemes using finite element spatial discretizations are derived nume

regularized seismic full waveform inversion with prior model information full waveform inversion fwi delivers high resolution quantitative images and is a promising technique to obtain macroscale physical property model of the subsurface in most geophysical applications prior information such as that collected in wells is available and should be used to increase the image reliability for this we propose to introduce three terms in the definition of the fwi misfit function the data misfit itself the first order tikhonov regularization term acting as a smoothing operator and a prior model norm term this last term is the way to smoothly introduce prior information into the fwi workflow on a selected target of the marmousi synthetic example significant improvement was obtained when using the prior model term for noise free and noisy synthetic data the prior model term may significantly reduce the inversion sensi tivity to incorrect initial conditions the limited range of spatial wavenumber sampling by the acquisition may be compensated with the prior model information for multiple free and multiplecontaminated data prior and initial models play different roles in the inversion scheme the starting model is used for wave propagation and therefore drives the data misfit gradient whereas the prior model is never explicitly used for solving the wave equation and only drives the optimization step as an additional constraint to minimize the total objective function thus the prior model is not required to follow kinematic properties as precisely as the initial model except in zones of poor illumination in addition we investigate the influence of a simple dynamic decreasing weighting of the prior model term once the cycle skipping problem has been solved the impact of the prior model term is gradually reduced within the misfit function to be driven by seismic data only

boundary integral formulation and semi implicit scheme coupling for modeling cells under electrical stimulation we model the electrical activity of biological cells under external stimuli via a novel boundary integral bi formulation together with a suitable time space numerical discretization scheme ionic channels follow a non linear dynamic behavior commonly described by systems of ordinary differential equations dependent on the electric potential jump across the membrane since potentials in both intra and extracellular domains satisfy an electrostatic approximation we represent them using solely dirichlet and neumann traces over the membrane via boundary potentials hence the volume problem is condensed to one posed over the cell boundary a second order time stepping semi implicit numerical galerkin scheme is proposed and analyzed wherein bi operators are approximated by low order basis functions with stability independent of space discretization

an analysis of discretizations of the helmholtz equation in l2 and in negative norms for a model helmholtz problem at high wavenumber k we present a wavenumber explicit error analysis in the l 2 and h 1 norms for the galerkin fem for the convergence in l 2 we show that the lowest order case p 1 is special in that the relative error in l 2 scales at best with k whereas it does not for higher order discretizations an alternative to the galerkin method with better dispersion properties is the optimally blended spectral finite element scheme of ainsworth and wajid 2010 for this method we present an error analysis in l 2 for the lowest order case p 1 in one dimension showing that the l 2 error is improved by a factor k compared to the lowest order galerkin fem

numerical solution of an inverse obstacle scattering problem with near field data consider the scattering of an arbitrary time harmonic incident wave by a sound soft obstacle in this paper a novel method is presented for solving the inverse obstacle scattering problem of the two dimensional helmholtz equation which is to reconstruct the obstacle surface by using the near field data the obstacle is assumed to be a small and smooth perturbation of a disc the method uses the transformed field expansion to reduce the boundary value problem into a successive sequence of one dimensional problems which are solved in closed forms by dropping the higher order terms in the power series expansion and truncating the infinite linear system for the first order term the inverse problem is linearized and an approximate but explicit formula is obtained between the fourier coefficients of the solution and data a nonlinear correction algorithm is introduced to improve the accuracy of the reconstructions for large deformations numerical examples show that the method is simple efficient and stable to reconstruct the obstacle with subwavelength resolution

the positive false discovery rate a bayesian interpretation and the q value multiple hypothesis testing is concerned with controlling the rate of false positives when testing several hypotheses simultaneously one multiple hypothesis testing error measure is the false discovery rate fdr which is loosely defined to be the expected proportion of false positives among all significant hypotheses the fdr is especially appropriate for exploratory analyses in which one is interested in finding several significant results among many tests in this work we introduce a modified version of the fdr called the positive false discovery rate pfdr we discuss the advantages and disadvantages of the pfdr and investigate its statistical properties when assuming the test statistics follow a mixture distribution we show that the pfdr can be written as a bayesian posterior probability and can be connected to classification theory these properties remain asymptotically true under fairly general conditions even under certain forms of dependence also a new quantity called the q value is introduced and investigated which is a natural bayesian posterior p value or rather the pfdr analogue of the p value 1 introduction when testing a single hypothesis one is usually concerned with controlling the false positive rate while maximizing the probability of detecting an effect when one really exists in statistical terms we maximize the power conditional on the type i error rate being at or below some level the field of multiple hypothesis testing tries to extend this basic paradigm to the situation where several hypotheses are tested simultaneously one must define an appropriate compound error measure according to the rate of false positives one is willing to encounter then a procedure is developed that allows one to control the error rate at a desired level while maintaining the power of each test as much as possible the most commonly controlled quantity when testing multiple hypotheses is the family wise error rate fwer which is the probability of yielding one or more false positives out of all hypotheses tested the most familiar example of this is the bonferroni method if there are m hypothesis tests each test is controlled so that the probability of a false positive is less than or equal to m for some chosen value of it then follows that the overall fwer is less than or equal to m any

controlling the false discovery rate a practical and powerful approach to multiple testing summary the common approach to the multiplicity problem calls for controlling the familywise error rate fwer this approach though has faults and we point out a few a different approach to problems of multiple significance testing is presented it calls for controlling the expected proportion of falsely rejected hypotheses the false discovery rate this error rate is equivalent to the fwer when all hypotheses are true but is smaller otherwise therefore in problems where the control of the false discovery rate rather than that of the fwer is desired there is potential for a gain in power a simple sequential bonferronitype procedure is proved to control the false discovery rate for independent test statistics and a simulation study shows that the gain in power is substantial the use of the new procedure and the appropriateness of the criterion are illustrated with examples

implicit explicit multistep methods for quasilinear parabolic equations efficient combinations of implicit and explicit multistep methods for nonlinear parabolic equations were recently studied in 1 in this note we present a refined analysis to allow more general nonlinearities the abstract theory is applied to a quasilinear parabolic equation

the dynamics and pinning of a spike for a reaction diffusion system the motion of a one spike solution to a simplified form of the gierer meinhardt activator inhibitor model is studied in both a one dimensional and a two dimensional domain the pinning effect on the spike motion associated with the presence of spatially varying coefficients in the differential operator referred to as precursor gradients is studied in detail in the one dimensional case we derive a differential equation for the trajectory of the spike in the limit e 0 where e is the activator diffusivity a similar differential equation is derived for the two dimensional problem in the limit for which e 1 and d 1 where d is the inhibitor diffusivity a numerical finite element method is presented to track the motion of the spike for the full problem in both one and two dimensions finally the numerical results for the spike motion are compared with corresponding asymptotic results for various examples

implicit explicit multistep finite element methods for nonlinear parabolic problems we approximate the solution of initial boundary value problems for nonlinear parabolic equations in space we discretize by finite element methods the discretization in time is based on linear multistep schemes one part of the equation is discretized implicitly and the other explicitly the resulting schemes are stable consistent and very efficient since their implementation requires at each time step the solution of a linear system with the same matrix for all time levels we derive optimal order error estimates the abstract results are applied to the kuramoto sivashinsky and the cahn hilliard equations in one dimension as well as to a class of reaction diffusion equations in r v v 2 3

linearly implicit methods for nonlinear evolution equations we construct and analyze combinations of rational implicit and explicit multistep methods for nonlinear evolution equations and extend thus recent results concerning the discretization of nonlinear parabolic equations the resulting schemes are linearly implicit and include as particular cases implicit explicit multistep schemes as well as the combination of implicit runge kutta schemes and extrapolation we establish optimal order error estimates the abstract results are applied to a third order evolution equation arising in the modelling of flow in a fluidized bed we discretize this equation in space by a petrov galerkin method the resulting fully discrete schemes require solving some linear systems to advance in time with coefficient matrices the same for all time levels

choosing the forcing terms in an inexact newton method an inexact newton method is a generalization of newton s method for solving f x 0 f mathbb r n to mathbb r n in which at the kth iteration the step s k from the current approximate solution x k is required to satisfy a condition f x k f x k s k leqslant eta k f x k for a forcing term eta k in 0 1 in typical applications the choice of the forcing terms is critical to the efficiency of the method and can affect robustness as well promising choices of the forcing terms are given their local convergence properties are analyzed and their practical performance is shown on a representative set of test problems