a proof of simon s theorem on piecewise testable languages abstract we give a new proof of the theorem of i simon that a language is piecewise testable if and only if it is recognized by a finite f trivial monoid our proof is based on representations by certain types of decreasing mappings

adaptive distributed parameter systems identification with enforceable identifiability conditions and reduced order spatial differentiation presents the synthesis of adaptive identifiers for distributed parameter systems dps with spatially varying parameters described by partial differential equations pdes of parabolic elliptic and hyperbolic type the features of the pde setting are utilized to obtain the not directly intuitive parameter estimation algorithms that use spatial derivatives of the output data with the order reduced from that of the highest spatial plant derivative the tunable identifier parameters are passed through the integrator block which forms their orthogonal expansions the latter are shown to be pointwise plant parameter estimates in this regard the approach of the paper is in the spirit of finite dimensional observer realization in integrating rather than differentiating the output data only applied to the spatial rather than temporal domain the constructively enforceable identifiability conditions formulated in terms of the sufficiently rich input signals referred to as generators of persistent excitation are shown to guarantee the existence of a unique zero steady state for the parameter errors under such inputs the tunable parameters in the adaptive identifiers proposed are shown to converge to plant parameters in l sub 2 and the orthogonal expansions of these tunable parameters pointwise

application of unsplit convolutional perfectly matched layer for scalar arbitrarily wide angle wave equation abstracta classical split perfectly matched layer pml method has recently been applied to the scalar arbitrarily wide angle wave equation awwe in terms of displacement however the classical split pml obviously increases computational cost and cannot efficiently absorb waves propagating into the absorbing layer at grazing incidence our goal was to improve the computational efficiency of awwe and to enhance the suppression of edge reflections by applying a convolutional pml cpml we reformulated the original awwe as a first order formulation and incorporated the cpml with a general complex frequency shifted stretching operator into the renewed formulation a staggered grid finite difference fd method was adopted to discretize the first order equation system for wavefield depth continuation the first order awwe with the cpml saved memory compared with the original second order awwe with the conventional split pml with the help of numerical examples we verified the correctness of the staggered

assessment of rod behaviour theories used in spectral finite element modelling abstract in this work different theories of rods have been discussed and compared the investigated theories are widely used in spectral finite element modelling of rod behaviour associated with propagation of symmetric longitudinal waves these are various single two mode and three mode theories including the elementary classical love and mindlin herrmann approaches as well as new two three and four mode theories proposed by the authors dispersion curves associated with each theory obtained by the use of hamilton s principle have been presented and discussed in the paper the investigation programme carried out by the authors aimed to show major differences and similarities between the rod theories and to discuss certain numerical aspects of their application great attention has been paid on properties limitations as well as difficulties associated with the use of the theories the results obtained from a wide program on numerical tests allowed the authors to draw certain general conclusions that are valid not only in the field of the spectral finite element method but also in the field of dynamics of engineering rod structures

wave equation processing using finite difference propagators part 1 wavefield dissection and imaging of marine multicomponent seismic data abstractmethods for wavefield injection are used in for instance reverse time extrapolation of shot gathers in reverse time migration for correct injection of recorded data without any ambiguity of the propagation direction the wavefield injection methodology requires pressure and particle velocity data such as multicomponent towed marine or seabed seismic recordings we discovered that by carefully considering the models medium parameters and boundary conditions for injection wavefield injection of multicomponent data can also be used to solve several long standing challenges in marine seismic data processing by means of conventional time space domain finite difference propagators we outlined and demonstrated several of these important applications including up down separation of wavefields deghosting direct wave removal source signature estimation multiple removal and imaging using primaries and multiples only acoustic models are considered but the concepts are straightforward to general

multidirectional vector based elastic reverse time migration and angle domain common image gathers with approximate wavefield decomposition of p and s waves abstractelastic reverse time migration e rtm has limitations when the migration velocities contain strong contrasts first the traditional scheme of p s wave mode separation is based on helmholtz s equations which ignore the conversion between p and s waves at the current separation time thus it contains an implicit assumption of the constant shear modulus and requires smoothing the heterogeneous model to approximately satisfy a locally constant condition second the vector based imaging condition needs to use the reflection image normal and it also cannot give the correct polarity of the pp image in all possible conditions third the angle domain common image gathers adcigs calculated using the poynting vectors pvs do not consider the wave interferences that happen at each reflector therefore smooth models are often used for e rtm we relax this condition by proposing an improved data flow that involves three new contributions the first contribution is an improved system of p s wave mode

direct adaptive control of parabolic systems algorithm synthesis and convergence and stability analysis this paper presents a model reference adaptive control of a class of distributed parameter systems described by linear n dimensional parabolic partial differential equations unknown parameters appearing in the system equation are either constant or spatially varying distributed sensing and actuation are assumed to be available adaptation laws are obtained by the lyapunov redesign method it is shown that the concept of persistency of excitation which guarantees the parameter error convergence to zero in finite dimensional adaptive systems in infinite dimensional adaptive systems should be investigated in relation to time variable spatial variable and also boundary conditions unlike the finite dimensional case in infinite dimensional adaptive systems even a constant input is shown to be persistently exciting in the sense that it guarantees the convergence of parameter errors to zero averaging theorems for two time scale systems which involve a finite dimensional slow system and an infinite dimensional fast system are developed the exponential stability of the adaptive system which is critical in finite dimensional adaptive control in terms of tolerating disturbances and unmodeled dynamics is shown by applying averaging

elastic wave propagation using fully vectorized high order finite difference algorithms two finite difference schemes for solving the elastic wave equation in heterogeneous two dimensional media are implemented on a vector computer a modified lax wendroff scheme that is second order accurate both in time and space and is a version of the maccormack scheme that is second order accurate in time and fourth order in space the algorithms are based on the matrix times vector by diagonals technique that is fully vectorized and is described using a novel notation for vector supercomputer operations the technique described can be implemented on a vector processor of modest dimensions and increase the applicability of finite differences the two difference operators are compared and the programs are tested for a simple case of standing sinusoidal waves for which the exact solution is known and also for a two layer model with a line source a comparison of the results for an actual well to well experiment verifies the usefulness of the two dimensional approach in modeling the results

mimetic seismic wave modeling including topography on deformed staggered grids abstractfinite difference methods for modeling seismic waves are known to be inaccurate when including a realistic topography due to the large dispersion errors that appear in the modelled surface waves and the scattering introduced by the staircase approximation to the topography as a consequence alternatives to finite difference methods have been proposed to circumvent these issues we present a new numerical scheme for 3d elastic wave propagation in the presence of strong topography this finite difference scheme is based upon a staggered grid of the lebedev type or fully staggered grid fsg it uses a grid deformation strategy to make a regular cartesian grid conform to a topographic surface in addition the scheme uses a mimetic approach to accurately solve the free surface condition and hence allows for a less restrictive grid spacing criterion in the computations the scheme can use high order operators for the spatial derivatives and obtain low dispersion results with as few as six points pe

grid dispersion and stability criteria of some common finite element methods for acoustic and elastic wave equations purely numerical methods based on finite element approximation of the acoustic or elastic wave equation are becoming increasingly popular for the generation of synthetic seismograms we present formulas for the grid dispersion and stability criteria for some popular finite element methods fem for wave propagation namely classical and spectral fem we develop an approach based on a generalized eigenvalue formulation to analyze the dispersive behavior of these fems for acoustic or elastic wave propagation that overcomes difficulties caused by irregular node spacing within the element and the use of high order polynomials as is the case for spectral fem analysis reveals that for spectral fem of order four or greater dispersion is less than 0 2 at four to five nodes per wavelength and dispersion is not angle dependent new results can be compared with grid dispersion results of some classical finite difference methods fdm used for acoustic or elastic wave propagation analysis reveals that fdm and classical fem require a larger sampling ratio than a spectral fem to obtain results with the same degree of accuracy the staggered grid fdm is an efficient scheme but the dispersion is angle dependent with larger values along the grid axes on the other hand spectral fem of order four or greater is isotropic with small dispersion making it attractive for simulations with long propagation times

sh wave propagation in heterogeneous media velocity stress finite difference method a new finite difference fd method is presented for modeling sh wave propagation in a generally heterogeneous medium this method uses both velocity and stress in a discrete grid density and shear modulus are similarly discretized avoiding any spatial smoothing therefore boundaries will be correctly modeled under an implicit formulation standard problems quarter plane propagation sedimentary basin propagation are studied to compare this method with other methods finally a more complex example a salt dome inside a two layered medium shows the effect of lateral propagation on seismograms recorded at the surface a corner wave always in phase with the incident wave and a head wave will appear which will pose severe problems of interpretation with the usual vertical migration methods

seismic wave synthesis by gaussian beam summation a comparison with finite differences we apply gaussian beam summation to the calcula tion of seismic reflections from complex interfaces introducing several modifications of the original method first we use local geographical coordinates for the representation of paraxial rays in the vicinity of the recording surface in this way we avoid the time consuming evaluation of the ray centered coordinates of the observation points second we propose a method for selecting the beams that ensures numerical stability of the synthetic seismograms third we introduce a simple source wave packet that simplifies and stabilizes the calculations of inverse fourier transforms we compare reflection seismograms computed using the gaussian beam summation method with those cal culated by finite differences two simple models are used the first is a continuous curved interface separ

going higher in first order quantifier alternation hierarchies on words we investigate quantifier alternation hierarchies in first order logic on finite words levels in these hierarchies are defined by counting the number of quantifier alternations in formulas we prove that one can decide membership of a regular language in the levels b 2 finite boolean combinations of formulas having only one alternation and 3 formulas having only two alternations and beginning with an existential block our proofs work by considering a deeper problem called separation which once solved for lower levels allows us to solve membership for higher levels

source independent full waveform inversion of seismic data a set of seismic trace data is collected in an input data set that is first fourier transformed in its entirety into the frequency domain a normalized wavefield is obtained for each trace of the input data set in the frequency domain normalization is done with respect to the frequency response of a reference trace selected from the set of seismic trace data the normalized wavefield is source independent complex and dimensionless the normalized wavefield is shown to be uniquely defined as the normalized impulse response provided that a certain condition is met for the source this property allows construction of the inversion algorithm disclosed herein without any source or source coupling information the algorithm minimizes the error between data normalized wavefield and the model normalized wavefield the methodology is applicable to any 3 d seismic problem and damping may be easily included in the process

semigroups and languages of dot depth two this paper is a contribution to the problem of effectively determining the dot depth of a star free language a problem in the theory of automata and formal languages with close connections to algebra and formal logic we conjecture an effective criterion for determining whether a given language has dot depth 2 the condition is shown to be necessary in general and sufficient for languages over a two letter alphabet the condition involves a novel use of categories in the study of semigroup theoretic problems

monoids respectingn chains of intervals abstract a new strict hierarchy for the pseudovariety of aperiodic monoids is described the n th level of the hierarchy is the pseudovariety of all divisors of monoids of full transformations of finite chains respecting some n chain of intervals the levels can be obtained by iterated semidirect product of the first level the associated hierarchy corresponding to taking partial transformations is also considered and appears intermingled with the above hierarchy

partially ordered finite monoids and a theorem of i simon abstract we use semigroup expansions to show that every finite j trivial monoid is a quotient of a finite monoid admitting a partial order that is compatible with multiplication as a result we obtain a radically new proof of the theorem of i simon characterizing the recognizable languages whose syntactic monoids are j trivial

elastic wavefield separation based on the helmholtz decomposition abstractdivergence and curl operators used for the decomposition of p and s wave modes in elastic reverse time migration rtm change the amplitudes units and phases of extrapolated wavefields i separate the p and s waves in elastic media based on the helmholtz decomposition the decomposed wavefields based on this approach have the same amplitudes units and phases as the extrapolated wavefields to avoid expensive multidimensional integrals in the helmholtz decomposition i introduce a fast poisson solver to efficiently solve the vector poisson s equation this fast algorithm allows us to reduce computational complexity from o n2 to o n log n where n is the total number of grid points because the decomposed p and s waves are vector fields i use vector imaging conditions to construct pp ps ss and sp images several 2d numerical examples demonstrate that this approach allows us to accurately and efficiently decompose p and s waves in elastic media in addition elastic rtm images based

fourth order finite difference p sv seismograms i describe the properties of a fourth order accurate space second order accurate time two dimensional p sk finite difference scheme based on the madariagavirieux staggered grid formulation the numerical scheme is developed from the first order system of hyperbolic elastic equations of motion and constitutive laws expressed in particle velocities and stresses the madariaga virieux staggered grid scheme has the desirable quality that it can correctly model any variation in material properties including both large and small poisson s ratio materials with minimal numerical dispersion and numerical anisotropy dispersion analysis indicates that the shortest wavelengths in the model need to be sampled at 5 gridpoints wavelength the scheme can be used to accurately simulate wave propagation in mixed acoustic elastic media making it ideal for modeling marine problems explicitly calculating both velocities and stresses makes it relatively simple to initiate a source at the free surface or within a layer and to satisfy free surface boundary conditions benchmark comparisons of finite difference and analytical solutions to lamb s problem are almost identical as are comparisons of finite difference and reflectivity solutions for elastic elastic and acoustic elastic layered models

eulerian gaussian beams for high frequency wave propagation we design an eulerian gaussian beam summation method for solving helmholtz equations in the high frequency regime the traditional gaussian beam summation method is based on lagrangian ray tracing and local ray centered coordinates we propose a new eulerian formulation of gaussian beam theory which adopts global cartesian coordinates level sets and liouville equations yielding uniformly distributed eulerian traveltimes and amplitudes in phase space simultaneously for multiple sources the time harmonic wavefield can be constructed by summing up gaussian beams with ingredients provided by the new eulerian formulation the conventional gaussian beam summation method can be derived from the proposed method there are three advantages of the new method 1 we have uniform resolution of ray distribution 2 we can obtain wavefields excited at different sources by varying only source locations in the summation formula 3 we can obtain wavefields excited at different frequencies by varying only frequencies in the summation formula numerical experiments indicate that the gaussian beam summation method yields accurate asymptotic wavefields even at caustics the new method may be used for seismic modeling and migration

classification of finite monoids the language approach abstract the concept of a variety of congruences is introduced and related to variety of languages and variety of monoids a systematic construction of increasingly complex varieties of congruences is presented this construction is powerful enough to generate all monoids containing solvable groups some hierarchies occurring through this process are shown to correspond to well known hierarchies of monoids thus indicating that our construction is natural from an algebraic point of view some problems that remain open are also discussed

time space domain mesh free finite difference based on least squares for 2d acoustic wave modeling abstractfinite difference fd methods approximate derivatives through a weighted summation of function values from neighboring nodes traditionally these neighboring nodes are assumed to be distributed regularly such as in square or rectangular lattices to improve geometric flexibility one option is to develop fd in a mesh free discretization in which scattered nodes can be placed suitably with respect to irregular boundaries or arbitrarily shaped anomalies without coordinate transformation or forming any triangles or tetrahedra etc these mesh free fds have had successful applications especially in computational geoscience however they are all space domain fd schemes in which fd coefficients are derived by approximating spatial derivatives individually in the space domain for acoustic wave modeling it has been proven that space domain fd methods normally have higher dispersion error than time space domain fd methods which determine fd coefficients by approximating the time space domain disp

identifiability of linear systems in hilbert spaces the identifiability problem is discussed for linear dynamical systems described by first and second order evolution equations in hilbert spaces the unknowns are the initial values and the operators appearing in the system equations and a number of identifiability conditions are established within the framework of linear operator theory these are applied to various classes of partial differential equations on bounded and unbounded spatial domains to obtain the constant and spatially varying parameter identifiability conditions for such systems

adaptive projection based observers and rm l _ 1 adaptive controllers for infinite dimensional systems with full state measurement adaptive observers using projection operator based parameter update laws are considered for a class of linear infinite dimensional systems with bounded input operator and full state measurement and subject to time varying matched uncertainties and disturbances the l1 adaptive control architecture introduced recently for finite dimensional plants to provide guaranteed transient performance via fast adaptation is then extended to this class using the proposed observers existence and uniqueness of solutions for the resulting closed loop system and uniform boundedness of the observation error are established first then provided certain assumptions on the plant transfer function and the solution of a lyapunov inequality hold uniform guaranteed transient performance bounds on the plant state and control signal under the l1 architecture are derived two examples satisfying the assumptions control of a heat equation and a wave equation are presented reference input tracking simulation results for the heat equation under the l1 adaptive control subject to time varying matched uncertainties and disturbances are presented in support of the theory

love wave waveform inversion in time domain for shallow shear wave velocity abstracthigh frequency surface wave techniques are widely used to estimate s wave velocity of near surface materials surface wave methods based on inversions of dispersion curves are only suitable to laterally homogeneous or smoothly laterally varying heterogeneous earth models due to the layered model assumption during calculation of dispersion curves waveform inversion directly fits the waveform of observed data and it can be applied to any kinds of earth models we have used the love wave waveform inversion in the time domain to estimate near surface s wave velocity we used the finite difference method as the forward modeling method the source effect was removed by the deconvolution technique which made our method independent of the source wavelet we defined the difference between the deconvolved observed and calculated waveform as the misfit function we divided the model into different sizes of blocks depending on the resolution of the love waves and we updated the s wave velocity of each blo

development of objective oriented groundwater models 2 robust experimental design 1 this paper continues the discussion in part 1 by considering the data collection strategy problem when the existing data are judged to be insufficient for constructing a reliable model designing an experiment for identifying a distributed parameter is very difficult because the identification of a more complex parameter structure requires more data moreover without knowing the sufficiency of a design finding an optimal design becomes meaningless these difficulties can be avoided if we turn to the construction of objective oriented models the identifiability of a distributed parameter as defined in this paper contains the reducibility of parameter structure sufficient conditions for this kind of identifiability are given when the structure error associated with a structure reduction is too large these conditions may not be satisfied no matter how much data are collected in this paper we formulate a new experimental design problem that consists of two objectives minimizing the cost and maximizing the information content with robustness and feasibility as constraints we develop an algorithm that can find a cost effective robust design for objective oriented parameter identification we also present a heuristic algorithm that can find a suboptimal design with less computational effort for real case studies the proposed methodology is used to design a pumping test for identifying a distributed hydraulic conductivity we verify the robustness of the obtained design by assuming that the true parameter may have continuous discrete random and fractured structures finally the presented procedure of constructing objective oriented models is described step by step

on line parameter estimation for infinite dimensional dynamical systems the on line or adaptive identification of parameters in abstract linear and nonlinear infinite dimensional dynamical systems is considered an estimator in the form of an infinite dimensional linear evolution system having the state and parameter estimates as its states is defined convergence of the state estimator is established via a lyapunov estimate the finite dimensional notion of a plant being sufficiently rich or persistently excited is extended to infinite dimensions convergence of the parameter estimates is established under the additional assumption that the plant is persistently excited a finite dimensional approximation theory is developed and convergence results are established numerical results for examples involving the estimation of both constant and functional parameters in one dimensional linear and nonlinear heat or diffusion equations and the estimation of stiffness and damping parameters in a one dimensional wave equation with kelvin voigt viscoelastic damping are presented

a constant fractional order viscoelastic wave equation and its numerical simulation scheme abstractefficient modeling schemes currently exist to handle the spatially variable order fractional laplacians in the fractional laplacian viscoacoustic wave equation the simplest approach is to change the spatially variable order fractional laplacians into a linear combination of several constant fractional order laplacians we generalize the constant fractional order scheme to a spatially variable fractional order viscoelastic wave equation and develop an almost equivalent constant fractional order viscoelastic wave equation our constant fractional order scheme avoids the simulation error introduced by directly averaging the spatially varying fractional order thus our scheme simulates seismic wave propagation in viscoelastic media with sharp q contrasts well the fast fourier transform is used in the approximation of the fractional laplacians which improves the spectral accuracy in space several simulation examples are performed to verify that the numerical solution of a homogeneous q model obtai

identifiability of piecewise constant conductivity in a heat conduction process we study the identification and identifiability problems for heat conduction in a nonhomogeneous rod the identifiability results are established for two different sets of observations given a sequence of distributed type observations the identifiability is proved for conductivities in a piecewise smooth class of functions in the case of observations taken at finitely many points the identifiability is established for piecewise constant conductivities such conductivities can be uniquely identified using the proposed marching algorithm

application of averaging method for integro differential equations to model reference adaptive control of parabolic systems abstract an averaging theorem for integro differential equations is applied to the convergence analysis of controller parameters of a model reference adaptive control algorithm for a class of parabolic partial differential equations with constant coefficients the stability of an adaptive control algorithm is proven as well

wave equation processing using finite difference propagators part 2 deghosting of marine hydrophone seismic data abstractwe have developed a new and simple method for deghosting of conventional hydrophone streamer data towed at arbitrary variable depths the method uses a time space domain finite difference fd solution to the wave equation with pressure field boundary conditions to predict and remove ghosts because it operates in the time domain our method is unaffected by any number of notches in the frequency spectrum of the data and therefore will deghost through notches apart from the acquired hydrophone data the method relies on the depth profile of the streamer recording the data beneath a sea surface with a known reflection coefficient as well as the propagation velocity in the water above the streamer the method was applied to simple and more complex synthetic data which illustrated its ability to deal with complex data and any acquisition geometry

on the persistence of excitation in the adaptive estimation of distributed parameter systems persistence of excitation is a sufficient condition for parameter convergence in adaptive identification schemes for dynamical systems for abstract parabolic and hyperbolic distributed parameter systems this condition requires that a family of bounded linear functionals be norm bounded away from zero the level of persistence of excitation of the plant and its implications are considered for a simple parabolic and hyperbolic system its effect on the qualitative and quantitative behavior of the estimators is investigated

stable and accurate wave propagation in discontinuous media a time stable discretization is derived for the second order wave equation with discontinuous coefficients the discontinuity corresponds to inhomogeneity in the underlying medium and is treated by splitting the domain each homogeneous sub domain is discretized using narrow diagonal summation by parts operators and then patched to its neighbors by using a penalty method leading to fully explicit time integration this discretization yields a time stable and efficient scheme the analysis is verified by numerical simulations in one dimension using high order finite difference discretizations and in three dimensions using an unstructured finite volume discretization

asymptotic behavior analysis of a coupled time varying system application to adaptive systems asymptotic behavior of a partial state of a coupled ordinary and or partial differential equation is investigated it is specifically shown that if a signal x t is a solution to a dynamic system existing for all t spl ges 0 in an abstract banach space and pth p spl ges 1 power integrable then x t spl rarr 0 as t spl rarr spl infin the system is allowed to be nonautonomous and assumes the existence of a lyapunov function since the derivative of the lyapunov function is negative semidefinite stability or uniform stability in the sense of lyapunov would be concluded however this paper further asserts that the partial state which remains in the time derivative of the lyapunov function converges to zero asymptotically

multiscale modeling of acoustic wave propagation in 2d media abstractconventional finite difference methods produce accurate solutions to the acoustic and elastic wave equation for many applications but they face significant challenges when material properties vary significantly over distances less than the grid size this challenge is likely to occur in reservoir characterization studies because important reservoir heterogeneity can be present on scales of several meters to ten meters here we describe a new multiscale finite element method for simulating acoustic wave propagation in heterogeneous media that addresses this problem by coupling fine and coarse scale grids the wave equation is solved on a coarse grid but it uses basis functions that are generated from the fine grid and allow the representation of the fine scale variation of the wavefield on the coarser grid time stepping also takes place on the coarse grid providing further speed gains another important property of the method is that the basis functions are only computed once and time savin

adaptive boundary control for unstable parabolic pdes part ii estimation based designs the certainty equivalence approach to adaptive control is commonly used with two types of identifiers passivity based identifiers and swapping identifiers the passive also known as observer based approach is the prevalent identification technique in existing results on adaptive control for pdes but has so far not been used in boundary control problems the swapping approach prevalent in finite dimensional adaptive control is employed here for the first time in adaptive control of pdes for a class of unstable parabolic pdes we prove a separation principle result for both the passive and swapping identifiers combined with the backstepping boundary controllers the result is applicable in any dimension for physical reasons we restrict our attention to dimensions no higher than three the results of the paper are illustrated by simulations

identification of parameters in distributed parameter systems by regularization identification of spatially varying parameters in distributed parameter systems from noisy data is an ill posed problem the concept of regularization widely used in solving linear fredholm integral equations is developed for the identification of parameters in distributed parameter systems a general regularization identification theory is first presented and then applied to the identification of parabolic systems the performance of the regularization identification method is evaluated by numerical experiments on the identification of a spatially varying diffusivity in the diffusion equation