multidimensional global extremum seeking via the direct optimisation algorithm direct is a sample based global optimisation method for lipschitz continuous functions defined over compact multidimensional domains this paper adapts the direct method with a modified termination criterion for global extremum seeking control of multivariable dynamical plants finite time semi global practical convergence is established based on a periodic sampled data control law whose sampling period is a parameter which determines the region and accuracy of convergence a crucial part of the development is dedicated to a robustness analysis of the direct method against bounded additive perturbations on the objective function extremum seeking involving multiple units is also considered within the same context as a means to increase the speed of convergence numerical examples of global extremum seeking based on direct are presented at the end

application of the perfectly matched layer pml absorbing boundary condition to elastic wave propagation a method is presented for application of the perfectly matched layer pml absorbing boundary condition abc to the p sv velocity stress finite difference method the pml consists of a nonphysical material containing both passive loss and dependent sources that provides active absorption of fields it has been used in electromagnetic applications where it has provided excellent results for a wide range of angles and frequencies in this work numerical simulations are used to compare the pml and an optimal second order elastic abc peng and toksoz j acoust soc am 95 733 745 1994 reflection factors are used to compare angular performance for continuous wave illumination snapshots of potentials are used to compare performance for broadband illumination these comparisons clearly demonstrate the superiority of the pml formulation within the pml there is a 60 increase in the number of unknowns per grid cell relative to the velocity stress formulation however the high quality of the pml abc allows the use of a smaller grid which can result in a lower overall computational cost

almost sure convergence of extremum seeking algorithm using stochastic perturbation abstract this paper presents an extremum seeking es algorithm where the perturbation signal is a martingale difference sequence m d s with a vanishing variance the measurement noise at the plant output is modeled by a superposition of deterministic component and a non stationary colored noise signal the optimizing set point of the uncertain reference to output equilibrium map is estimated by a stochastic approximation sa type algorithm the algorithm has a vanishing gain sequence dependent on the set point estimates by utilizing powerful tools of the martingale convergence theory it is proved that with probability one the set point estimates converge to the optimizing equilibrium point in spite of the presence of a measurement noise this result is derived without requiring boundedness or any prior condition on the set point estimates

p sv wave propagation in heterogeneous media velocity stress finite difference method i present a finite difference method for modeling p sv wave propagation in heterogeneous media this is an extension of the method i previously proposed for modeling sh wave propagation by using velocity and stress in a discrete grid the two components of the velocity cannot be defined at the same node for a complete staggered grid the stability condition and the p wave phase velocity dispersion curve do not depend on the poisson s ratio while the s wave phase velocity dispersion curve behavior is rather insensitive to the poisson s ratio therefore the same code used for elastic media can be used for liquid media where s wave velocity goes to zero and no special treatment is needed for a liquid solid interface typical physical phenomena arising with p sv modeling such as surface waves are in agreement with analytical results the weathered layer and corner edge models show in seismograms the same converted phases obtained by previous authors this method gives stable results for step discontinuities as shown for a liquid layer above an elastic half space the head wave preserves the correct amplitude finally the corner edge model illustrates a more complex geometry for the liquid solid interface as the poisson s ratio v increases from 0 25 to 0 5 the shear converted phases are removed from seismograms and from the time section of the wave field

global optimization of univariate lipschitz functions ii new algorithms and computational comparison we consider the following global optimization problems for a lipschitz functionf implicitly defined on an interval a b problem p find a globallye optimal value off and a corresponding point problem q find a set of disjoint subintervals of a b containing only points with a globallye optimal value and the union of which contains all globally optimal points a two phase algorithm is proposed for problem p in phase i this algorithm obtains rapidly a solution which is often globallye optimal moreover a sufficient condition onf for this to be the case is given in phase ii the algorithm proves thee optimality of the solution obtained in phase i or finds a sequence of points of increasing value containing one with a globallye optimal value the new algorithm is empirically compared on twenty problems from the literature with a best possible algorithm for which the optimal value is assumed to be known with a passive algorithm and with the algorithms of evtushenko galperin shen and zhu piyavskii timonov and schoen for smalle the new algorithm requires only a few percent more function evaluations than the best possible one an extended version of piyavskii s algorithm is proposed for problem q a sufficient condition onf is given for the globally optimal points to be in one to one correspondance with the obtained intervals this result is achieved for all twenty test problems

factorization of asplund operators abstract we give necessary and sufficient conditions for an operator a x y on a banach space having a shrinking fdd to factor through a banach space z such that the szlenk index of z is equal to the szlenk index of a we also prove that for every ordinal 0 1 1 a limit ordinal there exists a banach space g having a shrinking basis and szlenk index such that for any separable banach space x and any operator a x y having szlenk index less than a factors through a subspace and through a quotient of g and if x has a shrinking fdd a factors through g

extremum seeking based adaptive control for electromagnetic actuators in this paper we present a learning based adaptive method to solve the problem of robust trajectory tracking for electromagnetic actuators we merge a nonlinear backstepping controller that ensures bounded input bounded states stability with a multi variable extremum seeking model free learning algorithm the learning algorithm is used to estimate online the uncertain parameters of the model in this sense we propose a learning based adaptive controller we present a proof of stability of this learning based nonlinear controller when considering uncertainties with linear parametrisation the efficiency of this approach is shown on a numerical example

newton like extremum seeking for the control of thermoacoustic instability in practice the convergence rate and stability of perturbation based extremum seeking schemes can be very sensitive to the curvature of the plant map an example of this can be seen in the use of extremum seeking to reduce the amplitude of thermoacoustic oscillations in premixed gas turbine combustors this sensitivity to the plant map curvature arises from the use of a gradient descent adaptation algorithm such extremum seeking schemes may need to be conservatively tuned in order to maintain stability over a wide range of operating conditions resulting in slower optimization than could be achieved for a fixed operating condition this can severely reduce the effectiveness of perturbation based extremum seeking schemes in some applications in this paper a sinusoidally perturbed extremum seeking scheme using a newton like step is developed non local stability results for the scheme are formulated using a lyapunov analysis a local analysis of the scheme is given to investigate the influence of plant dynamics and to show that the local rate of convergence is independent of the plant map curvature the benefit of this plant map curvature independence is then experimentally demonstrated in minimizing the thermoacoustic oscillations in a model premixed combustor

three space property for asymptotically uniformly smooth renormings abstract we prove that if y is a closed subspace of a banach space x such that y and x y admit an equivalent asymptotically uniformly smooth norm then x also admits an equivalent asymptotically uniformly smooth norm the proof is based on the use of the szlenk index and yields a few other applications to renorming theory

an efficient complex frequency shifted perfectly matched layer for second order acoustic wave equation summary in the context of simulations of wave propagations in unbounded domain absorbing boundary conditions are often used to truncate the simulation domain to a finite space perfectly matched layer pml has proven to be an excellent absorbing boundary conditions however as this technique was primarily designed for the first order equation system it cannot be applied to the second order equation system directly in this paper based on a complex coordinate stretching technique we developed a novel efficient auxiliary differential equation form of the complex frequency shifted pml for the second order equation system this facilitates the use of complex frequency shifted pml in acoustic simulations based upon wave equations of second order form compared with previous state of the art methods the proposed one has the advantage of simpler implementation it is an unsplit field scheme that can be extended to higher order discretization schemes conveniently numerical results from both homogeneous and heterogeneous computational domains are provided to illustrate the validity of the method copyright 2013 john wiley sons ltd

perfectly matched absorbing layers for the paraxial equations a new absorbing boundary technique for the paraxial wave equations is proposed and analyzed numerical results show the efficiency of the method

explicit finite element perfectly matched layer for transient three dimensional elastic waves the use of a perfectly matched layer pml model is an efficient approach toward the bounded domain modelling of wave propagation on unbounded domains this paper formulates a three dimensional pml for elastic waves by building upon previous work by the author and implements it in a displacement based finite element setting the novel contribution of this paper over the previous work is in making this finite element implementation suitable for explicit time integration thus making it practicable for use in large scale three dimensional dynamic analyses an efficient method of calculating the strain terms in the pml is developed in order to take advantage of the lack of the overhead of solving equations at each time step the pml formulation is studied and validated first for a semi infinite bar and then for the classical soil structure interaction problems of a square flexible footing on a i half space ii layer on a half space and iii layer on a rigid base numerical results for these problems demonstrate that the pml models produce highly accurate results with small bounded domains and at low computational cost and that these models are long time stable with critical time step sizes similar to those of corresponding fully elastic models copyright 2008 john wiley sons ltd

a robust event triggered approach for fast sampled data extremization and learning this paper presents a general framework for the analysis and design of a class of model free robust and efficient sampled data based algorithms for extremization and learning in continuous time nonlinear systems that generate response maps with an optimal operational set in particular we consider plants described by differential inclusions interconnected in a sampled data setting with a robust learning algorithm characterized by a constrained difference inclusion in contrast to standard sampled data based approaches where the learning dynamics are updated after a fixed sufficiently long sampling time has passed we design a robust dynamic event based mechanism that triggers the control action as soon as the rate of change of the output of the plant is sufficiently small by using this event based update rule a significant improvement in the convergence time of the closed loop system can be achieved using the framework of set valued hybrid dynamical systems we establish for the closed loop system the existence of a uniformly asymptotically stable compact set which by an appropriate tuning of the control parameters can be made arbitrarily close to the optimal operational set our results generalize existing results for periodic sampled data extremum seeking and can be used to solve model free multivariable smooth nonsmooth constrained optimization problems as well as learning problems in game theoretical scenarios

solution of two dimensional riemann problems of gas dynamics by positive schemes the positivity principle and positive schemes to solve multidimensional hyperbolic systems of conservation laws have been introduced in x d liu and p d lax j fluid dynam 5 1996 pp 133 156 some numerical experiments presented there show how well the method works in this paper we use positive schemes to solve riemann problems for two dimensional gas dynamics

an information based learning approach to dual control dual control aims to concurrently learn and control an unknown system however actively learning the system conflicts directly with any given control objective for it will disturb the system during exploration this paper presents a receding horizon approach to dual control where a multiobjective optimization problem is solved repeatedly and subject to constraints representing system dynamics balancing a standard finite horizon control objective a knowledge gain objective is defined to explicitly quantify the information acquired when learning the system dynamics measures from information theory such as entropy based uncertainty fisher information and relative entropy are studied and used to quantify the knowledge gained as a result of the control actions the resulting iterative framework is applied to markov decision processes and discrete time nonlinear systems thus the broad applicability and usefulness of the presented approach is demonstrated in diverse problem settings the framework is illustrated with multiple numerical examples

on non local stability properties of extremum seeking control in this paper we consider several extremum seeking schemes and show under appropriate conditions that these schemes achieve extremum seeking from an arbitrarily large domain of initial conditions if the parameters in the controller are appropriately adjusted this non local stability result is proved by showing semi global practical stability of the closed loop system with respect to the design parameters we show that reducing the size of the parameters typically slows down the convergence rate of the extremum seeking controllers and enlarges the domain of the attraction our results provide guidelines on how to tune the controller parameters in order to achieve extremum seeking simulation examples illustrate our results

time varying boundary conditions in simulation of seismic wave propagation we propose two new boundary conditions to regulate coherent reflections from the model boundaries in numerical solutions of wave equations both boundary conditions have the common feature that the boundary condition is varied with respect to time the first boundary condition expands or contracts the computational model during a modeling simulation the effect is to cause a doppler shift in the reflected wavefield that can be used to shift energy outside a frequency band of interest in addition when the computational domain is expanding the range of possible incidence angles on the boundary is restricted this can be used to increase the effectiveness of many existing absorbing boundary conditions that are more effective for incidence angles close to normal the second boundary condition is an extension of random boundaries by carefully changing the realization of a random boundary over time a more diffusive wavefield can be simulated we show results with 2d numerical simulations of the scalar wave equation for both these boundary conditions the first boundary condition has application to modeling but both these boundary conditions have potential application within algorithms that rely upon modeling kernels such as reverse time migration and full waveform inversion

extremum seeking of dynamical systems via gradient descent and stochastic approximation methods this paper examines the use of gradient based methods for extremum seeking control of possibly infinite dimensional dynamic nonlinear systems with general attractors within a periodic sampled data framework first discrete time gradient descent method is considered and semi global practical asymptotic stability with respect to an ultimate bound is shown next under the more complicated setting where the sampled measurements of the plant s output are corrupted by an additive noise three basic stochastic approximation methods are analysed namely finite difference random directions and simultaneous perturbation semi global convergence to an optimum with probability one is established a tuning parameter within the sampled data framework is the period of the synchronised sampler and hold device which is also the waiting time during which the system dynamics settle to within a controllable neighbourhood of the steady state input output behaviour

continued fraction absorbing boundary conditions for convex polygonal domains continued fraction absorbing boundary conditions cfabcs are highly effective boundary conditions for modelling wave absorption into unbounded domains they are based on rational approximation of the exact dispersion relationship and were originally developed for straight computational boundaries in this paper cfabcs are extended to the more general case of polygonal computational domains the key to the current development is the surprising link found between the cfabcs and the complex co ordinate stretching of perfectly matched layers pmls this link facilitates the extension of cfabcs to oblique corners and thus to polygonal domains it is shown that the proposed cfabcs are easy to implement expected to perform better than pmls and are effective for general polygonal computational domains in addition to the derivation of cfabcs a novel explicit time stepping scheme is developed for efficient numerical implementation numerical examples presented in the paper illustrate that effective absorption is attained with a negligible increase in the computational cost for the interior domain although this paper focuses on wave propagation its theoretical development can be easily extended to the more general class of problems where the governing differential equation is second order in space with constant coefficients copyright 2005 john wiley sons ltd

a conservative numerical scheme for modeling nonlinear acoustic propagation in thermoviscous homogeneous media abstract a nonlinear system of partial differential equations capable of describing the nonlinear propagation and attenuation of finite amplitude perturbations in thermoviscous media is presented this system constitutes a full nonlinear wave model that has been formulated in the conservation form initially this model is investigated analytically in the inviscid limit where it has been found that the resulting flux function fulfills the lax wendroff theorem and the scheme can match the solutions of the westervelt and burgers equations numerically here high order numerical descriptions of strongly nonlinear wave propagations become of great interest for that matter we consider finite difference formulations of the weighted essentially non oscillatory weno schemes associated with explicit strong stability preserving runge kutta ssp rk time integration methods although this strategy is known to be computationally demanding it is found to be effective when implemented to be solved in graphical processing units gpus as we consider wave propagations in unbounded domains perfectly matching layers pml have been also considered in this work the proposed system model is validated and illustrated by using one and two dimensional benchmark test cases proposed in the literature for nonlinear acoustic propagation in homogeneous thermoviscous media

lattice uehling uhlenbeck boltzmann bhatnagar gross krook hydrodynamics of quantum gases we present a semiclassical lattice boltzmann method based on quantum kinetic theory the method is directly derived by projecting the uehling uhlenbeck boltzmann bhatnagar gross krook equations onto the tensor hermite polynomials following grad s moment expansion method the intrinsic discrete nodes of the gauss hermite quadrature provide the natural lattice velocities for the semiclassical lattice boltzmann method gases of particles of arbitrary statistics can be considered simulation of one dimensional compressible gas flow and two dimensional hydrodynamic flows are shown the results indicate distinct characteristics of the effects of quantum statistics

absorbing boundary conditions for elastic waves absorbing boundary conditions are needed for computing numerical models of wave motions in unbounded spatial domains the boundary conditions developed here for elastic waves are generalizations of ones developed earlier for acoustic waves these conditions are based on compositions of simple first order differential operators the formulas can be applied without modification to problems in both two and three dimensions the boundary conditions are stable for all values of the ratio of p wave velocity to s wave velocity and they are effective near a free surface and in a horizontally stratified medium the boundary conditions are approximated with simple finite difference equations that use values of the solution only along grid lines perpendicular to the boundary this property facilitates implementation especially near a free surface and at other corners of the computational domain

global optimization of univariate lipschitz functions i survey and properties we consider the following global optimization problems for a univariate lipschitz functionf defined on an interval a b problem p find a globally optimal value off and a corresponding point problem p find a globally optimal value off and a corresponding point problem q localize all globally optimal points problem q find a set of disjoint subintervals of small length whose union contains all globally optimal points problem q find a set of disjoint subintervals containing only points with a globally optimal value and whose union contains all globally optimal points we present necessary conditions onf for finite convergence in problem p and problem q recall the concepts necessary for a worst case and an empirical study of algorithms i e those ofpassive and ofbest possible algorithms summarize and discuss algorithms of evtushenko piyavskii shubert timonov schoen galperin shen and zhu presenting them in a simplified and uniform way in a high level computer language we address in particular the problems of using an approximation for the lipschitz constant reducing as much as possible the expected length of the region of indeterminacy which contains all globally optimal points and avoiding remaining subintervals without points with a globally optimal value new algorithms for problems p and q and an extensive computational comparison of algorithms are presented in a companion paper

power type ξ asymptotically uniformly smooth and ξ asymptotically uniformly flat norms abstract for each ordinal and each 1 p we offer a natural isomorphic characterization of those spaces and operators which admit an equivalent p asymptotically uniformly smooth norm we also introduce the notion of asymptotically uniformly flat norms and provide an isomorphic characterization of those spaces and operators which admit an equivalent asymptotically uniformly flat norm given a compact hausdorff space k we prove an optimal renorming theorem regarding the asymptotic smoothness of c k in terms of the cantor bendixson index of k we also prove that for all ordinals both the isomorphic properties and isometric properties pass from banach spaces to their injective tensor products we study the classes of p asymptotically uniformly smooth p asymptotically uniformly smoothable asymptotically uniformly flat and asymptotically uniformly flattenable operators we show that these classes are either a banach ideal or a right banach ideal when assigned an appropriate ideal norm

an efficient direct solver for rarefied gas flows with arbitrary statistics a new numerical methodology associated with a unified treatment is presented to solve the boltzmann bgk equation of gas dynamics for the classical and quantum gases described by the bose einstein and fermi dirac statistics utilizing a class of globally stiffly accurate implicit explicit runge kutta scheme for the temporal evolution associated with the discrete ordinate method for the quadratures in the momentum space and the weighted essentially non oscillatory method for the spatial discretization the proposed scheme is asymptotic preserving and imposes no non linear solver or requires the knowledge of fugacity and temperature to capture the flow structures in the hydrodynamic euler limit the proposed treatment overcomes the limitations found in the work by yang and muljadi 2011 33 due to the non linear nature of quantum relations and can be applied in studying the dynamics of a gas with internal degrees of freedom with correct values of the ratio of specific heat for the flow regimes for all knudsen numbers and energy wave lengths the present methodology is numerically validated with the unified treatment by the one dimensional shock tube problem and the two dimensional riemann problems for gases of arbitrary statistics descriptions of ideal quantum gases including rotational degrees of freedom have been successfully achieved under the proposed methodology

a spectral fc solver for the compressible navier stokes equations in general domains i explicit time stepping we present a fourier continuation fc algorithm for the solution of the fully nonlinear compressible navier stokes equations in general spatial domains the new scheme is based on the recently introduced accelerated fc method which enables use of highly accurate fourier expansions as the main building block of general domain pde solvers previous fc based pde solvers are restricted to linear scalar equations with constant coefficients the fc methodology presented in this text thus constitutes a significant generalization of the previous fc schemes as it yields general domain fc solvers for nonlinear systems of pdes while not restricted to periodic boundary conditions and therefore applicable to general boundary value problems on arbitrary domains the proposed algorithm inherits many of the highly desirable properties arising from rapidly convergent fourier expansions including high order convergence essentially spectrally accurate dispersion relations and much milder cfl constraints than those imposed by polynomial based spectral methods since for example the spectral radius of the fc first derivative grows linearly with the number of spatial discretization points we demonstrate the accuracy and optimal parallel efficiency of the algorithm in a variety of scientific and engineering contexts relevant to fluid dynamics and nonlinear acoustics

discretizing singular point sources in hyperbolic wave propagation problems we develop high order accurate source discretizations for hyperbolic wave propagation problems in first order formulation that are discretized by finite difference schemes by studying the fourier series expansions of the source discretization and the finite difference operator we derive sufficient conditions for achieving design accuracy in the numerical solution only half of the conditions in fourier space can be satisfied through moment conditions on the source discretization and we develop smoothness conditions for satisfying the remaining accuracy conditions the resulting source discretization has compact support in physical space and is spread over as many grid points as the number of moment and smoothness conditions in numerical experiments we demonstrate high order of accuracy in the numerical solution of the 1 d advection equation both in the interior and near a boundary the 3 d elastic wave equation and the 3 d linearized euler equations

unified frameworks for sampled data extremum seeking control global optimisation and multi unit systems two frameworks are proposed for extremum seeking of general nonlinear plants based on a sampled data control law within which a broad class of nonlinear programming methods is accommodated it is established that under some generic assumptions semi global practical convergence to a global extremum can be achieved in the case where the extremum seeking algorithm satisfies a stronger asymptotic stability property the converging sequence is also shown to be stable using a trajectory based proof as opposed to a lyapunov function type approach the former is more straightforward and insightful this allows for more general optimisation algorithms than considered in existing literature such as those which do not admit a state update realisation and or lyapunov functions lying at the heart of the analysis throughout is robustness of the optimisation algorithms to additive perturbations of the objective function multi unit extremum seeking is also investigated with the objective of accelerating the speed of convergence

application of the perfectly matched absorbing layer model to the linear elastodynamic problem in anisotropic heterogeneous media we present and analyze a perfectly matched absorbing layer model for the velocity stress formulation of elastodynamics the principal idea of this method consists of introducing an absorbing layer in which we decompose each component of the unknown into two auxiliary components a component orthogonal to the boundary and a component parallel to it a system of equations governing these new unknowns then is constructed a damping term finally is introduced for the component orthogonal to the boundary this layer model has the property of generating no reflection at the interface between the free medium and the artificial absorbing medium in practice both the boundary condition introduced at the outer boundary of the layer and the dispersion resulting from the numerical scheme produce a small reflection which can be controlled even with very thin layers as we will show with several experiments this model gives very satisfactory results namely the reflection coefficient even in the case of heterogeneous anisotropic media is about 1 for a layer thickness of five space discretization steps

simulation of shock wave diffraction over 90 sharp corner in gases of arbitrary statistics the unsteady shock wave diffraction over a 90 sharp corner in gases of arbitrary particle statistics is simulated using an accurate and direct algorithm for solving the semiclassical boltzmann equation with relaxation time approximation in phase space the numerical method is based on the usage of discrete ordinate method for discretizing the velocity space of the distribution function whereas a second order accurate tvd scheme harten in j comput phys 49 3 357 393 1983 with van leer s limiter j comput phys 32 1 101 136 1979 is used for evolving the solution in physical space and time the specular reflection surface boundary condition is assumed the complete diffraction patterns are recorded using various flow property contours different range of relaxation times approximately corresponding to continuum slip and transitional regimes are considered and the equilibrium euler limit solution is also computed for comparison the effects of gas particles that obey the maxwell boltzmann bose einstein and fermi dirac statistics are examined and depicted

rarefied flow computations using nonlinear model boltzmann equations high resolution finite difference schemes for solving the nonlinear model boltzmann equations are presented for the computations of rarefied gas flows the discrete ordinate method is first applied to remove the velocity space dependency of the distribution function which renders the model boltzmann equation in phase space to a set of hyperbolic conservation laws with source terms in physical space then a high order essentially nonoscillatory method due to harten et al j comput phys 71 231 1987 is adapted and extended to solve them explicit methods using operator splitting and implicit methods using the lower upper factorization are described to treat multidimensional problems the methods are tested for both steady and unsteady rarefied gas flows to illustrate its potential use the computed results using model boltzmann equations are found to compare well both with those using the direct simulation monte carlo results in the transitional regime flows and those with the continuum navier stokes calculations in near continuum regime flows

a unified gas kinetic scheme for continuum and rarefied flows iv fluid dynamic equations are valid in their respective modeling scales such as the particle mean free path scale of the boltzmann equation and the hydrodynamic scale of the navier stokes ns equations with a variation of the modeling scales theoretically there should have a continuous spectrum of fluid dynamic equations even though the boltzmann equation is claimed to be valid in all scales many boltzmann solvers including direct simulation monte carlo method require the cell resolution to the order of particle mean free path scale therefore they are still single scale methods in order to study multiscale flow evolution efficiently the dynamics in the computational fluid has to be changed with the scales a direct modeling of flow physics with a changeable scale may become an appropriate approach the unified gas kinetic scheme ugks is a direct modeling method in the mesh size scale and its underlying flow physics depends on the resolution of the cell size relative to the particle mean free path the cell size of ugks is not limited by the particle mean free path with the variation of the ratio between the numerical cell size and local particle mean free path the ugks recovers the flow dynamics from the particle transport and collision in the kinetic scale to the wave propagation in the hydrodynamic scale the previous ugks is mostly constructed from the evolution solution of kinetic model equations even though the ugks is very accurate and effective in the low transition and continuum flow regimes with the time step being much larger than the particle mean free time it still has space to develop more accurate flow solver in the region where the time step is comparable with the local particle mean free time in such a scale there is dynamic difference from the full boltzmann collision term and the model equations this work is about the further development of the ugks with the implementation of the full boltzmann collision term in the region where it is needed the central ingredient of the ugks is the coupled treatment of particle transport and collision in the flux evaluation across a cell interface where a continuous flow dynamics from kinetic to hydrodynamic scales is modeled the newly developed ugks has the asymptotic preserving ap property of recovering the ns solutions in the continuum flow regime and the full boltzmann solution in the rarefied regime in the mostly unexplored transition regime the ugks itself provides a valuable tool for the non equilibrium flow study the mathematical properties of the scheme such as stability accuracy and the asymptotic preserving will be analyzed in this paper as well

three dimensional perfectly matched layer for the absorption of electromagnetic waves the perfectly matched layer is a technique of free space simulation developed for solving unbounded electromagnetic problems with the finite difference time domain method referred to as pml this technique has been described in a previous paper for two dimensional problems the present paper is devoted to the three dimensional case the theory of the perfectly matched layer is generalized to three dimensions and some numerical experiments are shown to illustrate the efficiency of this new method of free space simulation

an optimal absorbing boundary condition for elastic wave modeling absorbing boundary conditions are widely used in numerical modeling of wave propagation in unbounded media to reduce reflections from artificial boundaries lindman 1975 clayton and engquist 1977 reynolds 1978 liao et al 1984 cerjan et al 1985 randall 1988 higdon 1991 we are interested in a particular absorbing boundary condition that has maximum absorbing ability with a minimum amount of computation and storage this is practical for 3 d simulation of elastic wave propagation by a finite difference method peng and toksoz 1994 developed a method to design a class of optimal absorbing boundary conditions for a given operator length in this short note we give a brief introduction to this technique and we compare the optimal absorbing boundary conditions against those by reynolds 1978 and higdon 1991 using examples of 3 d elastic finite difference modeling on an ncube 2 parallel computer in the appendix we also give explicit formulas for computing coefficients of the optimal abso

optimizing the perfectly matched layer the berenger perfectly matched layer or pml 4 3 is used in computational electromagnetism as a sponge layer to terminate finite element approximations of scattering problems an infinite perfectly matched layer creates no reflection for incident waves of any frequency or any incident direction and the waves decay exponentially in magnitude into the layer this property is lost when the layer is truncated since the truncation boundary generates a reflected wave discretization of the differential equations further changes the reflection coefficient of the layer in this talk we shall describe the pml in the simple case of a two dimensional electromagnetic problem the first part of this abstract is expository we shall show how to derive the layer using a change of variable approach 5 9 6 after this derivation we describe some new analysis to show the effects of truncating and discretizing the pml 9 we end by showing how to compute an optimal layer and make some remarks about pml for the full maxwell system

a perturbation based proportional integral extremum seeking control approach this paper proposes a proportional integral extremum seeking control technique the technique is a generalization of the standard perturbation based techniques which provides fast transient performance of the closed loop system to the optimum equilibrium of a measured cost function under the assumption that the equations describing the dynamics of the nonlinear system and the cost function to be minimized are unknown it is shown that for a class of minimum phase nonlinear system with a cost function of relative degree one the closed loop esc can practically stabilize the unknown equilibrium optimum the stabilization result provides an alternative approach to output feedback control of nonlinear systems the effectiveness of the proposed approach is demonstrated using simulation examples

proximity to l1and distortion in asymptotic l1spaces abstract for an asymptotic l 1 space x with a basis x i certain asymptotic l 1 constants x are defined for 1 x measures the equivalence between all normalized block bases y i k i 1 of x i which are s admissible with respect to x i s is the th schreier class of sets and the unit vector basis of l k 1 this leads to the concept of the delta spectrum of x x which reflects the behavior of stabilized limits of x the analogues of these constants under all renormings of x are also defined and studied we investigate x both in general and for spaces of bounded distortion we also prove several results on distorting the classical tsirelson s space t and its relatives

an algorithm for finding the global maximum of a multimodal multivariate function this algorithm for global optimization uses an arbitrary starting point requires no derivatives uses comparatively few function evaluations and is not side tracked by nearby relative optima the algorithm builds a gradually closer piecewise differentiable approximation to the objective function the computer program exhibits a theoretically expected strong tendency to cluster around relative optima close to the global results of testing with several standard functions are given

a non gradient approach to global extremum seeking an adaptation of the shubert algorithm the main purpose of this paper is to adapt the so called shubert algorithm for extremum seeking control of general dynamic plants this algorithm is a good representative of the sampling optimization methods that achieve global extremum seeking on compact sets in the presence of local extrema the algorithm applies to lipschitz mappings the model of the system is assumed unknown but the knowledge of its lipschitz constant is assumed the controller depends on a design parameter the waiting time and tuning guidelines that relate the design parameter and the region of convergence and accuracy of the algorithm are presented the analysis shows that semi global practical convergence in the initial states to the global extremum can be achieved in presence of local extrema if compact sets of inputs are considered numerical simulations for global optimization in the presence of local extrema are provided to demonstrate the proposed approach

dispersion reducing finite elements for transient acoustics this paper focuses on increasing the accuracy of low order four node quadrilateral finite elements for the transient analysis of wave propagation modified integration rules originally proposed for time harmonic problems provide the basis for the proposed technique the modified integration rules shift the integration points to locations away from the conventional gauss or gauss lobatto integration points with the goal of reducing the discretization errors specifically the dispersion error presented here is an extension of the idea to time dependent analysis using implicit as well as explicit time stepping schemes the locations of the stiffness integration points remain unchanged from those in time harmonic case on the other hand the locations of the integration points for the mass matrix depend on the time stepping scheme and the step size furthermore the central difference method needs to be modified from its conventional form to facilitate fully explicit computation the superior performance of the proposed algorithms is illustrated with the help of several numerical examples

absorbing boundary conditions and surface waves one of the major problems in numerically simulating waves traveling in the earth is that an artificial boundary must be introduced to produce unique solutions to eliminate the spurious reflections introduced by this artificial boundary we use a damping expression based on analogies to shock absorbers this method can reduce the amplitude of the reflected wave to any pre specified value and is successful for waves at any angle of incidence the method can eliminate unwanted reflections from the surface reflections at the corners of the model and waves reflected off an interface that strike the artificial boundary many of the boundary conditions currently used in the numerical solution of waves are approximations to perfectly absorbing boundary conditions and depend upon the angle of incidence of the incoming wave at the artificial boundary stability problems often occur with these boundary conditions the method we use at the artificial boundary allows use of stable dirichlet or von neumann condition

improved pml for the fdtd solution of wave structure interaction problems in a previous paper an optimum perfectly matched layer has been designed for the finite difference time domain fdtd solution of wave structure interaction problems the present paper shows the results of subsequent investigations that were done with the intention of reducing the thickness of this optimum pml four improvements to the pml are presented the resulting reductions of the thickness are discussed

a robust extremum seeking scheme for dynamic systems with uncertainties and disturbances this paper studies a numerical optimization based extremum seeking scheme for systems with unmodeled dynamics and unknown disturbances without using dither signals the robust extremum seeking scheme is composed of a numerical gradient estimator a numerical optimizer and an extended state observer based state regulator a conjugate gradient method is adopted to achieve extremum seeking a nonlinear extended state observer based state regulator is proposed to regulate the states the robust extremum seeker is based on the numerical optimization based extremum seeking framework sufficient conditions are derived to ensure the convergence of the overall extremum seeking scheme the ultimate convergence bound is quantified for a class of strongly convex functions a numerical example is presented to verify the proposed method

ξ completely continuous operators and ξ schur banach spaces abstract for each ordinal 0 1 we introduce the notion of a completely continuous operator and prove that for each ordinal 0 1 the class v of completely continuous operators is a closed injective operator ideal which is not surjective symmetric or idempotent we prove that for distinct 0 1 the classes of completely continuous operators and completely continuous operators are distinct we also introduce an ordinal rank v for operators such that v a 1 if and only if a is completely continuous and otherwise v a is the minimum countable ordinal such that a fails to be completely continuous we show that there exists an operator a such that v a if and only if 1 1 and there exists a banach space x such that v i x if and only if there exists an ordinal 1 such that finally prove that for every 0 1 the class a l v a is 1 1 complete in l the coding of all operators between separable banach spaces this is in contrast to the class v l which is 2 1 complete in l

absorbing boundary conditions for acoustic and elastic waves in stratified media abstract absorbing boundary conditions are needed for computing numerical models of wave motions in unbounded spatial domains prior progress on this problem for acoustic and elastic waves has generally been concerned with waves propagating through uniform media the present paper is concerned with waves in stratified media which are of interest for example in geophysical problems suppose that the medium consists of homogeneous layers separated by parallel horizontal interfaces and suppose that absorbing boundary conditions are needed along a vertical computational boundary the boundary conditions that are described in this paper are based on a quantity known as the ray parameter according to snell s law this parameter remains the same when a plane wave propagates through a stratified medium and undergoes reflection refraction and in the case of elastic waves conversion once can therefore use the same absorbing boundary conditions in all layers for acoustic waves the absorption properties are the same in all layers for elastic waves the absorption properties vary somewhat from one layer to another however one still obtains good absorption in all layers even in the presence of strong contrasts between layers the boundary conditions are also effective in absorbing rayleigh waves which propagate along free surfaces of elastic media the boundary formulas developed here can be applied without modification to problems in both two and three dimensions

szlenk indices and uniform homeomorphisms we prove some rather precise renorming theorems for banach spaces with szlenk index wo we use these theorems to show the invariance of certain quantitative szlenk type indices under uniform homeomorphisms

non three space properties and szlenk index abstract we prove that for any non zero countable ordinal which is not additively indecomposable the property of having szlenk index not exceeding is not a three space property this complements a result of brooker and lancien which states that if is additively indecomposable then having szlenk index not exceeding is a three space property

asymptotic preserving implicit explicit runge kutta methods for nonlinear kinetic equations we discuss implicit explicit imex runge kutta methods which are particularly adapted to stiff kinetic equations of boltzmann type we consider both the case of easy invertible collision operators and the challenging case of boltzmann collision operators we give sufficient conditions in order that such methods are asymptotic preserving and asymptotically accurate their monotonicity properties are also studied in the case of the boltzmann operator the methods are based on the introduction of a penalization technique for the collision integral this reformulation of the collision operator permits us to construct penalized imex schemes which work uniformly for a wide range of relaxation times avoiding the expensive implicit resolution of the collision operator finally we show some numerical results which confirm the theoretical analysis

discrete velocity models and numerical schemes for the boltzmann bgk equation in plane and axisymmetric geometries we present new numerical models for computing transitional or rarefied gas flows as described by the boltzmann bgk and bgk es equations we first propose a new discrete velocity model based on the entropy minimization principle this model satisfies the conservation laws and the entropy dissipation moreover the problem of conservation and entropy for axisymmetric flows is investigated we find algebraic relations that must be satisfied by the discretization of the velocity derivative appearing in the transport operator then we propose some models that satisfy these constraints owing to these properties we obtain numerical schemes that are economic in terms of discretization and robust in particular we develop a linearized implicit scheme for computing stationary solutions of the discrete velocity bgk and bgk es models this scheme is the basis of a code which can compute high altitude hypersonic flows in 2d plane and axisymmetric geometries our results are analyzed and compared to other methods

fourier continuation methods for high fidelity simulation of nonlinear acoustic beams on the basis of recently developed fourier continuation fc methods and associated efficient parallelization techniques this text introduces numerical algorithms that due to very low dispersive errors can accurately and efficiently solve the types of nonlinear partial differential equation pde models of nonlinear acoustics in hundred wavelength domains as arise in the simulation of focused medical ultrasound as demonstrated in the examples presented in this text the fc approach can be used to produce solutions to nonlinear acoustics pdes models with significantly reduced discretization requirements over those associated with finite difference finite element and finite volume methods especially in cases involving waves that travel distances that are orders of magnitude longer than their respective wavelengths in these examples the fc methodology is shown to lead to improvements in computing times by factors of hundreds and even thousands over those required by the standard approaches a variety of one and two dimensional examples presented in this text demonstrate the power and capabilities of the proposed methodology including an example containing a number of scattering centers and nonlinear multiple scattering events