the existence and uniqueness of solution to wavelet collocation the study of the numerical solutions of pdes with wavelet collocation has yielded a number of substantial results however the existence and uniqueness of solution has not been discussed yet in this paper the existence and uniqueness of solution to wavelet collocation for elliptic equations is established and discussed moreover wavelet collocation is applied to numerical example to examine its appropriateness according to numerical example and analysis it is seen that the existence and uniqueness of solution of this paper is feasible and the new theory is meaningful for developing wavelet collocation

w stability and local input output stability results a new type of input output stability is defined based on the use of a sobolev space w w is well suited like the lebesgue space l sub 2 to obtain stability characterizations in the time and frequency domains moreover if compared with l sub 2 w has additional properties which enable one to establish local stability results a local version of the small gain theorem is established in this framework as well as some consequences of this result in particular local versions of the passivity theorem and of the circle criterion the relationship between w stability and asymptotic stability is studied

fast multiresolution algorithms for solving linear equations a comparative study in g beylkin r coifman and v rokhlin comm pure appl math xliv 1991 pp 141 183 and b engquist s osher and s zhong siam j sci comput 15 1994 pp 755 775 orthonormal wavelet basis and the multiscale analysis they define are used as building blocks in the design of algorithms for the rapid numerical application of a number of linear operators to arbitrary vectors the algorithms may be viewed as a method for converting whenever possible dense matrices to sparse form we use the discrete multiresolution analysis described by harten in j appl numer math 12 1993 pp 153 193 as the building block in the algorithms and compare their performance with the wavelet based ones computationally both techniques give comparable results

glass transition of hard spheres in high dimensions we have investigated analytically and numerically the liquid glass transition of hard spheres for dimensions d ensuremath rightarrow ensuremath infty in the framework of mode coupling theory the numerical results for the critical collective and self nonergodicity parameters f c k d and f c s k d exhibit non gaussian k dependence even up to d 800 f c s k d and f c k d differ for k ensuremath sim d 1 2 but become identical on a scale k ensuremath sim d which is proven analytically the critical packing fraction ensuremath varphi c d ensuremath sim d 2 2 ensuremath d is above the corresponding kauzmann packing fraction ensuremath varphi k d derived by a small cage expansion its quadratic pre exponential factor is different from the linear one found earlier the numerical values for the exponent parameter and therefore the critical exponents a and b depend on d even for the largest values of d

quadrature formulae and asymptotic error expansions for wavelet approximations of smooth functions this paper deals with typical problems that arise when using wavelets in numerical analysis applications the first part involves the construction of quadrature formulae for the calculation of inner products of smooth functions and scaling functions several types of quadratures are discussed and compared for different classes of wavelets since their construction using monomials is ill conditioned also a modified well conditioned construction using chebyshev polynomials is presented the second part of the paper deals with pointwise asymptotic error expansions of wavelet approximations of smooth functions they are used to derive asymptotic interpolating properties of the wavelet approximation and to construct a convergence acceleration algorithm this is illustrated with numerical examples

a multilevel wavelet collocation method for solving partial differential equations in a finite domain abstract a multilevel wavelet collection method for the solution of partial differential equations is developed two different approaches of treating general boundary conditions are suggested both are based on the wavelet interpolation technique developed in the present research the first approach uses wavelets as a basis and results in a differential algebraic system of equations where the algebraic part arises from boundary conditions the second approach utilizes extended wavelets which satisfy boundary conditions exactly this approach results in a system of coupled ordinary differential equations the method is tested on the one dimensional burgers equation with small viscosity the solutions are compared with those resulting from the use of other numerical algorithms the present results indicate that the method is competitive with well established numerical algorithms

simultaneous space time adaptive wavelet solution of nonlinear parabolic differential equations dynamically adaptive numerical methods have been developed to efficiently solve differential equations whose solutions are intermittent in both space and time these methods combine an adjustable time step with a spatial grid that adapts to spatial intermittency at a fixed time the same time step is used for all spatial locations and all scales this approach clearly does not fully exploit space time intermittency we propose an adaptive wavelet collocation method for solving highly intermittent problems e g turbulence on a simultaneous space time computational domain which naturally adapts both the space and time resolution to match the solution besides generating a near optimal grid for the full space time solution this approach also allows the global time integration error to be controlled the efficiency and accuracy of the method is demonstrated by applying it to several highly intermittent 1d t dimensional and 2d t dimensional test problems in particular we found that the space time method uses roughly 18 times fewer space time grid points and is roughly 4 times faster than a dynamically adaptive explicit time marching method while achieving similar global accuracy

global stabilization with almost disturbance decoupling of a class of uncertain non linear systems in this paper we study the global stabilization of a class of non linear uncertain systems while decoupling the output from external disturbances to any specified degree of accuracy the class of non linear systems considered could be viewed as a generalization of uniform rank multivariable linear systems which have the property that all their infinite zeros are of the same order although the plants considered are non linear the compensator developed here is linear our compensator which achieves global exponential stability and almost disturbance decoupling of the output requires only output feedback rather than full state feedback model uncertainty is dealt with either by restricting the uncertain elements to particular subspaces satisfying the so called matching conditions and or by treating a family of plants for which a single controller simultaneously globally exponentially stabilizes all the plants in while achieving almost disturbance decoupling of the output

design of multivariable feedback systems with stable plant this paper considers in a general algebraic framework the design of a unity feedback multivariable system with a stable plant the method is based on a simple parameterization of the four closed loop transfer functions in terms of p the plant transfer function and q h y 1 u 1 in particular the i o transfer function q h y 2 u 1 pq using the framework of rational transfer functions we show that the closed loop system will be exponentially stable if and only if q is exponentially stable furthermore if both p and q are strictly proper then the controller is also strictly proper the basic result is design theorem 2 an algorithm is given for obtaining strictly proper controllers such that the resulting i o map is decoupled all its poles can be chosen by the designer and the same holds for zeros except of course for the c zeros prescribed by the c zeros of the plant a discussion is included to temper these results by the constraints imposed by noise and plant saturation

on the stabilization of nonlinear systems we extend the applicability of the global q parametrization method of controller design to a large class of unstable nonlinear plants the main result is a two step compensation theorem analogous to that of zames for unstable linear plants if p l e2 rightarrow l e1 is a nonlinear possibly unstable plant and f 0 is any incrementally stable controller such that p 1 p i f 0 p 1 is incrementally stable then the class of controllers f which yields an f g stable closed loop system in the unity feedback configuration for p is globally parametrized by finite gain stable maps q l e1 rightarrow l e2 with f f 0 q i p 1 q 1

new relationships between input output and lyapunov stability in this note we define a concept called small signal l p stability and show its relationship to lyapunov stability

computation of decaying turbulence in an adaptive wavelet basis abstract the paper presents computations of decaying two dimensional turbulence in an adaptive wavelet basis at each time step the vorticity is represented by an adaptively selected set of wavelet functions which adjusts to the instantaneous distribution of vorticity essential features are the use of operator adapted test functions and the adaptive evaluation of the convection term the results of this new algorithm are compared to a classical fourier method and a fourier method supplemented with wavelet compression in each time step they show that turbulent flows with a multitude of spatial scales can be computed with a reduced number of degrees of freedom the investigation of diverse spectral and statistical criteria validates the wavelet approach

an overview of wavelet based multiresolution analyses in this paper an overview of wavelet based multiresolution analyses is presented first the continuous wavelet transform in its simplest form is discussed then the definition of a multiresolution analysis is given and how wavelets fit into it is shown the authors take a closer look at orthogonal biorthogonal and semiorthogonal wavelets the fast wavelet transform wavelets on an interval multidimensional wavelets and wavelet packets are discussed several examples of wavelet families are introduced and compared finally the essentials of two major applications are outlined data compression and compression of linear operators

construction of the left coprime fractional representation for a class of nonlinear control systems abstract a constructive approach to the left coprime fractional representations for a class of stable and unstable nonlinear control systems described by certain multi input multi output nonlinear ordinary differential equations is proposed the method works for the coprimeness defined by verma 15

adaptive multiresolution collocation methods for initial boundary value problems of nonlinear pdes we have designed a cubic spline wavelet like decomposition for the sobolev space h 0 2 i where i is a bounded interval based on a special point value vanishing property of the wavelet basis functions a fast discrete wavelet transform dwt is constructed this dwt will map discrete samples of a function to its wavelet expansion coefficients in at most 7n log n operations using this transform we propose a collocation method for the initial boundary value problem of nonlinear partial differential equations pdes then we test the efficiency of the dwt and apply the collocation method to solve linear and nonlinear pdes

nonlinear q design for convex stochastic control in this note we describe a version of the q design method that can be used to design nonlinear dynamic controllers for a discrete time linear time varying plant with convex cost and constraint functions and arbitrary disturbance distribution choosing a basis for the nonlinear q parameter yields a convex stochastic optimization problem which can be solved by standard methods such as sampling in principle for a large enough basis and enough sampling this method can solve the controller design problem to any degree of accuracy in any case it can be used to find a suboptimal controller using convex optimization methods we illustrate the method with a numerical example comparing a nonlinear controller found using our method with the optimal linear controller the certainty equivalent model predictive controller and a lower bound on achievable performance obtained by ignoring the causality constraint

robust right coprime factorization and robust stabilization of nonlinear feedback control systems robust right coprime factorization and robust stabilization of nonlinear feedback control systems are studied the concept of robust right coprime factorization of nonlinear operators for feedback control systems is introduced some conditions for the robustness of a right coprime factorization of a nonlinear plant under unknown but bounded perturbations are derived an example with a closed form solution is included to illustrate the general theory and a step by step construction of the robust factorization and the robust stabilization

multiresolution representation of data a general framework in this paper we present a general framework for a multiresolution representation of data which is obtained by the discretization of mappings this framework which can be viewed as a generalization of the theory of wavelets includes discretization corresponding to unstructured grids in several space dimensions and thus is general enough to enable us to embed most numerical problems in a multiresolution setting furthermore this framework allows for nonlinear data dependent multiresolution representation schemes and thus enables us to design adaptive data compression algorithms in this paper we also study the stability of linear schemes for a multiresolution representation and derive sufficient conditions for existence of a multiresolution basis

an adaptive spline wavelet adi sw adi method for two dimensional reaction diffusion equations we study a spline wavelet alternative direction implicit sw adi algorithm for solving two dimensional reaction diffusion equations this algorithm is based on a collocation method for pdes with a specially designed spline wavelet for the sobolev spaceh2 i on a closed intervali by using the tensor product nature of adaptive wavelet meshes we propose a sw adi method for two dimensional problems the proposed sw adi method is an efficient time dependent adaptive method with second order accuracy for solutions with localized phenomena such as in flame propagations issues like new boundary wavelets for more accurate boundary conditions adaptive strategy for two dimensional meshes data structure and storage and implementation details and numerical results will be discussed

a necessary and sufficient condition for stability of a perturbed system the problem of determining the stability of a feedback system in the presence of perturbation is considered a necessary and sufficient condition under which the perturbed system remains stable is obtained this condition is very general in particular it does not require the perturbed plant to have the same number of poles as the nominal plant

a sampling theorem for wavelet subspaces the classical shannon sampling theorem is extended to the subspaces used in the multiresolution analysis in wavelet theory under weak hypotheses these subspaces are first shown to have a riesz basis formed from the reproducing kernels these in turn are used to construct the sampling sequences examples are given

wavelet methods for pdes some recent developments abstract this paper is concerned with recent developments of wavelet schemes for the numerical treatment of operator equations with special emphasis on two issues adaptive solution concepts and nontrivial domain geometries after describing a general multiresolution framework the key features of wavelet bases are highlighted namely locality norm equivalences and cancellation properties assuming first that wavelet bases with these properties are available on the relevant problem domains the relevance of these features for a wide class of stationary problems is explained in subsequent sections the main issues are preconditioning and the efficient adaptive application of wavelet representations of the involved operators we indicate then how these ingredients combined with concepts from nonlinear or best n term approximation culminate in an adaptive wavelet scheme for elliptic selfadjoint problems covering boundary value problems as well as boundary integral equations these schemes can be shown to exhibit convergence rates that are in a certain sense asymptotically optimal we conclude this section with some brief remarks on data structures and implementation interrelations with regularity in a certain scale of besov spaces and strategies of extending such schemes to unsymmetric or indefinite problems we address then the adaptive evaluation of nonlinear functionals of wavelet expansions as a central task arising in connection with nonlinear problems wavelet bases on nontrivial domains are discussed next the main issues are the development of fourier free construction principles and criteria for the validity of norm equivalences finally we indicate possible combinations of wavelet concepts with conventional discretizations such as finite element or finite volume schemes in connection with convection dominated and hyperbolic problems

second generation wavelet collocation method for the solution of partial differential equations an adaptive numerical method for solving partial differential equations is developed the method is based on the whole new class of second generation wavelets wavelet decomposition is used for grid adaptation and interpolation while a new o n hierarchical finite difference scheme which takes advantage of wavelet multilevel decomposition is used for derivative calculations the treatment of nonlinear terms and general boundary conditions is a straightforward task due to the collocation nature of the algorithm in this paper we demonstrate the algorithm for one particular choice of second generation wavelets namely lifted interpolating wavelets on an interval with uniform regular sampling the main advantage of using second generation wavelets is that wavelets can be custom designed for complex domains and irregular sampling thus the strength of the new method is that it can be easily extended to the whole class of second generation wavelets leaving the freedom and flexibility to choose the wavelet basis depending on the application

global parametrization of feedback systems with nonlinear plants for unity feedback systems with a plant p and controller c we show that given a nonlinear stable p the class of all controller stabilizing the feedback system is globally parametrized by a stable map q we also show how q is useful in studying the effects of modeling errors on the stability

wavelets and the numerical solution of partial differential equations we present a new numerical method for the solution of partial differential equations in nonseparable domains the method uses a wavelet galerkin solver with a nontrivial adaptation of the standard capacitance matrix method the numerical solutions exhibit spectral convergence with regard to the order of the compactly supported daubechies wavelet basis furthermore the rate of convergence is found to be independent of the geometry we solve the helmholtz equation since for variations in the parameter the solutions have qualitative properties that well illustrate the applications of our method

on the parametrization of nonlinear finite gain over set stabilizing controllers the concepts of gain and incremental gain over set have been introduced to alleviate some of the limitations of the conventional notions of gain and incremental gain in this paper the class of all finite gain over set stabilizing controllers is parametrized in terms of a finite gain over set stable map q provided that the plant p is finite incremental gain over set stable an alternate theorem for finite gain over ball stability of the closed loop is also developed which provides a superset and a subset of the set of all finite gain over set stabilizing compensators for the feedback system

wavelet methods for fast resolution of elliptic problems this paper shows that the use of wavelets to discretize an elliptic problem with dirichlet or neumann boundary conditions has two advantages an explicit diagonal preconditioning makes the condition number of the corresponding matrix become bounded by a constant and the order of approximation is locally of spectral type in contrast with classical methods using a conjugate gradient method one thus obtains fast numerical algorithms of resolution a comparison is also drawn between wavelet and classical methods

on robust stabilization on nonlinear control systems abstract a robust stabilization problem for general mimo nonlinear systems is formulated in a banach space setting in the time domain existence and uniqueness theorems are established a general procedure for solving the problem is illustrated and a simple example is included to show how the results can be applied

inner outer factorization of nonlinear operators abstract a fundamental result in the theory of hardy spaces of analytic matrix and operator valued functions on the unit disk is inner outer factorization for the case of rational matrix functions explicit formulas for the inner and outer factors have recently been obtained in these formulas the functions are presented in terms of a collection of finite matrices as the transfer function of a linear system the key ingredient in the construction is the solution of a linear matrix equation called a stein equation in this paper we obtain extensions of these results to the setting of input output maps of nonlinear discrete time systems a particaular computable class of examples is presented for illustration

multiobjective h problems linear and nonlinear control in this note we compare the value of nonlinear and linear control for a multiobjective h control problem with a linear time invariant lti plant we show that if the h performance of a nonlinear feedback controller is measured by the maximum incremental gain the optimal achievable performance with nonlinear time varying control is identical to that achievable by lti control

model reference control system design for norm uncertain plants given a general non linear plant a method is presented for designing model reference feedback controllers for the plant it is shown that the feedback controllers produced will guarantee prespecified closed loop properties if the error between the plant and model lies within computable bounds in particular an upper bound on the tracking error is established and conditions are given that ensure closed loop stability

nonlinear unity feedback systems and q parametrization this paper concerns nonlinear systems defines a new concept of stability and extends to unity feedback systems the technique of q parametrization introduced by zames and developed by desoer chen and gustafson we specify 1 a global parametrization of all controllers that s stabilize a given s stable plant 2 a parametrization of a class of controllers that stabilize an unstable plant 3 necessary and sufficient conditions for a nonlinear controller to simultaneously stabilize two nonlinear plants

stabilization and robustness of non linear unity feedback system factorization approach this paper is a self contained discussion of a right factorization approach in the stability analysis of the non linear continuous time or discrete time time invariant or time varying well posed unity feedback system s1 p c we show that a well posed stable feedback system s1 p c implies that p and c have right factorizations in the case where c is stable p has a normalized right coprime factorization the factorization approach is used in stabilization and simultaneous stabilization results

a state space approach to parameterization of stabilizing controllers for nonlinear systems a state space approach to the youla parameterization of stabilizing controllers for linear and nonlinear systems is suggested the stabilizing controllers or a class of stabilizing controllers for nonlinear systems are characterized as fractional transformations of stable parameters the main idea behind this approach is to decompose the output feedback stabilization problem into state feedback and state estimation problems the parameterized output feedback controllers have separation structures this machinery allows the parameterization of stabilizing controllers to be conducted directly in state space without using coprime factorization

on a state space approach to nonlinear h control we study the standard h optimal control problem using state feedback for smooth nonlinear control systems the main theorem obtained roughly states that the l2 induced norm from disturbances to inputs and outputs can be made smaller than a constant 0 if the corresponding h norm for the system linearized at the equilibrium can be made smaller than by linear state feedback necessary and sufficient conditions for the latter problem are by now well known e g from the state space approach to linear h optimal control our approach to the nonlinear h optimal control problem generalizes the state space approach to the linear h problem by replacing the hamiltonian matrix and corresponding riccati equation as used in the linear context by a hamiltonian vector field together with a hamiltonian jacobi equation corresponding to its stable invariant manifold

feedback and optimal sensitivity model reference transformations multiplicative seminorms and approximate inverses in this paper the problem of sensitivity reduction by feedback is formulated as an optimization problem and separated from the problem of stabilization stable feedback schemes obtainable from a given plant are parameterized salient properties of sensitivity reducing schemes are derived and it is shown that plant uncertainty reduces the ability of feedback to reduce sensitivity the theory is developed for input output systems in a general setting of banach algebras and then specialized to a class of multivariable time invariant systems characterized by n times n matrices of h infty frequency response functions either with or without zeros in the right half plane the approach is based on the use of a weighted seminorm on the algebra of operators to measure sensitivity and on the concept of an approximate inverse approximate invertibility of the plant is shown to be a necessary and sufficient condition for sensitivity reduction an indicator of approximate invertibility called a measure of singularity is introduced the measure of singularity of a linear time invariant plant is shown to be determined by the location of its right half plane zeros in the absence of plant uncertainty the sensitivity to output disturbances can be reduced to an optimal value approaching the singularity measure in particular if there are no right half plane zeros sensitivity can be made arbitrarily small the feedback schemes used in the optimization of sensitivity resemble the lead lag networks of classical control design some of their properties and methods of constructing them in special cases are presented

fast wavelet transforms and numerical algorithms i a class of algorithms is introduced for the rapid numerical application of a class of linear operators to arbitrary vectors previously published schemes of this type utilize detailed analytical information about the operators being applied and are specific to extremely narrow classes of matrices in contrast the methods presented here are based on the recently developed theory of wavelets and are applicable to all calderon zygmund and pseudo differential operators the algorithms of this paper require order o n or o n log n operations to apply an n n matrix to a vector depending on the particular operator and the version of the algorithm being used and our numerical experiments indicate that many previously intractable problems become manageable with the techniques presented here

an adaptive multilevel wavelet collocation method for elliptic problems an adaptive multilevel wavelet collocation method for solving multi dimensional elliptic problems with localized structures is described the method is based on multi dimensional second generation wavelets and is an extension of the dynamically adaptive second generation wavelet collocation method for evolution problems int j comp fluid dyn 17 2003 151 wavelet decomposition is used for grid adaptation and interpolation while a hierarchical finite difference scheme which takes advantage of wavelet multilevel decomposition is used for derivative calculations the multilevel structure of the wavelet approximation provides a natural way to obtain the solution on a near optimal grid in order to accelerate the convergence of the solver an iterative procedure analogous to the multigrid algorithm is developed the overall computational complexity of the solver is o n where n is the number of adapted grid points the accuracy and computational efficiency of the method are demonstrated for the solution of two and three dimensional elliptic test problems

string stability of interconnected systems introduces the notion of string stability of a countably infinite interconnection of a class of nonlinear systems intuitively string stability implies uniform boundedness of all the states of the interconnected system for all time if the initial states of the interconnected system are uniformly bounded it is well known that the input output gain of all the subsystems less than unity guarantees that the interconnected system is input output stable the authors derive sufficient weak coupling conditions which guarantee the asymptotic string stability of a class of interconnected systems under the same weak coupling conditions string stable interconnected systems remain string stable in the presence of small structural singular perturbations in the presence of parameter mismatch these weak coupling conditions ensure that the states of all the subsystems are all uniformly bounded when a gradient based parameter adaptation law is used and that the states of all the systems go to zero asymptotically

packing hyperspheres in high dimensional euclidean spaces we present a study of disordered jammed hard sphere packings in four five and six dimensional euclidean spaces using a collision driven packing generation algorithm we obtain the first estimates for the packing fractions of the maximally random jammed mrj states for space dimensions d 4 5 and 6t o be mrj0 46 0 31 and 0 20 respectively to a good approximation the mrj density obeys the scaling form mrj c1 2 d c2d 2 d where c1 2 72 and c2 2 56 which appears to be consistent with the highdimensional asymptotic limit albeit with different coefficients calculations of the pair correlation function g2r and structure factor sk for these states show that short range ordering appreciably decreases with increasing dimension consistent with a recently proposed decorrelation principle which among other things states that unconstrained correlations diminish as the dimension increases and vanish entirely in the limit d as in three dimensions where mrj0 64 the packings show no signs of crystallization are isostatic and have a power law divergence in g2r at contact with power law exponent 0 4 across dimensions the cumulative number of neighbors equals the kissing number of the conjectured densest packing close to where g2r has its first minimum additionally we obtain estimates for the freezing and melting packing fractions for the equilibrium hard sphere fluid solid transition f0 32 and m0 39 respectively for d 4 and f0 20 and m0 25 respectively for d 5 although our results indicate the stable phase at high density is a crystalline solid nucleation appears to be strongly suppressed with increasing dimension

fast wavelet based algorithms for linear evolution equations the authors devise a class of fast wavelet based algorithms for linear evolution equations whose coefficients are time independent the method draws on the work of beylkin coifman and rokhlin comm pure appl math 44 1991 pp 141 184 which they applied to general calderon zygmund type integral operators the authors apply a modification of their idea to linear hyperbolic and parabolic equations with spatially varying coefficients the complexity for hyperbolic equations in one dimension is reduced from o n 2 to o n log 3 n there are somewhat better gains for parabolic equations in multidimensions

discrete singular convolution for the sine gordon equation abstract this paper explores the utility of a discrete singular convolution dsc algorithm for the integration of the sine gordon equation the initial values are chosen close to a homoclinic manifold for which previous methods have encountered significant numerical difficulties such as numerically induced spatial and temporal chaos a number of new initial values are considered including a case where the initial value is exactly on the homoclinic orbit the present algorithm performs extremely well in terms of accuracy efficiency simplicity stability and reliability

simultaneous stabilization of nonlinear systems we study the problem of simultaneous stabilization of a given set of nonlinear plants by one nonlinear not necessarily stable compensator we obtain a necessary and sufficient condition under which there is a single compensator which stabilizes a given set of n nonlinear plants this note emphasizes the importance of the following problem when is a nonlinear unstable plant stabilizable by an incrementally stable compensator

dimensional study of the caging order parameter at the glass transition the glass problem is notoriously hard and controversial even at the mean field level little is agreed upon regarding why a fluid becomes sluggish while exhibiting but unremarkable structural changes it is clear however that the process involves self caging which provides an order parameter for the transition it is also broadly assumed that this cage should have a gaussian shape in the mean field limit here we show that this ansatz does not hold by performing simulations as a function of spatial dimension d we find the cage to keep a nontrivial form quantitative mean field descriptions of the glass transition such as mode coupling theory density functional theory and replica theory all miss this crucial element although the mean field random first order transition scenario of the glass transition is qualitatively supported here and non mean field corrections are found to remain small on decreasing d reconsideration of its implementation is needed for it to result in a coherent description of experimental observations

solving hyperbolic pdes using interpolating wavelets a method is presented for adaptively solving hyperbolic pdes the method is based on an interpolating wavelet transform using polynomial interpolation on dyadic grids the adaptability is performed automatically by thresholding the wavelet coefficients operations such as differentiation and multiplication are fast and simple due to the one to one correspondence between point values and wavelet coefficients in the interpolating basis treatment of boundary conditions is simplified in this sparse point representation spr numerical examples are presented for one and two dimensional problems it is found that the proposed method outperforms a finite difference method on a uniform grid for certain problems in terms of flops

on a wavelet based method for the numerical simulation of wave propagation a wavelet based method for the numerical simulation of acoustic and elastic wave propagation is developed using a displacement velocity formulation and treating spatial derivatives with linear operators the wave equations are rewritten as a system of equations whose evolution in time is controlled by first order derivatives the linear operators for spatial derivatives are implemented in wavelet bases using an operator projection technique with nonstandard forms of wavelet transform using a semigroup approach the discretized solution in time can be represented in an explicit recursive form based on taylor expansion of exponential functions of operator matrices the boundary conditions are implemented by augmenting the system of equations with equivalent force terms at the boundaries the wavelet based method is applied to the acoustic wave equation with rigid boundary conditions at both ends in 1 d domain and to the elastic wave equation with a traction free boundary conditions at a free surface in 2 d spatial media the method can be applied directly to media with plane surfaces and surface topography can be included with the aid of distortion of the grid describing the properties of the medium the numerical results are compared with analytic solutions based on the cagniard technique and show high accuracy the wavelet based approach is also demonstrated for complex media including highly varying topography or stochastic heterogeneity with rapid variations in physical parameters these examples indicate the value of the approach as an accurate and stable tool for the simulation of wave propagation in general complex media

fourier spectral and wavelet solvers for the incompressible navier stokes equations with volume penalization convergence of a dipole wall collision in this study we use volume penalization to mimic the presence of obstacles in a flow or a domain with no slip boundaries this allows in principle the use of fast fourier spectral methods and coherent vortex simulation techniques based on wavelet decomposition of the flow variables to compute turbulent wall bounded flow or flows around solid obstacles by simply adding one term in the equation convergence checks are reported using a recently revived and unexpectedly difficult dipole wall collision as a benchmark computation several quantities like the vorticity isolines truncation error kinetic energy and enstrophy are inspected for a collision of a dipole with a no slip wall and compared with available benchmark data obtained with a standard chebyshev pseudospectral method we quantify the possible deteriorating effects of the gibbs phenomenon present in the fourier based schemes due to continuity restrictions of the penalized navier stokes equations on the wall it is found that gibbs oscillations have a negligible effect on the flow evolution allowing higher order recovery of the accuracy on a fourier basis by means of postprocessing an advantage of coherent vortex simulations on the other hand is that the degrees of freedom of the flow computation can strongly be reduced in this study we quantify the possible reduction of degrees of freedom while keeping the accuracy for an optimal convergence scenario the penalization parameter has to scale with the number of fourier and wavelet modes in addition an implicit treatment of the darcy drag term in the penalized navier stokes equations is beneficial since this allows one to set the time step independent from the penalization parameter without additional computational or memory requirements

an adaptive meshfree diffusion wavelet method for partial differential equations on the sphere abstract an adaptive meshfree diffusion wavelet method for solving partial differential equations pdes on the sphere is developed approximation formulae for laplacian beltrami 2 and gradient operators are derived using radial basis functions rbfs and the convergence of these approximations to 2 and is verified for two test functions the matrix approximating the laplace beltrami operator is used for the construction of the diffusion wavelet the diffusion wavelet is used for the adaptation of node arrangement as well as for the fast computation of dyadic powers of the matrices involved in the numerical solution of the pde the efficiency of the method is that the same operator is used for the construction of the diffusion wavelet and for the approximation of the differential operators as a part of the wavelet method the behaviour of the compression error with respect to different parameters involved in the construction of the diffusion wavelet is tested the cpu time taken by the proposed method is compared with the cpu time taken by the rbf based collocation method and it is observed that the proposed method performs better the method is tested on three test problems namely spherical diffusion equation linear problem of computing a moving steep front nonlinear and problem of turing patterns system of nonlinear reaction diffusion equations

operator based nonlinear feedback control design using robust right coprime factorization in this note robust stabilization and tracking performance of operator based nonlinear feedback control systems are studied by using robust right coprime factorization specifically a new condition of robust right coprime factorization of nonlinear systems with unknown bounded perturbations is derived using the new condition a broader class of nonlinear plants can be controlled robustly when the spaces of the nonlinear plant output and the reference input are different a space change filter is designed and in this case this note considers tracking controller design using the exponential iteration theorem

discontinuous legendre wavelet element method for elliptic partial differential equations abstract by incorporating the legendre multiwavelet into the discontinuous galerkin dg method this paper presents a novel approach for solving poisson s equation with dirichlet boundary which is known as the discontinuous legendre multiwavelet element dlwe method derive an adaptive algorithm for the method and estimate the approximating error of its numerical fluxes one striking advantage of our method is that the differential operator boundary conditions and numerical fluxes involved in the elementwise computation can be done with lower time cost numerical experiments demonstrate the validity of this method furthermore this paper generalizes the dlwe method to the general elliptic equations defined on a bounded domain and describes the possibilities of constructing optimal adaptive algorithm the proposed method and its generalizations are also applicable to some other kinds of partial differential equations

an adaptive wavelet vaguelette algorithm for the solution of pdes the paper first describes a fast algorithm for the discrete orthonormal wavelet transform and its inverse without using the scaling function this approach permits to compute the decomposition of a function into a lacunary wavelet basis i e a basis constituted of a subset of all basis functions up to a certain scale without modification the construction is then extended to operator adapted biorthogonal wavelets this is relevant for the solution of certain nonlinear evolutionary pdes where a priori information about the significant coefficients is available we pursue the approach described in j frohlich and k schneider europ j mech b fluids13 439 1994 which is based on the explicit computation of the scalewise contributions of the approximated function to the values at points of hierarchical grids here we present an improved construction employing the cardinal function of the multiresolution the new method is applied to the helmholtz equation and illustrated by comparative numerical results it is then extended for the solution of a nonlinear parabolic pde with semi implicit discretization in time and self adaptive wavelet discretization in space results with full adaptivity of the spatial wavelet discretization are presented for a one dimensional flame front as well as for a two dimensional problem

multilevel preconditioning this paper is concerned with multilevel techniques for preconditioning linear systems arising from galerkin methods for elliptic boundary value problems a general estimate is derived which is based on the characterization of besov spaces in terms of weighted sequence norms related to corresponding multilevel expansions the result brings out clearly how the various ingredients of a typical multilevel setting affect the growth rate of the condition numbers in particular our analysis indicates how to realize even uniformly bounded condition numbers for example the general results are used to show that the bramble pasciak xu preconditioner for piecewise linear finite elements gives rise to uniformly bounded condition numbers even when the refinements of the underlying triangulations are highly nonuniform furthermore they are applied to a general multivariate setting of refinable shift invariant spaces in particular covering those induced by various types of wavelets

factorization and feedback stabilization for nonlinear systems we establish the equivalence of internal input out stability for two feedback configurations of a nonlinear time varying plant p for which a related plant g is assumed to have a factorization g r with both r and r 1 incrementally stable this extends a factorization principle for stabilizability previously given only for the linear time invariant case as an application of a special case we recover a version of the youla parametrization of stabilizing compensators for the nonlinear case previously presented in the literature we use degree theory to parametrize a collection of solutions of the h control problem for the case of a 1 gain stable or lossless plant in the case of a plant g having a j inner outer factorization this last result combined with the above mentioned factorization principle leads to results on the h control problem for p

optimal disturbance rejection for nonlinear control systems the problem of optimal disturbance rejection for a general class of multiinput multioutput mimo nonlinear systems achieved over a set of stabilizing compensator operators is formulated in a banach space setting conditions on the operators are imposed on the basis of the inputs as well as the outputs an existence theorem for optimal solutions is established under some realistic and verifiable conditions for solving the optimization problem a generalized recursive nonlinear programming algorithm is passed and a simple example of nonlinear optimal feedback design is given to show that the problem can be reduced to standard min max optimization and is hence solvable

two dimensional cascadic finite element computations of combustion problems we present an integrated time space adaptive finite element method for solving systems of two dimensional nonlinear parabolic systems in complex geometry the partial differential system is first discretized in time using a singly linearly implicit runge kutta method of order three local time errors for the step size control are defined by an embedding strategy these errors are used to propose a new time step by a pi controller algorithm a multilevel finite element method with piecewise linear functions on unstructured triangular meshes is subsequently applied for the discretization in space the local error estimate of the finite element solution steering the adaptive mesh refinement is obtained solving local problems with quadratic trial functions located essentially at the edges of the triangulation this two fold adaptivity successfully ensures an a priori prescribed tolerance of the solution the devised method is applied to laminar gaseous combustion and to solid solid alloying reactions we demonstrate that for such demanding applications the employed error estimation and adaption strategies generate an efficient and versatile algorithm

qualitative equivalences of iss and lp gain stability properties for discrete time nonlinear systems input to state stability iss and inputoutput l2 gain are two important robust stability concepts in analyzing the interconnections of nonlinear dynamical systems in this paper we demonstrate several qualitative equivalences between different iss related properties and lp gain properties for discrete time systems via nonlinear changes of coordinates prior to that certain summation to summation estimates are shown to be equivalent to the standard definitions of iss and integral iss iiss which lead to sufficient lyapunov function conditions to verify lp gain properties combined with known results on the equivalence of 0 input global asymptotic stability and iiss and a superposition principle we subsequently outline interesting implicative relationships between various discrete time robust stability properties

toy model for the mean field theory of hard sphere liquids we investigate a toy model of liquid based on simplified hypernetted chain equations in very large spatial dimension d the model does not exhibit a phase transition but several regimes of behavior when d infinity can be observed in different intervals of the density

multiresolution algorithms for the numerical solution of hyperbolic conservation laws given any scheme in conservation form and an appropriate uniform grid for the numerical solution of the initial value problem for one dimensional hyperbolic conservation laws we describe a multiresolution algorithm that approximates this numerical solution to a prescribed tolerance in an efficient manner to do so we consider the grid averages of the numerical solution for a hierarchy of nested diadic grids in which the given grid is the finest and introduce an equivalent multiresolution representation the multiresolution representation of the numerical solution consists of its grid averages for the coarsest grid and the set of errors in predicting the grid averages of each level of resolution in this hierarchy from those of the next coarser one once the numerical solution is resolved to our satisfaction in a certain locality of some grid then the prediction errors there are small for this particular grid and all finer ones this enables us to compress data by setting to zero small components of the representation which fall below a prescribed tolerance therefore instead of computing the time evolution of the numerical solution on the given grid we compute the time evolution of its compressed multiresolution representation algorithmically this amounts to computing the numerical fluxes of the given scheme at the points of the given grid by a hierarchical algorithm which starts with the computation of these numerical fluxes at the points of the coarsest grid and then proceeds through diadic refinements to the given grid at each step of refinement we add the values of the numerical flux at the center of the coarser cells the information in the multiresolution representation of the numerical solution is used to determine whether the solution is locally well resolved when this is the case we replace the costly exact value of the numerical flux with an accurate enough approximate value which is obtained by an inexpensive interpolation from the coarser grid the computational efficiency of this multiresolution algorithm is proportional to the rate of data compression for a prescribed level of tolerance that can be achieved for the numerical solution of the given scheme