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

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

a one time truncate and encode multiresolution stochastic framework in this work a novel adaptive strategy for stochastic problems inspired from the classical email protected s framework is presented the proposed algorithm allows building in a very general manner stochastic numerical schemes starting from a whatever type of deterministic schemes and handling a large class of problems from unsteady to discontinuous solutions its formulations permits to recover the same results concerning the interpolation theory of the classical multiresolution approach but with an extension to uncertainty quantification problems the present strategy permits to build numerical scheme with a higher accuracy with respect to other classical uncertainty quantification techniques but with a strong reduction of the numerical cost and memory requirements moreover the flexibility of the proposed approach allows to employ any kind of probability density function even discontinuous and time varying without introducing further complications in the algorithm the advantages of the present strategy are demonstrated by performing several numerical problems where different forms of uncertainty distributions are taken into account such as discontinuous and unsteady custom defined probability density functions in addition to algebraic and ordinary differential equations numerical results for the challenging 1d kraichnan orszag are reported in terms of accuracy and convergence finally a two degree of freedom aeroelastic model for a subsonic case is presented though quite simple the model allows recovering some physical key aspect on the fluid structure interaction thanks to the quasi steady aerodynamic approximation employed the injection of an uncertainty is chosen in order to obtain a complete parameterization of the mass matrix all the numerical results are compared with respect to classical monte carlo solution and with a non intrusive polynomial chaos method

multiresolution based on weighted averages of the hat function i linear reconstruction techniques in this paper we analyze a particular example of the general framework developed in a harten it siam j numer anal 33 1996 pp 1205 1256 the case in which the discretization operator is obtained by taking local averages with respect to the hat function we consider a class of reconstruction procedures which are appropriate for this multiresolution setting and describe the associated prediction operators that allow us to climb up the ladder from coarse to finer levels of resolution in part i we use data independent linear reconstruction techniques as our approximation tool we show how to obtain multiresolution transforms in bounded domains and analyze their stability with respect to perturbations

parallel adaptive wavelet collocation method for pdes a parallel adaptive wavelet collocation method for solving a large class of partial differential equations is presented the parallelization is achieved by developing an asynchronous parallel wavelet transform which allows one to perform parallel wavelet transform and derivative calculations with only one data synchronization at the highest level of resolution the data are stored using tree like structure with tree roots starting at a priori defined level of resolution both static and dynamic domain partitioning approaches are developed for the dynamic domain partitioning trees are considered to be the minimum quanta of data to be migrated between the processes this allows fully automated and efficient handling of non simply connected partitioning of a computational domain dynamic load balancing is achieved via domain repartitioning during the grid adaptation step and reassigning trees to the appropriate processes to ensure approximately the same number of grid points on each process the parallel efficiency of the approach is discussed based on parallel adaptive wavelet based coherent vortex simulations of homogeneous turbulence with linear forcing at effective non adaptive resolutions up to 20483 using as many as 2048 cpu cores

wavelet optimized finite difference mesh for mhd flow in a circular duct abstract there is a large class of important problems in scientific and engineering applications where the solution shows very little variation in the core region but very rapid changes close to the boundary of the domain of interest magnetohydrodynamic flow problems belong to this category treatment of such problems numerically therefore requires a non uniform mesh which is very fine near the boundary the determination of the size of the grid in different parts of the domain is therefore a challenging problem some recent studies suggest that wavelet methods can be effectively applied to create an optimized mesh the method has mostly been used for one dimensional problems the present paper investigates the problem for fully developed mhd duct flow problems which are basically two dimensional the boundary layer character is more pronounced in one direction as compared to the other the governing equations comprise a coupled system of two partial differential equations with dirichlet boundary conditions wavelet based adaptive mesh is first created and then the system is discretized over this mesh computations have been carried out at different precision levels for various hartmann numbers for a pipe of circular section contours are also drawn to depict the core and the boundary layer region

generalized multiresolution schemes abstract the concept of a multiresolution scheme as introduced by a harten see for instance cam report 93 03 1993 cam report 94 10 1994 icase report no 94 59 1994 j comput phys 115 1994 319 338 is generalized this generalization is based on a categorical setting which provides a much clearer and a more comprehensive understanding of the subject examples show the broad spectrum of applicability of generalized multiresolution schemes

multiresolution schemes for the reactive euler equations we present multiresolution mr schemes for the efficient numerical solution of the one dimensional system of the reactive euler equations which has possibly stiff source terms the original version of the method was developed by a harten 1995 comm pure appl math 48 12 1305 for homogeneous hyperbolic conservation laws by computing the cell average mr representation of the solution we obtain much information about the solution s regularity this description of smoothness is then used to reduce the number of direct flux computations as well as the expensive high order eno essentially nonoscillatory reconstruction both of which are now performed only near discontinuities thereby the numerical solution procedure becomes considerably more efficient in the present case of the reactive euler equations the average efficiency factor measured by counting the number of actual flux computations ranges from about 5 to 12 this is on the same order of and in some cases comes reasonably close to actual speed up factors obtained by code timings which were between 3 to 5 the mr overhead rate was about 10 for the eno and 36 for tvd schemes respectively the quality of the solution is shown to be the same as that of the finest grid detailed numerical and performance results are shown for up to fourth order accuracy for source terms ranging from moderate to extremely stiff

fully adaptive multiresolution finite volume schemes for conservation laws the use of multiresolution decompositions in the context of finite volume schemes for conservation laws was first proposed by a harten for the purpose of accelerating the evaluation of numerical fluxes through an adaptive computation in this approach the solution is still represented at each time step on the finest grid resulting in an inherent limitation of the potential gain in memory space and computational time the present paper is concerned with the development and the numerical analysis of fully adaptive multiresolution schemes in which the solution is represented and computed in a dynamically evolved adaptive grid a crucial problem is then the accurate computation of the flux without the full knowledge of fine grid cell averages several solutions to this problem are proposed analyzed and compared in terms of accuracy and complexity

application of generalized wavelets an adaptive multiresolution scheme abstract in this paper we add a new member to the family of adaptive multiresolution schemes these schemes employ adaptive data dependent reconstruction techniques which use the expensive eno essentially nonoscillatory interpolation only near discontinuities and a simple central stencil everywhere else in addition by recursive diadic coarsening of the original fine grid in smooth regions the fluxes can be interpolated from the coarser mesh instead of being computed directly we first present semi discrete versions of harten s multiresolution schemes harten 1994 to appear compare their performance in the nonlinear cases to their fully discrete counterparts and then test them in the linear case as well for the latter we find an unexpected oscillatory behavior which after a closer examination is shown to be related to the linear smearing of discontinuities we shall therefore discuss numerical issues related to the original adaptive multiresolution schemes in doing so we propose a new switch between eno and central interpolation which provides a truly tolerably oscillatory interpolation the resulting algorithm is termed the modified adaptive multiresolution scheme the new second order central eno switch is also proved to ensure that wherever the central stencil is used it will still result is an essentially nonoscillatory interpolation

multiresolution representation in unstructured meshes in this paper we describe techniques to represent data which originate from discretization of functions in unstructured meshes in terms of their local scale components to do so we consider a nested sequence of discretization which corresponds to increasing levels of resolution and we define the scales as the difference in information between any two successive levels we obtain data compression by eliminating scale coefficients which are sufficiently small this capability for data compression can be used to reduce the cost of numerical schemes by solving for the more compact representation of the numerical solution in terms of its significant scale coefficients

stochastic discrete equation method sdem for two phase flows a new scheme for the numerical approximation of a five equation model taking into account uncertainty quantification uq is presented in particular the discrete equation method dem for the discretization of the five equation model is modified for including a formulation based on the adaptive semi intrusive asi scheme thus yielding a new intrusive scheme sdem for simulating stochastic two phase flows some reference test cases are performed in order to demonstrate the convergence properties and the efficiency of the overall scheme the propagation of initial conditions uncertainties is evaluated in terms of mean and variance of several thermodynamic properties of the two phases

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

bilinear optimal control of the velocity term in a kirchhoff plate equation abstract we consider a bilinear optimal control problem where the state equation is a kirchhoff plate equation the control acts as a multiplier of the velocity term we prove the existence of an optimal control in a class h um h l 0 t m h t m and uniqueness of this optimal control for t sufficiently small

on the global finite time stabilization of bilinear systems by homogeneous feedback laws applications to some pde s abstract in this paper the problem of global finite time stabilization of bilinear control systems by means of homogeneous feedback law is investigated we prove under some reasonable assumptions on the operators a and b that continuous bounded and discontinuous unbounded feedbacks stabilize globally in finite time the closed loop system for illustrative the examples of heat schrodinger and transport equations are considered where homogeneous stabilizing feedback is built for these systems

multiresolution schemes for conservation laws with viscosity this paper presents multiresolution schemes for the efficient numerical solution of one dimensional conservation laws with viscosity the method originally developed by a harten commun pure appl math to appear for hyperbolic conservation laws computes the cell average multiresolution representation of the solution which provides much information about the solution s regularity as a consequence the possibly expensive eno essentially nonoscillatory reconstruction as well as numerous flux computations are performed only near discontinuities and thereby the numerical solution procedure becomes considerably more efficient the multiresolution scheme is also expected to follow possibly unsteady irregularities from one time step to the next when viscosity is added predicting the location of the irregularity becomes a problem of estimating the change in shock thickness to this end we derive shock width estimates for our 1d prototype equations which when combined with the stability restriction of the numerical scheme provide a reliable mechanism for enlarging the original multiresolution stencil the numerical experiments for scalar conservation laws indicate the feasibility of multiresolution schemes for the viscous case as well

stabilization of unbounded bilinear control systems in hilbert space abstract we consider bilinear control systems of the form y t a y t u t b y t where a generates a strongly continuous semigroup of contraction e t a t 0 on an infinite dimensional hilbert space y whose scalar product is denoted by we suppose that this system is unbounded in the sense that the linear operator b is unbounded from the state y into itself tacking into account eventual control saturation we study the problem of stabilization by possibly nonquadratic feedback of the form u t f b y t y t applications to the heat equation is considered

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

exponential and weak stabilization of constrained bilinear systems this paper considers feedback stabilization for the infinite dimensional bilinear system frac dz t dt az t v t bz t where a is the infinitesimal generator of a linear c 0 semigroup of contractions on a hilbert space h and b h rightarrow h is a linear bounded operator sufficient conditions for exponential and weak stabilization are given a minimization problem is also considered applications to parabolic and hyperbolic equations are presented

multiresolution schemes on triangles for scalar conservation laws this paper proposes a multiresolution procedure adapted to triangular cell averages to improve the performance of finite volume schemes by reducing flux evaluation cost using the approach introduced by a harten a specific coarse to fine prediction scheme is proposed that ensures the stability of the computations even when a large number of scales are involved numerical tests are presented that illustrate the computational gain as well as the order of accuracy of the scheme

stabilization and decay estimate for distributed bilinear systems abstract in this paper we consider the problem of feedback stabilization for the distributed bilinear system y t ay t u t by t here a is the infinitesimal generator of a linear c0 semigroup of contractions on a hilbert space h and b h h is a linear bounded operator a sufficient condition for feedback stabilization is given and explicit decay estimate is established applications to vibrating systems are presented

oscillation of a quasilinear impulsive delay parabolic equation with two different boundary conditions in this paper we discuss the oscillation for a class of quasilinear impulsive delay parabolic equations with two different boundary conditions and obtain several oscillation criteria

continuous and discontinuous homogeneous feedbacks finite time partially stabilizing controllable multichained systems this paper gives various constructions of stabilizing feedbacks of the well known chained systems by various approaches briefly the paper presents new alternatives to overcome the brockett s obstruction these alternatives are the finite time partially stabilizability by continuous or discontinuous state feedback laws

wavelet optimized finite difference method using interpolating wavelets for self adjoint singularly perturbed problems we design a wavelet optimized finite difference wofd scheme for solving self adjoint singularly perturbed boundary value problems the method is based on an interpolating wavelet transform using polynomial interpolation on dyadic grids small dissipation of the solution is captured significantly using an adaptive grid the adaptive feature is performed automatically by thresholding the wavelet coefficients numerical examples have been solved and compared with non standard finite difference schemes in j m s lubuma k c patidar uniformly convergent non standard finite difference methods for self adjoint singular perturbation problems j comput appl math 191 2006 228 238 the proposed method outperforms the non standard finite difference for studying singular perturbation problems for small dissipations very small e and effective grid generation therefore the proposed method is better for studying the more challenging cases of singularly perturbed problems

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

on the stabilization in finite time of locally controllable systems by means of continuous time varying feedback law it is proven that if for any positive time t there exists an open loop control u a t depending continuously on the initial data a vanishing for a 0 and steering a small neighborhood of 0 into 0 in time t then the control system can be locally stabilized in small time by means of continuous time varying feedback law provided that the dimension of the state space is at least 4 and the strong accessibility rank condition holds

interpolating wavelet collocation method of time dependent maxwell s equations characterization of electrically large optical waveguide discontinuities biorthogonal interpolating wavelets have been applied to electromagnetic field modeling through the wavelet collocation method in time domain yielding a versatile first principle algorithm for the solution of time dependent maxwell s equations with inhomogeneous media the resulting scheme maintains high accuracy while by virtue of its sub gridding capability significant reduction of the computational expenditure has been obtained the proposed method has been applied to the analysis of two dimensional dielectric waveguide discontinuities particularly for the modeling of electrically large optical waveguides where the dimension of the analyzed structure is much larger than the wavelength of the highest frequency content of the transmitted signal the proposed method has been proven to be highly efficient compared to the standard finite difference method

a fast bie iteration method for an arbitrary body in a flow of incompressible inviscid fluid the paper is concerned with the new iteration algorithm to solve boundary integral equations arising in boundary value problems of mathematical physics the stability of the algorithm is demonstrated on the problem of a flow around bodies placed in the incompressible inviscid fluid with a discrete numerical treatment we approximate the exact matrix by a certain toeplitz one and then apply a fast algorithm for this matrix on each iteration step we illustrate the convergence of this iteration scheme by a number of numerical examples both for hard and soft boundary conditions it appears that the method is highly efficient for hard boundaries being much less efficient for soft boundaries

partial stabilisation of semilinear systems using bounded controls this paper studies a partial stabilisation problem for distributed semilinear systems this consists of steering only a part of the system s state to its equilibrium constrained controls are proposed to force the desired part of the state to approach the equilibrium necessary and sufficient conditions for weak and strong partial stabilisation are given applications are presented

multiresolution schemes for the numerical solution of 2 d conservation laws i a generalization of harten s multiresolution algorithms to two dimensional 2 d hyperbolic conservation laws is presented given a cartesian grid and a discretized function on it the method computes the local scale components of the function by recursive diadic coarsening of the grid since the function s regularity can be described in terms of its scale or multiresolution analysis the numerical solution of conservation laws becomes more efficient by eliminating flux computations wherever the solution is smooth instead in those locations the divergence of the solution is interpolated from the next coarser grid level first the basic 2 d essentially nonoscillatory eno scheme is presented then the 2 d multiresolution analysis is developed and finally the subsequent scheme is tested numerically the computational results confirm that the efficiency of the numerical scheme can be considerably improved in two dimensions as well

convergence of an adaptive semi lagrangian scheme for the vlasov poisson system an adaptive semi lagrangian scheme for solving the cauchy problem associated to the periodic 1 1 dimensional vlasov poisson system in the two dimensional phase space is proposed and analyzed a key feature of our method is the accurate evolution of the adaptive mesh from one time step to the next one based on a rigorous analysis of the local regularity and how it gets transported by the numerical flow the accuracy of the scheme is monitored by a prescribed tolerance parameter e which represents the local interpolation error at each time step in the l metric the numerical solutions are proved to converge in l towards the exact ones as e and t tend to zero provided the initial data is lipschitz and has a finite total curvature or in other words that it belongs to w 1 infty cap w 2 1 the rate of convergence is mathcal o delta t 2 varepsilon delta t which should be compared to the results of besse who recently established in siam j numer anal 42 1 350 382 2004 similar rates for a uniform semi lagrangian scheme but requiring that the initial data are in mathcal c 2 several numerical tests illustrate the effectiveness of our approach for generating the optimal adaptive discretizations

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

existence theory for perturbed impulsive hyperbolic differential inclusions with variable times in this paper the existence of solutions for a perturbed impulsive hyperbolic differential inclusion with variable times is proved under the mixed generalized lipschitz and caratheodory s conditions

comparison principles for impulsive hyperbolic equations of first order strong and weak comparison principles for impulsive hyperbolic equations of first order are proved uniqueness criterion is obtained

exponential stabilization of distributed semilinear systems by optimal control this paper studies exponential stabilization of distributed semilinear systems the paper i gives a constrained feedback control that ensure the exponential stabilizability and ii shows that this control is the unique solution of an appropriate minimization problem examples of hyperbolic partial equations are provided

a multiresolution finite volume scheme for two dimensional hyperbolic conservation laws in this paper high resolution finite volume schemes are combined with an adaptive mesh technique inspired by multiresolution analysis to improve the computational efficiency for two dimensional hyperbolic conservation laws the method is conservative moreover it is stable which is proven numerically in this paper the computational grid is dynamically adapted so that higher spatial resolution is automatically allocated to regions where strong gradients are observed using this proposed scheme we compute several two dimensional model problems and a compressive rate ranging from about 5 10 is observed in all simulations

oscillatory solutions of delay hyperbolic system with distributed deviating arguments in this paper we consider the oscillatory solutions of delay hyperbolic system under robin and dirichlet boundary value conditions more precisely using eigenvalue method we can obtain several sufficient conditions of oscillation of solutions of hyperbolic functional differential system with continuous distributed deviating arguments

adaptive multiresolution schemes for shock computations in this paper we present adaptive multiresolution schemes for the computation of discontinuous solutions of hyperbolic conservation laws starting with the given grid we consider the grid averages of the numerical solution for a hierarchy of nested grids which is obtained by diadic coarsening and we compute its equivalent multiresolution representation this representation of the numerical solution consists of the grid averages of the coarsest grid and the set of errors in predicting the grid averages of each level of resolution from the next coarser one these errors depend on the size of the grid and the local regularity of the solution at a jump discontinuity they remain the size of the jump independent of the level of resolution this observation enables us to identify the location of discontinuities in the numerical solution in a region of smoothness once the numerical solution is resolved to our satisfaction at a certain locality of some grid then the prediction errors there for this grid and all finer ones are small this enables us to obtain data compression by setting to zero the terms of the multiresolution representation that fall below a specified tolerance the numerical flux of the adaptive scheme is taken to be that of a standard centered scheme unless it corresponds to an identified discontinuity in which case it is taken to be the flux of an eno scheme we use the data compression of the numerical solution in order to reduce the number of numerical flux evaluations as follows we start with the computation of the exact numerical fluxes at the few grid points of the coarsest grid and then we proceed through diadic refinement to the given grid at each step of refinement we add values for the numerical flux at the new grid points which are the centers of the coarser cells wherever the solution is locally well resolved i e the corresponding prediction error is below the specified tolerance the costly exact value of the numerical flux function is replaced by an accurate enough approximate value which is obtained by an inexpensive interpolation from the values of the coarser grid

a numerical strategy to discretize and solve the poisson equation on dynamically adapted multiresolution grids for time dependent streamer discharge simulations we develop a numerical strategy to solve multi dimensional poisson equations on dynamically adapted grids for evolutionary problems disclosing propagating fronts the method is an extension of the multiresolution finite volume scheme used to solve hyperbolic and parabolic time dependent pdes such an approach guarantees a numerical solution of the poisson equation within a user defined accuracy tolerance most adaptive meshing approaches in the literature solve elliptic pdes level wise and hence at uniform resolution throughout the set of adapted grids here we introduce a numerical procedure to represent the elliptic operators on the adapted grid strongly coupling inter grid relations that guarantee the conservation and accuracy properties of multiresolution finite volume schemes the discrete poisson equation is solved at once over the entire computational domain as a completely separate process the accuracy and numerical performance of the method are assessed in the context of streamer discharge simulations