over dispersed age period cohort models summarywe consider inference and forecasting for aggregate data organised in a two way table with age and cohort as indices but without measures of exposure this is modelled using a poisson likelihood with an age period cohort structure for the mean while allowing for over dispersion we propose a repetitive structure that keeps the dimension of table fixed while increasing the latent exposure for this we use a class of infinitely divisible distributions which include a variety of compound poisson models and poisson mixture models this results in asymptotic f inference and t forecast distributions

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

adaptive multiresolution approach for two dimensional pdes a multiresolution adaptive approach for the solution of two dimensional partial differential equations pdes is presented this methodology is a multidimensional extension of that presented in a previous work comput methods appl mech engrg 191 2002 3909 the method proposed is unconditionally stable by incorporating convection differencing schemes with the tvd property and the grid is dynamically adapted so that higher spatial resolution is automatically allocated to domain regions where strong gradients are observed the two desired properties of any pde solver stability and accuracy are therefore retained numerical results for four test problems are presented which serve to demonstrate the robustness and cost effectiveness of the 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

effortless critical representation of laplacian pyramid the laplacian pyramid lp is a multiresolution representation introduced originally for images and it has been used in many applications a major shortcoming of the lp representation is that it is oversampled the dependency among the lp coefficients is studied in this paper it is shown that whenever the lp compression filter is interpolatory the redundancy in the lp coefficients can be removed effortlessly by merely discarding some of the lp coefficients furthermore it turns out that the remaining now critically sampled lp coefficients are actually the coefficients of a wavelet filter bank as a result a new algorithm for designing a nonredundant wavelet filter bank from non biorthogonal lowpass filters is obtained our methodology presented in this paper does not depend on the spatial dimension of the data or the dilation matrix for sampling

adaptive multiresolution weno schemes for multi species kinematic flow models multi species kinematic flow models lead to strongly coupled nonlinear systems of first order spatially one dimensional conservation laws the number of unknowns the concentrations of the species may be arbitrarily high models of this class include a multi species generalization of the lighthill whitham richards traffic model and a model for the sedimentation of polydisperse suspensions their solutions typically involve kinematic shocks separating areas of constancy and should be approximated by high resolution schemes a fifth order weighted essentially non oscillatory weno scheme is combined with a multiresolution technique that adaptively generates a sparse point representation spr of the evolving numerical solution thus computational effort is concentrated on zones of strong variation near shocks numerical examples from the traffic and sedimentation models demonstrate the effectiveness of the resulting weno multiresolution weno mrs scheme

data sparse approximation to the operator valued functions of elliptic operator llin previous papers the arithmetic of hierarchical matrices has been described which allows us to compute the inverse for instance of finite element stiffness matrices discretising an elliptic operator l the required computing time is up to logarithmic factors linear in the dimension of the matrix in particular this technique can be used for the computation of the discrete analogue of a resolvent zi l 1 z c in the present paper we consider various operator functions the operator exponential e tl negative fractional powers l the cosine operator function cos t l l k and finally the solution operator of the lyapunov equation using the dunford cauchy representation we get integrals which can be discretised by a quadrature formula which involves the resolvents z k i l 1 mentioned above we give error estimates which are partly exponentially partly polynomially decreasing

backward euler method for abstract time dependent parabolic equations with variable domains we consider semidiscretizations in time based on the backward euler method of an abstract non autonomous parabolic initial value problem where a t d a t subset x to x 0 le t le t is a family of sectorial operators in a banach space x the domains d a t are allowed to depend on t our hypotheses are fulfilled for classical parabolic problems in the l p 1 p infty norms we prove that the semidiscretization is stable in a suitable sense we get optimal estimates for the error even when non homogeneous boundary values are considered in particular the results are applicable to the analysis of the semidiscretizations of time dependent parabolic problems under non homogeneous neumann boundary conditions

reconstruction of turbulent fluctuations using a hybrid rans les approach a hybrid rans les approach is presented and assessed considering several flows which can be interpreted as the most general case of the nlde approach as defined by morris et al p j morris l n long a bangalore and a wang j comput phys 133 56 1997 a decomposition into three parts of the exact solution of the navier stokes equations is considered mean flow resolved fluctuations and unresolved subgrid fluctuations the mean flow is computed using a classical rans method while resolved fluctuations are derived from a les method this approach is succesfully assessed on the stationary and the pulsed plane channel flow configurations the case of the flow around a low pressure turbine blade is discussed in a second step the hybrid method is also demonstrated to be robust with respect to several sources of error and provides good results in the case of a coarsened grid simulation a reduced computational domain simulation and nonconsistent mean flow fields

data sparse approximation to a class of operator valued functions in earlier papers we developed a method for the data sparse approximation of the solution operators for elliptic parabolic and hyperbolic pdes based on the dunford cauchy representation to the operator valued functions of interest combined with the hierarchical matrix approximation of the operator resolvents in the present paper we discuss how these techniques can be applied to approximate a hierarchy of the operator valued functions generated by an elliptic operator

exponentially convergent algorithms for the operator exponential with applications to inhomogeneous problems in banach spaces new exponentially convergent algorithms for the operator exponential generated by a strongly positive operator a in a banach space x are proposed these algorithms are based on representations by a dunford cauchy integral along paths enveloping the spectrum of a combined with a proper quadrature involving a short sum of resolvents where the choice of the integration path dramatically affects desired features of the algorithms a parabola and a hyperbola are analyzed as the integration paths and scales of estimates of dependence on the smoothness of initial data i e of the initial vector and of the inhomogeneous right hand side are obtained one of the algorithms possesses an exponential convergence rate for the operator exponential e at for all t ge 0 including the initial point this allows one to construct an exponentially convergent algorithm for inhomogeneous initial value problems the algorithm is parallelizable it turns out that the resolvent must be modified in order to get numerically stable algorithms near the initial point the efficiency of the proposed method is demonstrated by numerical examples

high order nonlinear interpolatory reconstruction operators and associated multiresolution schemes this paper is devoted to the construction and analysis of a new family of nonlinear reconstruction operators and associated interpolating subdivision schemes they are based on centered piecewise nonlinear polynomial interpolations adapted to discontinuities we analyze both subdivision schemes and multiresolution schemes associated to these reconstructions some practical properties of these schemes are demonstrated in various numerical examples

an adaptive multiresolution method on dyadic grids we propose a modified adaptive multiresolution scheme for solving d dimensional hyperbolic conservation laws which is based on cell average discretization in dyadic grids adaptivity is obtained by interrupting the refinement at the locations where appropriate scale wavelet coefficients are sufficiently small one important aspect of such a multiresolution representation is that we can use the same binary tree data structure for domains of any dimension the tree structure allows us to succinctly represent the data and efficiently navigate through it dyadic grids also provide a more gradual refinement as compared with the traditional quad trees 2d or oct trees 3d that are commonly used for multiresolution analysis we show some examples of adaptive binary tree representations with significant savings in data storage when compared to quad tree based schemes as a test problem we also consider this modified adaptive multiresolution method using a dynamic binary tree data structure applied to a transport equation in 2d domain based on a second order finite volume discretization highlights we describe an adaptive multiresolution method for d dimensional pdes the method is based on cell average discretization on a dyadic grid the grid refinement is dynamically adapted to the current solution the same binary tree structure is used for domains of any dimension the method is tested on a transport problem in a two dimensional domain

a nonlinear chaikin based binary subdivision scheme abstract in this work we introduce and analyze a new nonlinear subdivision scheme based on a nonlinear blending between chaikin s subdivision rules and the linear 3 cell subdivision scheme our scheme seeks to improve the lack of convergence in the uniform metric of the nonlinear scheme proposed in amat et al 2012 where the authors define a cell average version of the pph subdivision scheme amat et al 2006 the properties of the new scheme are analyzed and its performance is illustrated through numerical examples

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

biorthogonal bases of compactly supported wavelets orthonormal bases of compactly supported wavelet bases correspond to subband coding schemes with exact reconstruction in which the analysis and synthesis filters coincide we show here that under fairly general conditions exact reconstruction schemes with synthesis filters different from the analysis filters give rise to two dual riesz bases of compactly supported wavelets we give necessary and sufficient conditions for biorthogonality of the corresponding scaling functions and we present a sufficient condition for the decay of their fourier transforms we study the regularity of these biorthogonal bases we provide several families of examples all symmetric corresponding to linear phase filters in particular we can construct symmetric biorthogonal wavelet bases with arbitrarily high preassigned regularity we also show how to construct symmetric biorthogonal wavelet bases close to a nonsymmetric orthonormal basis

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

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

exponentially convergent duhamel like algorithms for differential equations with an operator coefficient possessing a variable domain in a banach space a suitable abstract setting of an initial value problem for the first order operator differential equation is introduced for the case when both the unbounded operator coefficient and its domain can depend on the differentiation variable t a new exponentially convergent algorithm is proposed the algorithm is based on a generalization of the duhamel integral for vector valued functions the duhamel like technique makes it possible to translate the initial problem to an integral equation and then approximate it with exponential accuracy the theoretical results are illustrated by examples associated with the heat transfer boundary value problems

a spectral order method for inverting sectorial laplace transforms laplace transforms which admit a holomorphic extension to some sector strictly containing the right half plane and exhibiting a potential behavior are considered a spectral order parallelizable method for their numerical inversion is proposed the method takes into account the available information about the errors arising in the evaluations several numerical illustrations are provided

generalized wavelets design using kernel methods application to signal processing abstract multiresolution representations of data are powerful tools in signal processing in harten s framework multiresolution transforms are defined by predicting finer resolution levels of information from coarser ones using an operator called the prediction operator and defining details or wavelet coefficients that are the difference between the exact values and the predicted values in this paper we present a multiresolution scheme using local polynomial regression theory in order to design a more accurate prediction operator the stability of the scheme is proved and the order of the method is calculated finally some results are presented comparing our method with the classical methods

kernel based adaptive approximation of functions with discontinuities one of the basic principles of approximation theory is that the quality of approximations increase with the smoothness of the function to be approximated functions that are smooth in certain subdomains will have good approximations in those subdomains and these sub approximations can possibly be calculated efficiently in parallel as long as the subdomains do not overlap this paper proposes an algorithm that first calculates sub approximations on non overlapping subdomains then extends the subdomains as much as possible and finally produces a global solution on the given domain by letting the subdomains fill the whole domain consequently there will be no gibbs phenomenon along the boundaries of the subdomains the method detects faults and gradient faults with good accuracy throughout the algorithm works for fixed scattered input data of the function itself not on spectral data and it does not resample

on a class of l1 stable nonlinear cell average multiresolution schemes this paper is devoted to the analysis of a general family of nonlinear cell average multiresolution schemes the l 1 stability of this multiresolution transform is analyzed some examples are presented

h matrix approximation for the operator exponential with applications we develop a data sparse and accurate approximation to parabolic solution operators in the case of a rather general elliptic part given by a strongly p positive operator 4

a multilevel algorithm for large eddy simulation of turbulent compressible flows abstract a multilevel method for large eddy simulation of turbulent compressible flows is proposed the method relies on the splitting of the turbulent flowfield into several frequency bands in space and time each band being associated to a specific computational grid in physical space this allows to take into account in a deterministic way the information contained on finer grids a subgrid model adapted to such a decomposition based on a generalization of the germano s identity to multilevel decomposition is also introduced the approach is validated by several multilevel simulations in a subsonic plane channel flow configuration for a low and a high value of the reynolds number while reductions of the cpu times up to 80 are obtained

nonlinear harten s multiresolution on the quincunx pyramid multiresolution transforms provide useful tools for image processing applications for an optimal representation of the edges it is crucial to develop nonlinear schemes which are not based on tensor product this paper links the nonseparable quincunx pyramid and the nonlinear discrete harten s multiresolution framework in order to obtain the stability of these representations an error control multiresolution algorithm is introduced a prescribed accuracy in various norms is ensured by these strategies explicit error bounds are presented finally a nonlinear reconstruction is proposed and tested

adaptive rational interpolation for point values abstract g ramponi et al introduced in carrato et al 1997 1998 castagno and ramponi 1996 and ramponi 1995 a non linear rational interpolator of order two in this paper we extend this result to get order four we observe the gibbs phenomenon that is obtained near discontinuities with its weights with the weights we propose we obtain approximations of order four in smooth regions and three near discontinuities we also introduce a rational nonlinear extrapolation which is also of order four in the smooth region of the given function in the experiments we calculate numerically approximation orders for the different methods described in this paper and see that they coincide with those that have been obtained theoretically we also present reconstructions in 1d and 2d with which we reach the same conclusions

on specific stability bounds for linear multiresolution schemes based on piecewise polynomial lagrange interpolation abstract the deslauriers dubuc symmetric interpolation process can be considered as an interpolatory prediction scheme within harten s framework in this paper we express the deslauriers dubuc prediction operator as a combination of either second order or first order differences through a detailed analysis of certain contractivity properties we arrive to specific l stability bounds for the multiresolution transform a variety of tests indicate that these l bounds are closer to numerical estimates than those obtained with other approaches

multiwavelet based grid adaptation with discontinuous galerkin schemes for shallow water equations we provide an adaptive strategy for solving shallow water equations with dynamic grid adaptation including a sparse representation of the bottom topography a challenge in computing approximate solutions to the shallow water equations including wetting and drying is to achieve the positivity of the water height and the well balancing of the approximate solution a key property of our adaptive strategy is that it guarantees that these properties are preserved during the refinement and coarsening steps in the adaptation process the underlying idea of our adaptive strategy is to perform a multiresolution analysis using multiwavelets on a hierarchy of nested grids this provides difference information between successive refinement levels that may become negligibly small in regions where the solution is locally smooth applying hard thresholding the data are highly compressed and local grid adaptation is triggered by the remaining significant coefficients furthermore we use the multiresolution analysis of the underlying data as an additional indicator of whether the limiter has to be applied on a cell or not by this the number of cells where the limiter is applied is reduced without spoiling the accuracy of the solution by means of well known 1d and 2d benchmark problems we verify that multiwavelet based grid adaptation can significantly reduce the computational cost by sparsening the computational grids while retaining accuracy and keeping well balancing and positivity

the simplest decomposition of a turbulent field abstract in this paper we will present some recent results concerning the simplest decomposition of a turbulent field the large scale filtering operator is simply given by the two point sum in space and the associated fluctuation is given by the two point difference in the paper we will present the general properties of this simple decomposition and their particular relations with the subgrid stresses and the generalized central moments associated to a generic filtering operator

the pchip subdivision scheme in this paper we propose and analyze a nonlinear subdivision scheme based on the monotononicity preserving third order hermite type interpolatory technique implemented in the pchip package in matlab we prove the convergence and the stability of the pchip nonlinear subdivision process by employing a novel technique based on the study of the generalized jacobian of the first difference scheme

some sets of indistinguishability operators as multiresolution families multiresolution is a general mathematical concept that allows us to study a property by means of several changes of resolution from a fixed resolution a coarser projection can be calculated and then the changes between a finer resolution and a coarser one can be studied that information can give a good knowledge about the problem under consideration also using multiresolution techniques it is possible to present information with a higher or a lower detail given a way to get the adequate granularity or abstraction for a context the granularity of a system can be obtained or modeled by the use of indistinguishability operators in this work the relation between indistinguishability operators and multiresolution theory is studied and several methods to build families of indistinguishability operators with multiresolution capacities are given

orthonormal bases of compactly supported wavelets we construct orthonormal bases of compactly supported wavelets with arbitrarily high regularity the order of regularity increases linearly with the support width we start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction the construction then follows from a synthesis of these different approaches

recursive biorthogonal interpolating wavelets and signal adapted interpolating filter banks in this paper by combining the ideas of the recursive wavelets with second generation wavelets a family of recursive biorthogonal interpolating wavelets rbiws is developed the rbiws have simple shape parameter vectors on each level which allows a multichannel decomposition algorithm and provides a flexible structure for designing signal adapted interpolating filter banks in the single level case an efficient approach to design an optimum two channel biorthogonal interpolating filter bank is proposed which maximizes the coding gain under the traditional quantization noise assumption furthermore in the multilevel case using level wise optimization of the shape parameter vectors signal adapted tree structured recursive biorthogonal interpolating filter banks rbifbs are designed which are efficient in computation and can remarkedly improve the coding gain finally numerical results demonstrate the effectiveness of the proposed methods

nonseparable multiresolution with error control multiresolution representations of data are a powerful tool in data compression for a proper adaptation to the edges it is crucial to develop nonlinear methods which are not based on tensor product thus one needs to control the stability of these representations in this paper two dimensional multiresolution processing algorithms that ensure this stability are introduced a prescribed accuracy in various norms is ensured by these strategies explicit error bounds are presented

multiresolution based on weighted averages of the hat function ii nonlinear reconstruction techniques in this paper we describe and analyze a class of nonlinear mr schemes for the multiresolution setting which corresponds to discretization by local averages with respect to the hat function these schemes are based on the essentially non oscillatory eno interpolatory procedure described in a harten b engquist s osher and s chakravarthy j comput phys 71 1987 pp 231 302 we show that by allowing the approximation to fit the local nature of the data one can improve the compression capabilities of the multiresolution algorithms the question of stability for nonlinear data dependent reconstruction techniques is also addressed

wavelets and dilation equations a brief introduction wavelets are new families of basis functions that yield the representation f x sum b jk w 2 j x k their construction begins with the solution phi x to a dilation equation with coefficients c k then w comes from phi and the basis comes by translation and dilation of w it is shown in part 1 how conditions on the c k lead to approximation properties and orthogonality properties of the wavelets part 2 describes the recursive algorithms also based on the c k that decompose and reconstruct f the object of wavelets is to localize as far as possible in both time and frequency with efficient algorithms

a family of stable nonlinear nonseparable multiresolution schemes in 2d multiresolution representations of data are powerful tools in data compression for a proper adaptation to the edges a good strategy is to consider a nonlinear approach thus one needs to control the stability of these representations in this paper 2d multiresolution processing algorithms that ensure this stability are introduced a prescribed accuracy is ensured by these strategies

adaptive multiresolution discontinuous galerkin schemes for conservation laws a multiresolution based adaptation concept is proposed that aims at accelerating discontinuous galerkin schemes applied to nonlinear hyperbolic conservation laws opposite to standard adaptation concepts no error estimates are needed to tag mesh elements for refinement instead of this a multiresolution analysis is performed on a hierarchy of nested grids for the data given on a uniformly refined mesh this provides difference information between successive refinement levels that may become negligibly small in regions where the solution is locally smooth applying hard thresholding the data are highly compressed and local grid adaptation is triggered by the remaining significant coefficients a central mathematical problem addressed in this work is then to show that the adaptive solution is of the same accuracy as the reference solution on a uniformly refined mesh numerical comparisons demonstrate the efficiency of the concept and provide reliable estimates of the actual error in the numerical solution

smoothing for signals with discontinuities using higher order mumford shah models minimizing the mumford shah functional is frequently used for smoothing signals or time series with discontinuities a significant limitation of the standard mumford shah model is that linear trends and in general polynomial trends in the data are not well preserved this can be improved by building on splines of higher order which leads to higher order mumford shah models in this work we study these models in the univariate situation we discuss important differences to the first order mumford shah model and we obtain uniqueness results for their solutions as a main contribution we derive fast minimization algorithms for mumford shah models of arbitrary orders we show that the worst case complexity of all proposed schemes is quadratic in the length of the signal remarkably they thus achieve the worst case complexity of the fastest solver for the piecewise constant mumford shah model which is the simplest model of the class further we obtain stability results for the proposed algorithms we complement these results with a numerical study our reference implementation processes signals with more than 10 000 elements in less than 1 s

an algebraic variational multiscale multigrid method based on plain aggregation for convection diffusion problems a new computational framework referred to as algebraic variational multiscale multigrid method is proposed representing the initial step of our work on merging the variational multiscale method with algebraic multigrid projection methods this new approach allows for a separation of resolved scales into various scale groups in a purely algebraic way that is with no need to utilize further discretizations beyond the basic one by this means it conveniently facilitates the application of modeling terms either to all scale groups or only to selected scale groups for instance only to the fine scales of the problem variational multigrid techniques form the basis of our new approach after presenting the cornerstones of the framework for an abstract problem formulation the close relationship of the variational multiscale method to projection methods in general and to variational multigrid techniques are discussed the algebraic variational multiscale multigrid method is then particularly analyzed for convection diffusion problems the present initial study focuses on exploiting the methodical aspects of the new framework by developing a fine scale discontinuity capturing approach to diminish oscillations at sharp layers our technique including a fine scale discontinuity capturing term is applied to numerical examples of strongly convection dominated convection diffusion problems the results demonstrate that it enables the diminuation of local oscillations at parabolic boundary and interior layers by about 60 80 percent without any notable smearing effect

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

generalized multiresolution analysis on unstructured grids efficiency of high order essentially non oscillatory eno approximations of conservation laws may be drastically improved if ideas of multiresolution analysis are taken into account these methods of data compression not only reduce the necessary amount of discrete data but can also serve as tools in detecting local low dimensional features in the numerical solution we describe the mathematical background of the generalized multiresolution analysis as developed by abgrall and harten in 2 5 6 the functional analytic background is ultimately reduced to matrix vector operations of linear algebra we consider the important case of multiresolution analysis of cell average data which is used in finite volume approximations