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Sparse evaluation of compositions of functions using multiscale expansions
 SIAM J. Math. Anal
"... Abstract. This paper is concerned with the estimation and evaluation of wavelet coefficients of the composition F◦u of two functions F and u from the wavelet coefficients of u. Our main objective is to show that certain sequence spaces that can be used to measure the sparsity of the arrays of wavele ..."
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Abstract. This paper is concerned with the estimation and evaluation of wavelet coefficients of the composition F◦u of two functions F and u from the wavelet coefficients of u. Our main objective is to show that certain sequence spaces that can be used to measure the sparsity of the arrays of wavelet coefficients are stable under a class of nonlinear mappings F that occur naturally, e.g., in nonlinear PDEs. We indicate how these results can be used to facilitate the sparse evaluation of arrays of wavelet coefficients of compositions at asymptotically optimal computational cost. Furthermore, the basic requirements are verified for several concrete choices of nonlinear mappings. These results are generalized to compositions by a multivariate map F of several functions u1,...,un and their derivatives, i.e., F(Dα1u1,...,Dαnun).
New resolution strategy for multiscale reaction waves using time operator splitting, space adaptive multiresolutionanddedicatedhighorderimplicit/explicittimeintegrators
 29 version 2  8 Feb 2013
"... Abstract. We tackle the numerical simulation of reactiondiffusion equations modeling multiscale reaction waves. This type of problems induces peculiar difficulties and potentially large stiffness which stem from the broad spectrum of temporal scales in the nonlinear chemical source term as well as ..."
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Abstract. We tackle the numerical simulation of reactiondiffusion equations modeling multiscale reaction waves. This type of problems induces peculiar difficulties and potentially large stiffness which stem from the broad spectrum of temporal scales in the nonlinear chemical source term as well as from the presence of steep spatial gradients in the reaction fronts, spatially very localized. In this paper, we introduce a new resolution strategy based on time operator splitting and space adaptive multiresolution in the context of very localized and stiff reaction fronts. It considers a high order implicit time integration of the reaction and an explicit one for the diffusion term in order to build a time operator splitting scheme that exploits efficiently the special features of each problem. Based on recent theoretical studies of numerical analysis such a strategy leads to a splitting time step which is not restricted neither by the fastest scales in the source term nor by stability constraints of the diffusive steps, but only by the physics of the phenomenon. We aim thus at solving complete models including all time and space scales within a prescribed accuracy, considering large simulation domains with conventional computing resources. The efficiency is evaluated through the numerical simulation of configurations which were so far, out of reach of standard methods in the field of nonlinear chemical dynamics for 2D spiral waves and 3D scroll waves as an illustration. Future extensions of the proposed strategy to more complex configurations involving other physical phenomena as well as optimization capability on new computer architectures are finally discussed. Key words. Reactiondiffusion equations, multiscale reaction waves, operator splitting, adaptive multiresolution AMS subject classifications. 33K57, 35A18, 65M50, 65M08 1. Introduction. Numerical
A parallel Vlasov solver using a wavelet based adaptive mesh refinement
 In 7th Workshop on High Perf. Scientific and Engineering Computing (ICPP’2005
, 2005
"... We are interested in solving the Vlasov equation used to describe collective effects in plasmas. This nonlinear partial differential equation coupled with Maxwell equation describes the time evolution of the particle distribution in phase space. The numerical solution of the full threedimensional V ..."
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We are interested in solving the Vlasov equation used to describe collective effects in plasmas. This nonlinear partial differential equation coupled with Maxwell equation describes the time evolution of the particle distribution in phase space. The numerical solution of the full threedimensional VlasovMaxwell system represents a considerable challenge due to the huge size of the problem. A numerical method based on wavelet transform enables to compute the distribution function on an adaptive mesh from a regular discretization of the phase space. In this paper, we evaluate the costs of this recently developed adaptive scheme applied on a reduced onedimensional model, and its parallelization. We got a fine grain parallel application that achieves a good scalability up to 64 processors on a shared memory architecture. 1
Adaptive Multiresolution Methods for the Simulation of Waves in Excitable Media
 J SCI COMPUT (2010 ) 43 : 261–290
, 2010
"... We present fully adaptive multiresolution methods for a class of spatially twodimensional reactiondiffusion systems which describe excitable media and often give rise to the formation of spiral waves. A novel model ingredient is a strongly degenerate diffusion term that controls the degree of spat ..."
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Cited by 7 (1 self)
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We present fully adaptive multiresolution methods for a class of spatially twodimensional reactiondiffusion systems which describe excitable media and often give rise to the formation of spiral waves. A novel model ingredient is a strongly degenerate diffusion term that controls the degree of spatial coherence and serves as a mechanism for obtaining sharper wave fronts. The multiresolution method is formulated on the basis of two alternative reference schemes, namely a classical finite volume method, and Barkley’s approach (Barkley in Phys. D 49:61–70, 1991), which consists in separating the computation of the nonlinear reaction terms from that of the piecewise linear diffusion. The proposed methods are enhanced with local time stepping to attain local adaptivity both in space and time. The computational efficiency and the numerical precision of our methods are assessed. Results illustrate that the fully adaptive methods provide stable approximations and substantial savings in memory storage and CPU time while preserving the accuracy of the discretizations on the corresponding finest uniform grid.
An adaptive numerical method for the Vlasov equation based on a multiresolution analysis
 Numerical Mathematics and Advanced Applications ENUMATH 2001
, 2001
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Multiresolution technique and explicitimplicit scheme for multicomponent flows
 J. of Num. Math
, 2005
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PARALLELISATION OF MULTISCALEBASED GRID ADAPTATION USING SPACEFILLING CURVES ∗
"... Abstract. The concept of fully adaptive multiscale finite volume schemes has been developed and investigated during the past decade. By now it has been successfully employed in numerous applications arising in engineering. In order to perform 3D computations for complex geometries in reasonable CPU ..."
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Cited by 6 (1 self)
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Abstract. The concept of fully adaptive multiscale finite volume schemes has been developed and investigated during the past decade. By now it has been successfully employed in numerous applications arising in engineering. In order to perform 3D computations for complex geometries in reasonable CPU time, the underlying multiscalebased grid adaptation strategy has to be parallelised via MPI for distributed memory architectures. In view of a proper scaling of the computational performance with respect to CPU time and memory, the load of data has to be wellbalanced and communication between processors has to be minimised. This has been realised using spacefilling curves.
Local time stepping for implicitexplicit methods on time varying grids
"... In the context of nonlinear conservation laws a model for multiphase flows where slow kinematic waves coexist with fast acoustic waves is discretized with an implicitexplicit time scheme. Space adaptivity of the grid is implemented using multiresolution techniques and local time stepping further ..."
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Cited by 6 (1 self)
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In the context of nonlinear conservation laws a model for multiphase flows where slow kinematic waves coexist with fast acoustic waves is discretized with an implicitexplicit time scheme. Space adaptivity of the grid is implemented using multiresolution techniques and local time stepping further enhances the computing time performances. A parametric study is presented to illustrate the robustness of the method.
Convergence of an adaptive semiLagrangian scheme for the VlasovPoisson system
, 2007
"... An adaptive semiLagrangian scheme for solving the Cauchy problem associated to the periodic 1+1dimensional VlasovPoisson system in the twodimensional 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 on ..."
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Cited by 6 (3 self)
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An adaptive semiLagrangian scheme for solving the Cauchy problem associated to the periodic 1+1dimensional VlasovPoisson system in the twodimensional 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 ε 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 ε 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, ∞ ∩W 2,1. The rate of convergence is O(∆t 2 + ε/∆t), which should be compared to the results of Besse, who recently established [6] similar rates for a uniform semiLagrangian scheme, but requiring that the initial data are in