adaptive solution of operator equations using wavelet frames in adaptive wavelet methods ii beyond the elliptic case of cohen dahmen and devore found comput math 2 2002 pp 203 245 an adaptive method has been developed for solving general operator equations using a riesz basis of wavelet type for the energy space the operator equation is transformed into an equivalent matrix vector system this system is solved iteratively where the application of the infinite stiffness matrix is replaced by an adaptive approximation assuming that the stiffness matrix is sufficiently compressible i e that it can be sufficiently well approximated by sparse matrices it was proved that the adaptive method has optimal computational complexity in the sense that it converges with the same rate as the best n term approximation for the solution assuming that the latter would be explicitly available the condition concerning compressibility requires that dependent on their order the wavelets have sufficiently many vanishing moments and that they be sufficiently smooth however except on tensor product domains wavelets that satisfy this smoothness requirement are not easy to construct in this paper we write the domain or manifold on which the operator equation is posed as an overlapping union of subdomains each of them being the image under a smooth parametrization of the hypercube by lifting wavelets on the hypercube to the subdomains we obtain a em frame for the energy space with this frame the operator equation is transformed into a matrix vector system after which this system is solved iteratively by an adaptive method similar to the one from the work of cohen dahmen and devore with this approach frame elements that have sufficiently many vanishing moments and are sufficiently smooth something needed for the compressibility are easily constructed by handling additional difficulties due to the fact that a frame gives rise to an underdetermined matrix vector system we prove that this adaptive method has optimal computational complexity

an optimal adaptive wavelet method without coarsening of the iterands in this paper an adaptive wavelet method for solving linear operator equations is constructed that is a modification of the method from math comp 70 2001 pp 27 75 by cohen dahmen and devore in the sense that there is no recurrent coarsening of the iterands despite this it will be shown that the method has optimal computational complexity numerical results for a simple model problem indicate that the new method is more efficient than an existing alternative adaptive wavelet method

finite element wavelets with improved quantitative properties in w dahmen r stevenson element by element construction of wavelets satisfying stability and moment conditions siam j numer anal 37 1 1999 319 352 electronic finite element wavelets were constructed on polygonal domains or lipschitz manifolds that are piecewise parametrized by mappings with constant jacobian determinants the wavelets could be arranged to have any desired order of cancellation properties and they generated stable bases for the sobolev spaces h s for s 32 or s 1 on manifolds unfortunately it appears that the quantitative properties of these wavelets are rather disappointing in this paper we modify the construction from the above mentioned work to obtain finite element wavelets which are much better conditioned

parallel space time hp adaptive discretization scheme for parabolic problems abstract we introduce an integration scheme for parabolic problems our parallelizable method uses adaptive h p finite elements in space and finite differences in time the strategy can also be combined with a classical finite element method parallelization technique based on domain decomposition we verified the performance of our method against two different benchmarks in both two dimensional model problem on an l shaped domain and three dimensional pennes bioheat equation settings results show a significant speedup in computational time when compared with the sequential version of the algorithm moreover we develop a mathematical framework to analyze similar schemes which include h p spatial adaptivity our framework describes error propagation rigorously and as such allows to analyze convergence properties of these mixed methods

computation of differential operators in wavelet coordinates in found comput math 2 2002 pp 203 245 cohen dahmen and devore proposed an adaptive wavelet algorithm for solving general operator equations assuming that the operator defines a boundedly invertible mapping between a hilbert space and its dual and that a riesz basis of wavelet type for this hilbert space is available the operator equation is transformed into an equivalent well posed infinite matrix vector system this system is solved by an iterative method where each application of the infinite stiffness matrix is replaced by an adaptive approximation it was shown that if the errors of the best linear combinations from the wavelet basis with n terms are o n s for some s 0 which is determined by the besov regularity of the solution and the order of the wavelet basis then approximations yielded by the adaptive method with n terms also have errors of o n s moreover their computation takes only 0 n operations provided s s assuming that each entry of the stiffness matrix is exactly available at unit cost generally these entries have to be approximated using numerical quadrature in this paper restricting ourselves to differential operators we develop a numerical integration scheme that computes these entries giving an additional error that is consistent with the approximation error whereas in each column the average computational cost per entry is 0 1 as a consequence we can conclude that the adaptive wavelet algorithm has optimal computational complexity

nonvanishing of hecke l functions and the bloch kato conjecture in this paper we study the central values of l functions associated to a large class of algebraic hecke characters of imaginary quadratic fields when these central values are nonzero the bloch kato conjecture predicts an exact formula for the algebraic parts of the central values in terms of periods and arithmetic data most notably the selmer groups corresponding to the hecke characters we investigate the nonvanishing of these central values and prove the p part of the bloch kato conjecture in these cases for primes p which split in k

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

compression techniques for boundary integral equations asymptotically optimal complexity estimates matrix compression techniques in the context of wavelet galerkin schemes for boundary integral equations are developed and analyzed that exhibit optimal complexity in the following sense the fully discrete scheme produces approximate solutions within discretization error accuracy offered by the underlying galerkin method at a computational expense that is proven to stay proportional to the number of unknowns key issues are the second compression which reduces the near field complexity significantly and an additional a posteriori compression the latter is based on a general result concerning an optimal work balance that applies in particular to the quadrature used to compute the compressed stiffness matrix with sufficient accuracy in linear time

bounds for eigenforms on arithmetic hyperbolic 3 manifolds on a family of arithmetic hyperbolic 3 manifolds of square free level we prove an upper bound for the sup norm of hecke maas cusp forms with a power saving over the local geometric bound simultaneously in the laplacian eigenvalue and the volume by a novel combination of diophantine and geometric arguments in a noncommutative setting we obtain bounds as strong as the best corresponding results on arithmetic surfaces

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

on interpolatory divergence free wavelets we construct interpolating divergence free multiwavelets based on cubic hermite splines we give characterizations of the relevant function spaces and indicate their use for analyzing experimental data of incompressible flow fields we also show that the standard interpolatory wavelets based on the deslauriers dubuc interpolatory scheme or on interpolatory splines cannot be used to construct compactly supported divergence free interpolatory wavelets

an optimal adaptive wavelet method for nonsymmetric and indefinite elliptic problems in this paper we modify the adaptive wavelet algorithm from gantumur et al an optimal adaptive wavelet method without coarsening of the iterands technical report 1325 department of mathematics utrecht university march 2005 math comp to appear so that it applies directly i e without forming the normal equation not only to self adjoint elliptic operators but also to operators of the form l a b where a is self adjoint elliptic and b is compact assuming that the resulting operator equation is well posed we show that the algorithm has optimal computational complexity

element by element construction of wavelets satisfying stability and moment conditions in this paper we construct a class of locally supported wavelet bases for c0 lagrange finite element spaces on possibly nonuniform meshes on n dimensional domains or manifolds the wavelet bases are stable in the sobolev spaces mathcal h s for s frac 3 2 s leq 1 on lipschitz manifolds and the wavelets can in principle be arranged to have any desired order of vanishing moments as a consequence these bases can be used e g for constructing an optimal solver of discretized mathcal h s elliptic problems for s in above ranges the construction of the wavelets consists of two parts an implicit part involves some computations on a reference element which for each type of finite element space have to be performed only once in addition there is an explicit part which takes care of the necessary adaptations of the wavelets to the actual mesh the only condition we need for this construction to work is that the refinements of initial elements are uniform we will show that the wavelet bases can be implemented efficiently

wavelets with patchwise cancellation properties we construct wavelets on general n dimensional domains or manifolds via a domain decomposition technique resulting in so called composite wavelets with this construction wavelets with supports that extend to more than one patch are only continuous over the patch interfaces normally this limited smoothness restricts the possibility for matrix compression and with that the application of these wavelets in adaptive methods for solving operator equations by modifying the scaling functions on the interval and with that on the n cube that serves as parameter domain we obtain composite wavelets that have patchwise cancellation properties of any required order meaning that the restriction of any wavelet to each patch is again a wavelet this is also true when the wavelets are required to satisfy zeroth order homogeneous dirichlet boundary conditions on part of the boundary as a result compression estimates now depend only on the patchwise smoothness of the wavelets that one may choose also taking stability into account our composite wavelets have all the properties for the application to the adaptive solution of well posed operator equations of orders 2t for t 3 2

l 4 norms of hecke newforms of large level we prove a new upper bound for the l 4 norm of a holomorphic hecke newform of large fixed weight and prime level q rightarrow infty this is achieved by proving a sharp mean value estimate for a related l function on gl 6

wavelet bases in h div and h curl some years ago compactly supported divergence free wavelets have been constructed which also give rise to a stable biorthogonal wavelet splitting of h div these bases have successfully been used both in the analysis and numerical treatment of the stokes and navier stokes equations in this paper we construct stable wavelet bases for the stream function spaces h curl moreover curl free vector wavelets are constructed and analysed the relationship between h div and h curl are expressed in terms of these wavelets we obtain discrete orthogonal hodge decompositions our construction works independently of the space dimension but in terms of general assumptions on the underlying wavelet systems in l 2 omega that are used as building blocks we give concrete examples of such bases for tensor product and certain more general domains omega subset er n as an application we obtain wavelet multilevel preconditioners in h div and h curl email urban dragon ian pv cnr it keywords h div h curl stream function spaces wavelets

wavelet adaptive method for second order elliptic problems boundary conditions and domain decomposition summary wavelet methods allow to combine high order accuracy multilevel preconditioning techniques and adaptive approximation in order to solve efficiently elliptic operator equations one of the main difficulty in this context is the efficient treatment of non homogeneous boundary conditions in this paper we propose a strategy that allows to append such conditions in the setting of space refinement i e adaptive discretizations of second order problems our method is based on the use of compatible multiscale decompositions for both the domain and its boundary and on the possibility of characterizing various function spaces from the numerical properties of these decompositions in particular this allows the construction of a lifting operator which is stable for a certain range of smoothness classes and preserves the compression of the solution in the wavelet basis an explicit construction of the wavelet bases and the lifting is proposed on fairly general domains based on c 0 conforming domain decomposition techniques

mollification of the fourth moment of automorphic l functions and arithmetic applications we compute the asymptotics of twisted fourth power moments of modular l functions of large prime level near the critical line this allows us to prove some new non vanishing results on the central values of automorphic l functions in particular those obtained by base change from gl 2 q to gl 2 k for k a cyclic field of low degree

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

concentrated ring loadings in a full space or half space solutions for transverse isotropy and isotropy in this analysis solutions for concentrated ring loading in a transversely isotropic full space or half space are found the elastic field is derived by integrating the known fundamental point force solutions along a circular ring three cases of ring loading are considered taking the z axis along the material axis the z 0 plane is the isotropic plane the first case studied is when the ring load is in the z direction referred to as normal loading the other two cases are shear ring loads directed in the plane of isotropy in the first instance the ring load is unidirectional and independently applied in either the x or y direction the second case considers the axisymmetric radial and axisymmetric torsional ring loads the solution for ring loading applied to the surface of a half space is first obtained subsequently the solutions for ring loading in a full space and buried ring loads in a half space are found in all cases the elastic displacement and stress fields are evaluated in terms of closed form expressions containing complete elliptic integrals of the first second and third kinds an interesting feature of the full space solution is that the potential function and its radial derivatives exhibit a cylindrical discontinuity for negative z values however it is shown that these discontinuous functions do provide continuous displacement and stress fields a limiting form of the solutions for transverse isotropy also provides the corresponding results for isotropic materials

the subconvexity problem for rankin selberg l functions and equidistribution of heegner points ii we prove a general subconvex bound in the level aspect for rankin selberg l functions associated with two primitive holomorphic or maass cusp forms over q we use this bound to establish the equidistribution of incomplete galois orbits of heegner points on shimura curves associated with indefinite quaternion algebras over q

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

symmetry transitions in random matrix theory l functions we compute the moments of the characteristic polynomials of random orthogonal and symplectic matrices defined by averages with respect to haar measure on so 2n and usp 2n to leading order as n on the unit circle as functions of the angle measured from one of the two symmetry points in the eigenvalue spectrum exp pm i theta n 1 le n le n our results extend previous formulae that relate just to the symmetry points i e to 0 local spectral statistics are expected to converge to those of random unitary matrices in the limit as n when is fixed and to show a transition from the orthogonal or symplectic to the unitary forms on the scale of the mean eigenvalue spacing if y n they become functions of y in the limit when n we verify that this is true for the spectral two point correlation function but show that it is not true for the moments of the characteristic polynomials for which the leading order asymptotic approximation is a function of rather than y symmetry points therefore influence the moments asymptotically far away on the scale of the mean eigenvalue spacing we also investigate the moments of the logarithms of the characteristic polynomials in the same context the moments of the characteristic polynomials of random matrices are conjectured to be related to the moments of families of l functions previously moments at the symmetry point 0 have been related to the moments of families of l functions evaluated at the centre of the critical strip our results motivate general conjectures for the moments of orthogonal and symplectic families of l functions evaluated at a fixed height t up the critical line these conjectures suggest that the symmetry of the non trivial zeros of the l functions influences the moments asymptotically far on the scale of the mean zero spacing from the centre of the critical strip we verify that the second moments of real quadratic dirichlet l functions and a family of automorphic l functions are consistent with our conjectures

adaptive wavelet methods for elliptic operator equations convergence rates this paper is concerned with the construction and analysis of wavelet based adaptive algorithms for the numerical solution of elliptic equations these algorithms approximate the solution u of the equation by a linear combination of n wavelets therefore a benchmark for their performance is provided by the rate of best approximation to u by an arbitrary linear combination of n wavelets so called n term approximation which would be obtained by keeping the n largest wavelet coefficients of the real solution which of course is unknown the main result of the paper is the construction of an adaptive scheme which produces an approximation to u with error o n s in the energy norm whenever such a rate is possible by n term approximation the range of s 0 for which this holds is only limited by the approximation properties of the wavelets together with their ability to compress the elliptic operator moreover it is shown that the number of arithmetic operations needed to compute the approximate solution stays proportional to n the adaptive algorithm applies to a wide class of elliptic problems and wavelet bases the analysis in this paper puts forward new techniques for treating elliptic problems as well as the linear systems of equations that arise from the wavelet discretization

composite wavelet bases with extended stability and cancellation properties the efficient solution of operator equations using wavelets requires that they generate a riesz basis for the underlying sobolev space and that they have cancellation properties of a sufficiently high order suitable biorthogonal wavelets were constructed on reference domains as the n cube via a domain decomposition approach these bases have been used as building blocks to construct biorthogonal wavelets on general domains or manifolds where in order to end up with local wavelets biorthogonality was realized with respect to a modified l 2 scalar product the use of this modified scalar product restricts the application of these so called composite wavelets to problems of orders strictly larger than 1 moreover those wavelets with supports that extend to more than one patch generally have no cancellation properties in this paper we construct local composite wavelets that are close to being biorthogonal with respect to the standard l 2 scalar product as a consequence they generate riesz bases for the sobolev spaces h s for the full range of s allowed by the continuous gluing of functions over the patch interfaces the properties of the primal and dual approximation spaces on the reference domain and in the manifold case by the regularity of the manifold moreover all these wavelets have cancellation properties of the full order induced by the approximation properties of the dual spaces on the reference domain we illustrate our findings by a concrete realization of wavelets on a perturbed sphere