multilevel frames for sparse tensor product spaces for au f with an elliptic differential operator a mathcal h rightarrow mathcal h and stochastic data f the m point correlation function mathcal m m u of the random solution u satisfies a deterministic equation with the m fold tensor product operator a m of a sparse tensor products of hierarchic fe spaces in mathcal h are known to allow for approximations to mathcal m m u which converge at essentially the rate as in the case m 1 i e for the deterministic problem they can be realized by wavelet type fe bases von petersdorff and schwab in appl math 51 2 145 180 2006 schwab and todor in computing 71 43 63 2003 if wavelet bases are not available we show here how to achieve the fast computation of sparse approximations of mathcal m m u for galerkin discretizations of a by multilevel frames such as bpx or other multilevel preconditioners of any standard fem approximation for a numerical examples illustrate feasibility and scope of the method

correspondence between berry s phase and lewis s phase for quadratic hamiltonians the authors show the existence of a relation between berry s phase and lewis s phase for the case of quadratic hamiltonians

fast evaluation of system matrices w r t multi tree collections of tensor product refinable basis functions an algorithm is presented that for a local bilinear form evaluates in linear complexity the application of the stiffness matrix w r t a collection of tensor product multiscale basis functions assuming that this collection has a multi tree structure it generalizes an algorithm for sparse grid index sets r balder ch zenger the solution of multidimensional real helmholtz equations on sparse grids siam j sci comput 17 3 1996 631 646 and it finds its application in adaptive tensor product approximation methods

optimal control problems with set valued control and state constraints in this paper a general optimal control problem with pure state and mixed control state constraints is considered these constraints are of the form of set inclusions second order necessary optimality conditions for weak local minimum are derived for this problem in terms of the original data in particular the nonemptiness of the set of critical directions and the evaluation of its support function are expressed in terms of the given functions and set valued maps in order that the lagrange multiplier corresponding to the mixed control state inclusion constraint be represented via an integrable function a strong normality condition involving the notion of the critical tangent cone is introduced

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

adaptive wavelet schemes for nonlinear variational problems we develop and analyze wavelet based adaptive schemes for nonlinear variational problems we derive estimates for convergence rates and corresponding work counts that turn out to be asymptotically optimal our approach is based on a new paradigm that has been put forward recently for a class of linear problems the original problem is transformed first into an equivalent one which is well posed in the euclidean metric ell 2 then conceptually one seeks iteration schemes for the infinite dimensional problem that exhibits at least a fixed error reduction per step this iteration is then realized approximately through an adaptive application of the involved operators with suitable dynamically updated accuracy tolerances the main conceptual ingredients center around nonlinear tree approximation and the sparse evaluation of nonlinear mappings of wavelet expansions we prove asymptotically optimal complexity for adaptive realizations of first order iterations and of newton s method

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

a comparative study of the hannay s angles associated with a damped harmonic oscillator and a generalized harmonic oscillator in this paper we show that a damped harmonic oscillator the simplest dissipative physical system is canonically equivalent to a generalized harmonic oscillator a conservative system even for time dependent parameters as a consequence the hannay s angles and the adiabatic invariants of the two systems appear to be the same modulo this equivalence this raises the question of whether this analogy can be extended to other nonlinear dissipative systems

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

berry s phase as the asymptotic limit of an exact evolution an example the authors exhibit for arbitrary time variations of the parameters of the generalised harmonic oscillator hamiltonian a canonical transformation which facilitates an exact analysis of quantal phase and classical angle formulae for berry s phase and hannay s angle are readily obtained by taking the adiabatic limit of the exact theory

classical adiabatic angles and quantal adiabatic phase a semiclassical connection is established between quantal and classical properties of a system whose hamiltonian is slowly cycled by varying its parameters round a circuit the quantal property is a geometrical phase shift gamma n associated with an eigenstate with quantum numbers n nl the classical property is a shift delta theta l i in the lth angle variable for motion round a phase space torus with actions i il the connection is delta theta l delta gamma delta nl two applications are worked out in detail the generalised harmonic oscillator with quadratic hamiltonian whose parameters are the coefficients of q2 qp and p2 and the rotated rotator consisting of a particle sliding freely round a non circular hoop slowly turned round once in its own plane

second order necessary optimality conditions for a class of semilinear elliptic optimal control problems with mixed pointwise constraints this paper deals with first and second order necessary optimality conditions for a class of optimal control problems governed by semilinear elliptic partial differential equations with mixed pointwise state control constraints

composite wavelet bases for operator equations this paper is concerned with the construction of biorthogonal wavelet bases defined on a union of parametric images of the unit n cube these bases are to satisfy certain requirements imposed by applications to a class of operator equations acting on such domains this covers also elliptic boundary value problems although this study is primarily motivated by our previous analysis of wavelet methods for pseudo differential equations with special emphasis on boundary integral equations in this case it is natural to model the boundary surface as a union of parametric images of the unit cube it will be shown how to construct wavelet bases on the surface which are composed of wavelet bases defined on each surface patch here the relevant properties are the validity of norm equivalences in certain ranges of sobolev scales as well as appropriate moment conditions

invariance of the generalised berry phase under unitary transformations application to the time dependent generalised harmonic oscillator a generalisation of the berry phase which reduces to the usual berry phase in the adiabatic and cyclic case is derived in a gauge invariant way this generalised berry phase unlike the berry phase is invariant under unitary transformations a time dependent generalised harmonic oscillator with a cross term in the hamiltonian is taken as an example the hamiltonian of this system is not in general the energy an energy the time derivative of which is the power is obtained from the equation of motion when the system is quantised the generalised berry phase is zero and remains invariant under unitary transformations on the other hand if the energy is chosen to be the hamiltonian a non zero generalised berry phase is obtained but the power is given incorrectly in this case the total phase which includes the dynamical phase and the generalised berry phase is equal in the adiabatic limit to the correct total phase

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

adaptive wavelet algorithms for elliptic pde s on product domains with standard isotropic approximation by piecewise polynomials of fixed order in a domain d subset of r d the convergence rate in terms of the number n of degrees of freedom is inversely proportional to the space dimension d this so called curse of dimensionality can be circumvented by applying sparse tensor product approximation when certain high order mixed derivatives of the approximated function happen to be bounded in l 2 it was shown by nitsche 2006 that this regularity constraint can be dramatically reduced by considering best n term approximation from tensor product wavelet bases when the function is the solution of some well posed operator equation dimension independent approximation rates can be practically realized in linear complexity by adaptive wavelet algorithms assuming that the infinite stiffness matrix of the operator with respect to such a basis is highly compressible applying piecewise smooth wavelets we verify this compressibility for general non separable elliptic pdes in tensor domains applications of the general theory developed include adaptive galerkin discretizations of multiple scale homogenization problems and of anisotropic equations which are robust i e independent of the scale parameters resp of the size of the anisotropy

characterization of l1 closed decomposable sets in l abstract in this paper the set of essentially bounded measurable selections of a measurable set valued map is considered one of the main results states that only those subsets of l that are l 1 closed in l and decomposable can be represented in this way the connection between topological and convexity properties of the set valued map and its l selection set is also studied in details the adjacent and clarke s tangent cones to l 1 closed and decomposable sets and their polar cones are described in terms of the corresponding set valued map as well

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

generalized berry phase for the most general time dependent damped harmonic oscillator the most general time dependence hamiltonian for a harmonic oscillator is both linear and quadratic in the coordinate and the canonical momentum it describes in general a harmonic oscillator with a time dependent mass a time dependent friction or antifriction constant and a time dependent spring constant acted upon by a time dependent force the energy whose time derivative is the power is in general different from the hamiltonian the generalized berry phase for a given energy eigenstate has a state dependent part which vanishes if there is no damping and an arbitrary state independent part if the hamiltonian is identified as the energy both the energy eigenvalues and the generalized berry phase are different in the adiabatic limit both approaches give the same total phase which is the sum of the dynamical and berry phases

efficient spectral sparse grid methods and applications to high dimensional elliptic problems we develop in this paper some efficient algorithms which are essential to implementations of spectral methods on the sparse grid by smolyak s construction based on a nested quadrature more precisely we develop a fast algorithm for the discrete transform between the values at the sparse grid and the coefficients of expansion in a hierarchical basis and by using the aforementioned fast transform we construct two very efficient sparse spectral galerkin methods for a model elliptic equation in particular the chebyshev legendre galerkin method leads to a sparse matrix with a much lower number of nonzero elements than that of low order sparse grid methods based on finite elements or wavelets and can be efficiently solved by a suitable sparse solver ample numerical results are presented to demonstrate the efficiency and accuracy of our algorithms

exact evolution of the generalized damped harmonic oscillator in this paper we use the algebraic and the invariant method to study the time dependent damped harmonic oscillator from classical and quantum points of view the solution of the classical equation of motion and the wavefunction solving the time dependent schrodinger equation are found explicitly we show that the original time dependent quantum mechanical problem is completely related to the well known time independent harmonic oscillator in addition we elucidate the intimate connection between the damped harmonic oscillator dho and the generalized harmonic oscillator gho more importantly the evolution of the states of the dho cannot be cyclic in contradistinction with the states of the gho explicit expressions for both the dynamical and the geometric angles and phases are deduced in the adiabatic limit the coherent states describing the invariant angle variables of the classical dho are constructed they allow us to recover the classical evolution and invariant from the quantum evolution

on the removability of berry s phase the authors present a thorough analysis of the removability of the berry phase they show that for any system they can by properly restricting the parameter space turn the geometrical phase into an extra shift of the dynamical one however this extra shift in the dynamical phase retains its truly geometrical character the removability is thus only apparent and the geometrical phase can still be measured in interference experiments as indeed has already been done several times

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

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

optimum problems with measurable set valued constraints in this paper we provide a complete analysis of second order admissible variations to inequality type constraints which are given in terms of measurable set valued functions whose images are closed convex sets with nonempty interior as an application we consider optimization problems where such constraints are present and we deduce second order necessary conditions for optimality

quantal phase factors accompanying adiabatic changes a quantal system in an eigenstate slowly transported round a circuit c by varying parameters r in its hamiltonian r will acquire a geometrical phase factor exp i c in addition to the familiar dynamical phase factor an explicit general formula for c is derived in terms of the spectrum and eigenstates of r over a surface spanning c if c lies near a degeneracy of c takes a simple form which includes as a special case the sign change of eigenfunctions of real symmetric matrices round a degeneracy as an illustration c is calculated for spinning particles in slowly changing magnetic fields although the sign reversal of spinors on rotation is a special case the effect is predicted to occur for bosons as well as fermions and a method for observing it is proposed it is shown that the aharonov bohm effect can be interpreted as a geometrical phase factor

adaptive methods for boundary integral equations complexity and convergence estimates this paper is concerned with developing numerical techniques for the adaptive application of global operators of potential type in wavelet coordinates this is a core ingredient for a new type of adaptive solvers that has so far been explored primarily for pdes we shall show how to realize asymptotically optimal complexity in the present context of global operators asymptotically optimal means here that any target accuracy can be achieved at a computational expense that stays proportional to the number of degrees of freedom within the setting determined by an underlying wavelet basis that would ideally be necessary for realizing that target accuracy if full knowledge about the unknown solution were given the theoretical findings are supported and quantified by the first numerical experiments

the solution of multidimensional real helmholtz equations on sparse grids sparse grids provide a very efficient method for the multilinear approximation of functions especially in higher dimensional spaces in the d dimensional space the nodal multilinear basis on a grid with mesh size h 2 n consists of o 2 nd basis functions and leads to an l 2 error of order o 4 n and an h 1 error of order o 2 n with sparse grids we get an l 2 error of order o 4 n n d 1 and an h 1 error of order o 2 n with only o 2 n n d 1 basis functions if the function u fulfills the condition partial 2d over partial x 1 2 partial x 2 2 ldots partial x d 2 u infty therefore we can achieve much more accurate approximations with the same amount of storage a data structure for the sparse grid representation of functions defined on cubes of arbitrary dimension and a finite element approach for the helmholtz equation with sparse grid functions are introduced special emphasis is taken in the development of an efficient algorithm for the multiplication with the stiffness matrix with an appropriate preconditioned conjugate gradient method cg method the linear systems can be solved efficiently numerical experiments are presented for helmholtz equations and eigenvalue problems for the laplacian in two and three dimensions and for a six dimensional poisson problem the results support the assertion that the l 2 error bounds for the sparse grid approximation are also valid for sparse grid finite element solutions of elliptic differential equations problems with nonsmooth solutions are treated with adaptive sparse grids

so 2 1 invariant systems and the berry phase the general quadratic time dependent quantum hamiltonian is analysed using a time dependent realisation of its so 2 1 invariance lie algebra several interesting features of this procedure are pointed out as an example the authors easily calculate dynamical and geometrical berry phases using only an algebraic procedure an incorrect result previously published by another author is also pointed out

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