multiwavelets for second kind integral equations we consider a galerkin method for an elliptic pseudodifferential operator of order zero on a two dimensional manifold we use piecewise linear discontinuous trial functions on a triangular mesh and describe an orthonormal wavelet basis using this basis we can compress the stiffness matrix from n2 to o n log n nonzero entries and still obtain up to log n terms the same convergence rates as for the exact galerkin method

differential identities of prime rings with involution main theorem let r be a prime ring with involution suppose that 0 x5 xa 0 is a differential identity for r where a are distinct regular words of derivations in a basis m with respect to a linear order u e w be such that vl vs is dindependent in v and ul ut is d independent in w then there exists a e sw v such that via v i 1 s and a uj u j 1 t if and only if v uj v u for i 1 s and j 1 t proposition 4 suppose that r is a prime ring with involution and that f is a generalized polynomial if f vanishes on a nonzero ideal of r than f vanishes on q the two sided quotient ring of r the objective of this paper is to prove the following generalization of kharchenko s theorem on differential identities to prime rings with involution the notation here is explained in i below main theorem let r be a prime ring with involution suppose that x ai xj 0 is a differential identity for r where a1 are distinct regular words of derivations in a basis m with respect to a linear order on m then 0 zi iz 0 is a generalized identity for r where zij are distinct indeterminates received by the editors april 18 1988 1980 mathematics subject classification 1985 revision primary 16a38 secondary 16a28 16a72 16a12 16a48

adaptive multiscale moment method ammm for analysis of scattering from perfectly conducting plates adaptive multiscale moment method ammm is presented for the analysis of scattering from a thin perfectly conducting plate this algorithm employs the conventional moment method and a special matrix transformation which is derived from the tensor products of the two one dimensional 1 d multiscale triangular basis functions that are used for expansion and testing functions in the conventional moment method the special feature of these new basis functions introduced through this transformation is that they are orthogonal at the same scale except at the initial scale and not between scales from one scale to another scale the initial estimate for the solution can be predicted using this multiscale technique hence the compression is applied directly to the solution and the size of the linear equations to be solved is reduced thereby improving the efficiency of the conventional moment method the basic difference between this methodology and the other techniques that have been presented so far is that we apply the compression not to the impedance matrix but to the solution itself directly using an iterative solution methodology the extrapolated results at the higher scale thus provide a good initial guess for the iterative method typically when the number of unknowns exceeds a few thousand unknowns the matrix solution time exceeds generally the matrix fill time hence the goal of this method is directed in solving electrically larger problems where the matrix solution time is of concern two numerical results are presented which demonstrate that the ammm is a useful method to analyze scattering from perfectly conducting plates

fast collocation methods for second kind integral equations in this paper we develop fast collocation methods for integral equations of the second kind with weakly singular kernels for this purpose we construct multiscale interpolating functions and collocation functionals having vanishing moments moreover we propose a truncation strategy for the coefficient matrix of the corresponding discrete system which forms a basis for fast algorithms an optimal order of convergence of the approximate solutions obtained from the fast algorithms is proved and the computational complexity of the algorithms is estimated the stability of the numerical method and the condition number of the truncated coefficient matrix are analyzed

an efficient numerical method for studying interfacial motion in two dimensional creeping flows we present new methods for computing the motion of two dimensional closed interfaces in a slow viscous flow the interfacial velocity is found through the solution to an integral equation whose analytic formulation is based on complex variable theory for the biharmonic equation the numerical methods for solving the integral equations are spectrally accurate and employ a fast multipole based iterative solution procedure which requires only o n operations where n is the number of nodes in the discretization of the interface the interface is described spectrally and we use evolution equations that preserve equal spacing in arclength of the marker points a small scale decomposition is performed to extract the dominant term in the evolution of the interface and we show that this dominant term leads to a cfl type stability constraint when in an equal arclength frame this term is linear and we show that implicit time integration schemes that are explicit in fourier space can be formulated we verify this analysis through several numerical examples

a practical determination strategy of optimal threshold parameter for matrix compression in wavelet bem a practical strategy is developed to determine the optimal threshold parameter for wavelet based boundary element be analysis the optimal parameter is determined so that the amount of storage and computational work is minimized without reducing the accuracy of the be solution in the present study the beylkin type truncation scheme is used in the matrix assembly to avoid unnecessary integration concerning the truncated entries of a coefficient matrix a priori estimation of the matrix entries is introduced and thus the truncated entries are determined twice before and after matrix assembly the optimal threshold parameter is set based on the equilibrium of the truncation and discretization errors these errors are estimated in the residual sense for laplace problems the discretization error is in particular indicated with the potential s contribution c to the residual norm r used in error estimation for mesh adaptation since the normalized residual norm c u u the potential components of be solution cannot be computed without main be analysis the discretization error is estimated by the approximate expression constructed through subsidiary be calculation with smaller degree of freedom dof the matrix compression using the proposed optimal threshold parameter enables us to generate a sparse matrix with o n 1 7 0 1 non zero entries although the quasi optimal memory requirements and complexity are not attained the compression rate of a few per cent can be achieved for n 1000

estimates for the bar partial neumann problem in pseudoconvex domains of finite type in c2problem in pseudoconvex domains of finite type in c2

boundary element and finite element coupling for aeroacoustics simulations we consider the scattering of acoustic perturbations in the presence of a flow we suppose that the space can be split into a zone where the flow is uniform and a zone where the flow is potential in the first zone we apply a prandtl glauert transformation to recover the helmholtz equation the well known setting of boundary element method for the helmholtz equation is available in the second zone the flow quantities are space dependent we have to consider a local resolution namely the finite element method herein we carry out the coupling of these two methods and present various applications and validation test cases the source term is given through the decomposition of an incident acoustic field on a section of the computational domain s boundary validations against analytic another numerical method and measurements on different test cases are presented

wavelet galerkin schemes for boundary integral equations implementation and quadrature in the present paper we consider the fully discrete wavelet galerkin scheme for the fast solution of boundary integral equations in three dimensions it produces approximate solutions within discretization error accuracy offered by the underlying galerkin method at a computational expense that stays proportional to the number of unknowns we focus on algorithmical details of the scheme in particular on numerical integration of relevant matrix coefficients we illustrate the proposed algorithm by numerical results

multiscale bases for the sparse representation of boundary integral operators on complex geometry a multilevel transform is introduced to represent discretizations of integral operators from potential theory by nearly sparse matrices the new feature presented here is to construct the basis in a hierarchical decomposition of the three space and not as in previous approaches in a parameter space of the boundary manifold this construction leads to sparse representations of the operator even for geometrically complicated multiply connected domains we will demonstrate that the numerical cost to apply a vector to the operator using the nonstandard form is essentially equal to performing the same operation with the fast multipole method with a second compression scheme the multiscale approach can be further optimized the diagonal blocks of the transformed matrix can be used as an inexpensive preconditioner which is empirically shown to reduce the condition number of discretizations of the single layer operator so as to be independent of mesh size

carleson embedding theorem on convex finite type domains abstract this paper concerns certain geometric aspects of function theory on smoothly bounded convex domains of finite type in c n specifically we prove the carleson hormander inequality for this class of domains and provide examples of carleson measures improving a known result concerning such measures associated to bounded holomorphic functions

identities ore extensions and character skew derivations following 17 a derivation is called quasi algebraic if there exist a ai q such that k x i 1k 1ai i x ax k x a 0 for x q let k be the minimal such integer and g0 g1 mutually outer automorphisms of q let f zij be a generalized polynomial in variables zij where i 0 and 0 j k it is shown in 6 that if f gi j x or f jgi x is an identity then so is f zij we generalize this to sets of skew derivations in theorems 1 and 2 in theorem 3 we apply these results to the ore extensions of several variables of 2 and in theorem 4 to the character hopf algebras of the character skew derivations of 12

hardy spaces and the szegő projection of the non smooth worm domain dβ we define hardy spaces hp d p 1 on the non smooth worm domain d z1 z2 c2 imz1 log z2 2 2 log z2 2 2 and we prove a series of related results such as the existence of boundary values on the distinguished boundary d of the domain and a fatou type theorem i e pointwise convergence to the boundary values thus we study the szeg projection operator s and the associated szeg kernel kd more precisely if hp d denotes the space of functions which are boundary values for functions in hp d we prove that the operator s extends to a bounded linear operator s lp d hp d for every p 1 and s wk p d wk p d for every k 0 here wk p denotes the sobolev space of order k and underlying lp norm p 1 as a consequence of the lp boundedness of s we prove that hp d c d is a dense subspace of hp d

the cauchy integral in cn for domains with minimal smoothness abstract we prove l p b d regularity of the cauchy leray integral for bounded domains d c n whose boundary satisfies the minimal regularity condition of class c 1 1 together with a naturally occurring notion of convexity

morrey boundedness and compactness characterizations of integral commutators with singular kernel on strictly pseudoconvex domains in cn abstract we prove the morrey boundedness and compactness characterization of integral commutators with singular kernel of calderon zygmund type on pseudo convex domains in c n our results extend some of the ones of krantz li in 16 17

regular article a fast adaptive multipole algorithm in three dimensions we present an adaptive fast multipole method for the laplace equation in three dimensions it uses both new compression techniques and diagonal forms for translation operators to achieve high accuracy at a reasonable cost

tracking neumann data for stationary free boundary problems the present paper is dedicated to the verification of sufficient second order conditions for shape optimization problems that arise from stationary free boundary problems we assume that the state satisfies the dirichlet problem for the poisson equation and track the neumann data at the free boundary the gradient and hessian of the shape functional under consideration are computed by analyzing the shape hessian in case of matching data a sufficient criterion for its strict coercivity is derived strict coercivity implies stable minimizers and in case of a ritz galerkin method existence and convergence of approximate shapes by a fast boundary element method we realize an efficient numerical algorithm to solve the free boundary problem numerical experiments are carried out in three spatial dimensions

an efficient solution for cauchy likesystems of linear equations using the displacement approach and transformation of vectors the complexity ofsolving strongly nonsingular cauchy like systems of linear equations email protected email protected will be improved off a factor of log n from the known algorithms of the recursive triangular factorization where f is a field and email protected f n x 1 is a vector and c f n x n is a strongly nonsingular cauchy like matrix the efficient algorithms were presented by morf bitmead and anderson to solve strongly nonsingular toeplitz like equations of linear systems by using the recursive triangular factorization in 1980 recently this approach of the recursive triangular factorization was extended to solve cauchy like systems with the complexity of o n log 3n operations by pan et al this is the best known complexity bound by using the direct approach of recursive triangular factorization in cauchy like cases however these algorithms are still slower than the well known algorithms with the asymptotic bound of o n log 2n operations which have been proposed by the means of reducing cauchy like matrices into toeplitz like matrices in our present paper the algorithms of the recursive triangular factorization are modified so that the complexity bound of the direct recursive approach can be decreased to o n log 2n operations this matches the asymptotic bound without transforming to toeplitz like matrices our improvement of the direct recursive approach is by a factor of log n due to changing the original vectors expressed in the given cauchy like matrix into the special vectors where the entries are unit roots the applications of structured matrices include nevanlinna pick tangential interpolation problems

on maximal sobolev and holder estimates for the tangential cauchy riemann operator and boundary laplacian let m be the boundary of a smoothly bounded pseudoconvex domain in c n n 3 or more generally any compact pseudoconvex cr manifold of dimension 2 n 1 for which the range of inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i is closed in l 2 in this article we study the l p sobolev and holder regularity properties of inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i and b near a point of finite type under a comparable eigenvalues condition on the levi form we show that if all possible sums of q 0 eigenvalues of the levi matrix are comparable to its trace near a point of finite commutator type condition d q 0 then the inverse k q of b on 0 q forms for q 0 q n 1 q 0 satisfies sharp kernel estimates in terms of the quasi distance associated to the hormander sum of squares operator in particular we obtain the maximal l p estimates for b which were conjectured in the 1980s we also prove sharp estimates for certain parts of the kernels of k q 0 1 and k n q 0 and give some applications concerning domains with at most one degenerate eigenvalue finally we establish the composition and mapping properties of a class of singular integral nonisotropic smoothing operators that arises naturally in complex analysis these results yield optimal regularity of k q and related operators in the sobolev and lipschitz norms both isotropic and nonisotropic

robust exponential convergence of hp fem in balanced norms for singularly perturbed reaction diffusion problems corner domains abstract the h p version of the finite element method is applied to singularly perturbed reaction diffusion type equations on polygonal domains the solution exhibits boundary layers as well as corner layers on a class of meshes that are suitably refined near the boundary and the corners robust exponential convergence in the polynomial degree is shown in both a balanced norm and the maximum norm

short note moore s law and the saffman taylor instability ten years ago hou lowengrub and shelley 4 published a state of the art boundary integral simulation of a classical viscous fingering problem the saffman taylor instability 12 in terms of complexity and level of detail those computations 4 are still among the most ramified and accurately computed interfacial instability patterns that have appeared in the literature since 1994 the computational power of a standard workstation has increased a hundredfold as predicted by moore s law 7 the purpose of this note is to consider moore s law and its consequences in computational science and in particular its impact on studying the saffman taylor instability we illustrate moore s law and fast algorithms in action by presenting the worlds largest viscous fingering simulation to date viscous fingering is one of the fundamental interfacial instabilities in fluid dynamics perturbations to an expanding circular air bubble displacing a viscous fluid in a thin gap flow device become unstable resulting in intricate densely branched interfacial patterns the viscous fingering problem is governed by the three dimensional incompressible navier stokes equations with moving free boundaries at the fluid air interface without simplification this problem is computationally intractable and will remain so for the foreseeable future the simulations in this note build upon decades of advances in mathematical modeling and numerical methods i the fluid dynamics in a thin gap is reduced from the threedimensional navier stokes equations by asymptotic analysis to a darcy s law 12

identities with skew derivations abstract we solve affirmatively a problem raised by kharchenko on identities with compositions of skew derivations we define the notion of trivial identities with compositions of skew derivations which is unique in a certain sense it is proved that if a prime ring r satisfies a nontrivial identity with compositions of skew derivations then r also satisfies a generalized polynomial identity without skew derivations we actually work in a more general context in which higher skew derivations in the literature known to the author are all covered

data sparse approximation of non local operators by h2 matrices many of today s most efficient numerical methods are based on multilevel decompositions the multigrid algorithm is based on a hierarchy of grids wavelet techniques use a hierarchy of basis functions while fast panel clustering and multipole methods employ a hierarchy of clusters the high efficiency of these methods is due to the fact that the hierarchies are nested i e that the information present on a coarser level is also present on finer levels thus allowing efficient recursive algorithms h2 matrices employ nested local expansion systems in order to approximate matrices in optimal or for some problem classes at least optimal up to logarithmic factors order of complexity this paper presents a criterion for the approximability of general matrices in the h2 matrix format and an algorithm for finding good nested expansion systems and constructing the approximation efficiently

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

a parametrix for the neumann problem on pseudoconvex domains of finite type abstract we construct a parametrix for the neumann problem on any pseudoconvex domain of finite type which reduces completely the study of isotropic lp sobolev and holder estimates to those for b and b we also establish the maximal nonisotropic estimates in the case of d q domains and provide a precise comparison of the bergman and szego projections and of the canonical solutions to with respect to these two operators the argument is substantially simpler than the known ones for the cases of strongly pseudoconvex domains and pseudoconvex domains of finite type in c 2

identities with a single skew derivation abstract let r be a prime ring with extended centroid c and a continuous skew derivation of r we define the notion of k polynomials which in the case that is an ordinary derivation reduces to polynomials of the form x p n 1 x p n 1 n x where i c it is shown that all generalized identities with are consequences of gpis of r and an identity in the form x x m b b x where x is a k polynomial of minimal possible order m

analysis of arbitrarily shaped dielectric radomes using adaptive integral method based on volume integral equation the adaptive integral method aim is employed to solve the volume integral equation vie for analyzing the radiation of the antenna with an arbitrarily shaped radome small dipoles are used as exciting sources modeling the radomes by tetrahedron cells the induced volume current is determined by the aim based on vie the application of aim significantly reduces cpu time and computer memory requirement hence the method presented in the paper can be applied to simulate electrically large sized radomes finally the radiation patterns of small dipole arrays in the presence of spherical and conical radomes are calculated

an efficient surface integral algorithm applied to unsteady gravity waves a computationally fast method for calculating the unsteady motion of a surface on a two dimensional fluid is described cauchy s integral theorem is used iteratively to solve laplace s equation for successive time derivatives of the surface motion and time stepping is performed using truncated taylor series this allows fairly large time steps to be made for a given accuracy while the required number of spatial points is minimised by using high order differencing formulae this reduces the overall number of required calculations the numerical implementation of the method is found to be accurate and efficient a fairly thorough examination of this implementation is carried out revealing that high accuracies are often achievable using surprisingly few numerical surface points extensive calculations are also performed using modest computing resources some numerical instabilities are identified although these would not usually be significant in practical calculations a model analysis reveals that two of these instabilities can be eliminated by using suitable methods of time stepping should the third steep wave instability become significant it is shown that it can be completely controlled by using high order smoothing techniques at little cost to accuracy using a routine to ensure asymptotic conservation of energy this is confirmed by time stepping a very steep but stable wave over thousands of wave periods 0 1992 academx press inc since all of the interior properties of a body of fluid undergoing inviscid incompressible irrotational motion are fully determined by properties at its boundary it is possible to reduce the calculation of the motion of such a fluid to the evaluation of the motion of its surface alone in a numerical scheme the entire motion can thus be modelled using only a point discretisation of the surface such an approach was first suggested by svendsen in 1971 as a means of calculating the motion of two dimensional gravity waves on water 13 since then the idea has been implemented using a variety of boundary integral or conformal mapping techniques by longuet higgins and cokelet 2 vinje and brevig 3 baker meiron and orszag 4 roberts s fornberg 6 and based on his original formulation by svendsen and co workers 7 although this paper presents yet another numerical method the approach is aimed specifically at producing a computationally fast and efficient numerical scheme in order to be able to study problems of increased complexity in numerically representing complicated surface motions ranging from the breaking of waves to the evolution of instabilities in a train of travelling waves it typically becomes necessary to use a large number of surface points if running times tend to increase too rapidly with the number of points then this can destroy any apparent advantage in being able to treat the whole fluid only in terms of its moving surface the penalty is particularly severe for algorithms that require matrix inversion or factorisation techniques yielding running times per time step that increase like n3 the cube of the number of surface points by using iterative techniques the penalty can be reduced to running times per time step that increase in proportion to n2 while being less severe this still has the effect that increased numbers of surface points become increasingly expensive to calculate the use of a conformal mapping technique 6 can lead to running times per timestep that vary like n log n however this method is severely limited in its ability to resolve steep wave surfaces as are found in breaking waves requiring disproportionately many surface points to achieve a given accuracy it is therefore vital to ensure computational elliciency if it is to be possible accurately to describe wave processes for which large numbers of surface points may be required the most direct way of achieving this is to employ the lowest number of points that can adequately describe a fluid surface in this paper a further improvement is introduced by showing that higher time derivatives of the motion can be calculated directly using exactly the same technique that is used to determine the velocity of the surface at each timestep this yields significantly more information about the motion for relatively little extra computation in turn this allows larger time steps to be performed for a given accuracy so that the overall number of calculations can be reduced the method presented here combines these two ideas into producing a practical numerical algorithm the properties of which are examined in the following pages in appendix a the formulae are derived that make it

on the fast matrix multiplication in the boundary element method by panel clustering the boundary element method bem leads to a system of linear equations with a full matrix while fem yields sparse matrices this fact seems to require much computational work for the definition of the matrix for the solution of the system and in particular for the matrix vector multiplication which always occurs as an elementary in this paper a method for the approximate matrix vector multiplication is described which requires much less arithmetical work in addition the storage requirements are strongly reduced

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

the szegő projection sobolev estimates in regular domains the szego projection preserves global smoothness in weakly pseudoconvex domains that are regular in the sense of diederich fornxess and catlin it preserves local smoothness near boundary points of finite type the bergman projection being linked to the 8 problem plays an important role in several complex variables as demonstrated by recent applications to the boundary behavior of holomorphic maps 7 16 the szego projection is interesting both by analogy and by its connection with the ab problem and hence with unsolvable equations of lewy type while in strictly pseudoconvex domains the bergman and szego projections are both well understood in general much more is known about the bergman projection this is no surprise for the theory of the 8 problem is further advanced than the theory of the 8b problem nonetheless i believe that the szego projection ought not be any more mysterious than the bergman projection this paper is intended to support this contention the most important domains for which the bergman projection is known to be regular are domains with a good a neumann operator here it is shown that in such domains the szego projection also is regular more precisely in smooth bounded pseudoconvex domains satisfying catlin s property p the most general condition presently known that guarantees global regularity of the neumann operator the szego projection is shown to be globally regular theorem 4 1 moreover at boundary points of finite type the szego projection is locally regular theorem 5 1 in both cases the szego projection exactly preserves the degree of differentiability measured in sobolev norms the method of proof is not the obvious one of directly generalizing from the strictly peudoconvex case for there are serious obstacles to such a generalization for instance the approach of analysis on the heisenberg group used by phong and stein 29 to derive estimates for both the bergman and szego projections cannot yet be extended because the theory of singular integrals on weakly pseudoconvex surfaces is still in its early stages 28 15 similarly one does not know how to carry over the fourier integral operator approach of boutet de monvel and sjostrand 10 to the weakly pseudoconvex case explicit construction of cauchy fantappie forms received by the editors march 11 1986 paper presented april 11 1986 to the 826th meeting of the american mathematical society indianapolis special session on several complex variables 1980 mathematics subject classification 1985 revision primary 32a25 32h10 partially supported by national science foundation grants mcs 8201063 and dms 8501758 1987 american mathematical society 0002 9947 87 1 00 25 per page

wavelet methods for pdes some recent developments abstract this paper is concerned with recent developments of wavelet schemes for the numerical treatment of operator equations with special emphasis on two issues adaptive solution concepts and nontrivial domain geometries after describing a general multiresolution framework the key features of wavelet bases are highlighted namely locality norm equivalences and cancellation properties assuming first that wavelet bases with these properties are available on the relevant problem domains the relevance of these features for a wide class of stationary problems is explained in subsequent sections the main issues are preconditioning and the efficient adaptive application of wavelet representations of the involved operators we indicate then how these ingredients combined with concepts from nonlinear or best n term approximation culminate in an adaptive wavelet scheme for elliptic selfadjoint problems covering boundary value problems as well as boundary integral equations these schemes can be shown to exhibit convergence rates that are in a certain sense asymptotically optimal we conclude this section with some brief remarks on data structures and implementation interrelations with regularity in a certain scale of besov spaces and strategies of extending such schemes to unsymmetric or indefinite problems we address then the adaptive evaluation of nonlinear functionals of wavelet expansions as a central task arising in connection with nonlinear problems wavelet bases on nontrivial domains are discussed next the main issues are the development of fourier free construction principles and criteria for the validity of norm equivalences finally we indicate possible combinations of wavelet concepts with conventional discretizations such as finite element or finite volume schemes in connection with convection dominated and hyperbolic problems

biorthogonal wavelet approximation for the coupling of fem bem we apply multiscale methods to the coupling of finite and boundary element methods to solve an exterior dirichlet boundary value problem for the two dimensional poisson equation adopting biorthogonal wavelet matrix compression to the boundary terms with n degrees of freedom we show that the resulting compression strategy fits the optimal convergence rate of the coupling galerkin methods while the number of nonzero entries in the corresponding stiffness matrices is considerably smaller than n 2

minimal quasi algebraic relations of skew derivations in characteristic p 2 abstract let r be a prime ring of char r p 2 the monic minimal quasi algebraic relation of an ordinary derivation d of r takes the form d p s i 0 s 1 i d p i ad a where i are in the extended centroid of r and where a is in the symmetric martindale quotient ring of r among other things we extend this result to an arbitrary skew derivation d this is explained in details in 0 below

primitive identities of skew derivations abstract the commutator d d of two derivations d always yields a derivation and must be inner if one of d is inner we generalize this to primitive expressions of skew derivations theorem 3 2 theorem 3 3 our main results are theorem 2 6 theorem 2 7 as applications we first apply these to quantum lie operations theorem 3 6 then we extend koryukin s counterexample 14 by adding primitive identities involving inner skew derivations theorem 3 7 we work in the setting of hopf algebras and symmetric smash products theorem 1 3 is very useful in analyzing identities in this context

a fast high order solver for problems of scattering by heterogeneous bodies a new high order integral algorithm for the solution of scattering problems by heterogeneous bodies is presented here a scatterer is described by a continuously or discontinuously varying refractive index n x within a two dimensional 2d bounded region solutions of the associated helmholtz equation under given incident fields are then obtained by high order inversion of the lippmann schwinger integral equation the algorithm runs in o nlog n operations where n is the number of discretization points a wide variety of numerical examples provided include applications to highly singular geometries high contrast configurations as well as acoustically electrically large problems for which supercomputing resources have been used recently our method provides highly accurate solutions for such problems on small desktop computers in cpu times of the order of seconds

why finite element discretizations can be factored by triangular hierarchical matrices although the asymptotic complexity of direct methods for the solution of large sparse finite element systems arising from second order elliptic partial differential operators is far from being optimal these methods are often preferred over modern iterative methods this is mainly due to their robustness in this article it is shown that an approximate lu decomposition exists which can be computed in the algebra of hierarchical matrices with almost linear complexity and with the same robustness as the classical lu decomposition low precision approximations may be used for preconditioning iterative solvers as a byproduct we prove that schur complements of stiffness matrices can be approximated with almost linear complexity

sparse grids for boundary integral equations the potential of sparse grid discretizations for solving boundary integral equations is studied for the screen problem on a square in mathbb r 3 theoretical and numerical results on approximation rates preconditioning adaptivity and compression for piecewise constant and linear sparse grid spaces are obtained

wavelet methods for fast resolution of elliptic problems this paper shows that the use of wavelets to discretize an elliptic problem with dirichlet or neumann boundary conditions has two advantages an explicit diagonal preconditioning makes the condition number of the corresponding matrix become bounded by a constant and the order of approximation is locally of spectral type in contrast with classical methods using a conjugate gradient method one thus obtains fast numerical algorithms of resolution a comparison is also drawn between wavelet and classical methods

estimates for the bergman and szego kernels in mathbf c 2 the purpose of this paper some of whose conclusions were announced in nrsw is to study the bergman and szegd projection operators on pseudoconvex domains q of finite type in c2 the results we obtain are of three kinds i the size estimates of the bergman and szeg6 kernels and their derivatives ii the cancellation properties of those kernels expressed in terms of the actions of these operators on suitable bump functions these properties have an interest in their own right but are also crucial in iii below iii the sharp mapping properties of these operators on functions spaces such as lp the nonisotropic sobolev spaces and the nonisotropic lipschitz spaces which are naturally attached to the geometry of the boundary this also leads ultimately to sharp results in the regularity properties of solutions of the equation du f in q and 9bu f on the boundary let us describe these matters in more detail our starting points are certain basic geometric constructs associated to q a metric p defined in terms of the vector fields x1 and x2 which are the real and imaginary parts of the tangential levi vector field on d 2 and a function a p 8 a polynomial in 8 for p e d 2 which represents the higher levi invariant attached to d 9 the significance of these to the geometry of d q can be understood from the following facts the ball b p 8 c d q centered at p of radius 8 in the metric p when viewed in the appropriate coordinates has width 8 in the complex tangential directions but its width is a p 8 in the complementary direction thus the volume of the ball is 82a p 8 among other crucial properties

rapid solution of integral equations of classical potential theory an algorithm is described for rapid solution of classical boundary value problems dirichlet an neumann for the laplace equation based on iteratively solving integral equations of potential theory cpu time requirements for previously published algorithms of this type are proportional to n2 where n is the number of nodes in the discretization of the boundary of the region the cpu time requirements for the algorithm of the present paper are proportional to n making it considerably more practical for large scale problems

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

a fast algorithm for the multiplication of generalized hilbert matrices with vectors on presente un algorithme rapide de complexite logarithmique pour la multiplication des matrices de hilbert generalisees par des vecteurs

a fast multi level boundary element method for the helmholtz equation abstract most recently we have developed a novel multi level boundary element method mlbem for the solution of the steady heat diffusion equation involving asymptotically decaying non oscillatory log singular and strongly singular kernels this hierarchical approach generalizes the pioneering work of brandt and lubrecht on multi level multi integration mlmi and c cycle multi grid to the broader class of mixed boundary value problems the result is a fast accurate and efficient boundary element method the present paper extends this new computational methodology to the solution of the helmholtz equation involving oscillatory log singular and strongly singular kernels for two dimensional problems we consider a direct boundary element formulation and due to the nature of the fundamental solutions split the corresponding boundary integral equation into real and imaginary parts then we introduce double noded corners to facilitate a patch by patch application of the mlmi algorithm for fast matrix vector and matrix transpose vector multiplications within bi conjugate gradient methods the performance of the proposed fast mlbem is investigated using a numerical example that possesses an exact solution for wave numbers 20 and below we demonstrate that the fast mlbem algorithm for the helmholtz equation is robust accurate and exceptionally efficient

wavelet galerkin methods for second kind integral equations we use vector valued multiwavelets on compact sets to develop a galerkin method for systems of integral equations of the second kind we propose a compression strategy for the coefficient matrix of the linear system obtained from this method and show that the compressed scheme preserves almost optimal convergence rate of the original scheme and yields a sparse matrix with a bounded condition number

comparing the bergman and szego projections on domains with subelliptic boundary laplacian we prove that the difference between the bergman and szego projections on a bounded pseudoconvex domain with c boundary is smoothing whenever the boundary laplacian is subelliptic an equivalent statement is that the bergman projection can be represented as a composition of the szego and harmonic bergman projections along with the restriction and poisson extension operators modulo an error that is smoothing we give several applications to the study of optimal mapping properties for these projections and their difference

a wideband fast multipole method for the helmholtz kernel this work presents a new fast multipole method fmm based on plane wave expansions pwfmm combining the advantages of the low and high frequency formulations we revisit the method of greengard etal 1998 devoted to the low frequency regime and based on the splitting of the greens function into a propagative and an evanescent part more precisely we give an explicit formula of the filtered translation function for the propagative part we derive a new formula for the evanescent part and we provide a new interpolation algorithm at all steps we check the accuracy of the method by providing error estimates these theoretical developments are used to propose a wideband fmm based entirely on plane wave expansions the numerical efficiency and accuracy of this broadband pwfmm are illustrated with a numerical example

the fast solution of poisson s and the biharmonic equations on irregular regions we present fast methods for solving laplace s and the biharmonic equations on irregular regions with smooth boundaries the methods used for solving both equations make use of fast poisson solvers on a rectangular region in which the irregular region is embedded they also both use an integral equation formulation of the problem where the integral equations are fredholm integral equations of the second kind the main idea is to use the integral equation formulation to define a discontinuous extension of the solution to the rest of the rectangular region fast solvers are then used to compute the extended solution aside from solving the equations we have also been able to compute derivatives of the solutions with little loss of accuracy when the data was sufficiently smooth

fast wavelet transforms and numerical algorithms i a class of algorithms is introduced for the rapid numerical application of a class of linear operators to arbitrary vectors previously published schemes of this type utilize detailed analytical information about the operators being applied and are specific to extremely narrow classes of matrices in contrast the methods presented here are based on the recently developed theory of wavelets and are applicable to all calderon zygmund and pseudo differential operators the algorithms of this paper require order o n or o n log n operations to apply an n n matrix to a vector depending on the particular operator and the version of the algorithm being used and our numerical experiments indicate that many previously intractable problems become manageable with the techniques presented here

a multi level boundary element method for stokes flows in irregular two dimensional domains recently we have developed multi level boundary element methods mlbem for the solution of the laplace and helmholtz equations that involve asymptotically decaying non oscillatory and oscillatory singular kernels respectively the accuracy and efficiency of the fast boundary element methods for steady state heat diffusion and acoustics problems have been investigated for square domains the current work extends the mlbem methodology to the solution of stokes equation in more complex two dimensional domains the performance of the fast boundary element method for the stokes flows is first investigated for a model problem in a unit square then we consider an example problem possessing an analytical solution in a rectangular domain with 5 1 aspect ratio and finally we study the performance of the mlbem algorithm in a c shaped domain

stokes flow around cylinders in a bounded two dimensional domain using multipole accelerated boundary element methods the multipole technique has recently received attention in the field of boundary element analysis as a means of reducing the order of data storage and calculation time requirements from o n2 iterative solvers or o n3 gaussian elimination to o n log n or o n where n is the number of nodes in the discretized system such a reduction in the growth of the calculation time and data storage is crucial in applications where n is large such as when modelling the macroscopic behaviour of suspensions of particles in such cases a minimum of 1000 particles is needed to obtain statistically meaningful results leading to systems with n of the order of 10 000 for the smallest problems when only boundary velocities are known the indirect boundary element formulation for stokes flow results in fredholm equations of the second kind which generally produce a well posed set of equations when discretized a necessary requirement for iterative solution methods the direct boundary element formulation on the other hand results in fredholm equations of the first kind which upon discretization produce ill conditioned systems of equations the model system here is a two dimensional wide gap couette viscometer where particles are suspended in the fluid between the cylinders this is a typical system that is efficiently modelled using boundary element method simulations the multipolar technique is applied to both direct and indirect formulations it is found that the indirect approach is sufficiently well conditioned to allow the use of fast multipole methods the direct approach results in severe ill conditioning to a point where application of the multipole method leads to non convergence of the solution iteration copyright 1999 john wiley sons ltd

efficient numerical methods for multiple surfactant coated bubbles in a two dimensional stokes flow we present efficient and highly accurate numerical methods to compute the deformation of surfactant coated two dimensional bubbles in a slow viscous flow surfactant acts to locally alter the surface tension and thereby change the nature of the interface motion in this paper we restrict our attention to the case of a dilute insoluble surfactant the convection diffusion equation for the surfactant concentration on the interface is coupled with the stokes equations in the fluid domain through a boundary condition based on the laplace young condition the stokes equations are first recast as an integral equation and then solved using a fast multipole accelerated iterative procedure the computational cost per time step is only o n log n operations with n being the number of discretization points on the interface the bubble interfaces are described by a spectral mesh and is advected according to the fluid velocity in such a manner so as to preserve equal arc length spacing of marker points this equal arc length framework has the dual advantage of dynamically maintaining the spatial mesh and allowing efficient implicit treatment of the stiffest terms in the dynamics several phenomenologically different examples are presented

least squares methods for the coupling of fem and bem in the present paper we propose least squares formulations for the numerical solution of exterior boundary value problems the partial differential equation is a first order system in a bounded subdomain and the unbounded subdomain is treated by means of boundary integral equations the first order system is derived from a strongly elliptic second order system the analysis of the present least squares formulations is reduced to the analysis of the galerkin method for the coupling of finite element and boundary element methods fem and bem of the second order problem the least squares approach requires no stability condition however it requires the computation of negative as well as of half integer sobolev norms the arising linear systems can be preconditioned to have condition numbers sim 1 the present methods benefit strongly from the use of biorthogonal wavelets on the coupling boundary and the computation of corresponding equivalent norms in sobolev spaces in particular the application of green s formula leads to an efficient discretization of least squares formulations

multiply rooted multiscale models for large scale estimation divide and conquer or multiscale techniques have become popular for solving large statistical estimation problems the methods rely on defining a state which conditionally decorrelates the large problem into multiple subproblems each more straightforward than the original however this step cannot be carried out for asymptotically large problems since the dimension of the state grows without bound leading to problems of computational complexity and numerical stability in this paper we propose a new approach to hierarchical estimation in which the conditional decorrelation of arbitrarily large regions is avoided and the problem is instead addressed piece by piece the approach possesses promising attributes it is not a local method the estimate at every point is based on all measurements it is numerically stable for problems of arbitrary size and the approach retains the benefits of the multiscale framework on which it is based a broad class of statistical models a stochastic realization theory an algorithm to calculate statistical likelihoods and the ability to fuse local and nonlocal measurements

spatially adapted multiwavelets and sparse representation of integral equations on general geometries in this paper we develop irregular wavelet representations for complex domains with the goal of demonstrating their potential for three dimensional 3d scientific and engineering computing applications we show existence and construction of a large class of continuous spatially adapted multiwavelets in r eta with vanishing moments over complex geometries these wavelets share all of the major advantages of conventional wavelets in that they provide an analytical tool for studying data functions and operators at different scales however unlike conventional wavelets which are restricted to uniform grids spatially adapted multiwavelets allow fast transform localization and decorrelation on complex meshes such as those encountered in finite element modeling we show how these new constructions can be applied to partial differential equations cast in the integral form we implement the wavelet approach for several model two dimensional 2d and 3d potential problems it is shown that the optimal convergence rate is achieved with only cal o n rm log n alpha entries of the discrete operator matrix where alpha is a small number and n is the number of unknowns

discrete boundary element methods on general meshes in 3d abstract this paper is concerned with the stability and convergence of fully discrete galerkin methods for boundary integral equations on bounded piecewise smooth surfaces in mathbb r 3 our theory covers equations with very general operators provided the associated weak form is bounded and elliptic on h mu for some mu in 1 1 in contrast to other studies on this topic we do not assume our meshes to be quasiuniform and therefore the analysis admits locally refined meshes to achieve such generality standard inverse estimates for the quasiuniform case are replaced by appropriate generalised estimates which hold even in the locally refined case since the approximation of singular integrals on or near the diagonal of the galerkin matrix has been well analysed previously this paper deals only with errors in the integration of the nearly singular and smooth galerkin integrals which comprise the dominant part of the matrix our results show how accurate the quadrature rules must be in order that the resulting discrete galerkin method enjoys the same stability properties and convergence rates as the true galerkin method although this study considers only continuous piecewise linear basis functions on triangles our approach is not restricted in principle to this case as an example the theory is applied here to conventional triangle based quadrature rules which are commonly used in practice a subsequent paper 14 introduces a new and much more efficient node based approach and analyses it using the results of the present paper

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

nonlinear identities with skew derivations abstract let r be a prime ring with the extended centroid c suppose that r is acted by a pointed coalgebra with group like elements acting as automorphisms of r a generalized polynomial with variables acted by the coalgebra is called an identity if it vanishes on r we prove the following 1 if c is a perfect field then any such identity is a consequence of simple basic identities defined in 6 and gpis of r with variables acted by frobenius automorphisms 2 if c is not a perfect field then any such identity is a consequence of simple basic identities defined in 6 and gpis of r with this we extend yanai s result 25 to nonlinear identities these are actually special instances of our theorems 1 and 2 below respectively which extend kharchenko s theory of differential identities 14 15 to the context of expansion closed word sets introduced in 6

an adaptive multigrid technique for evaluating long range forces in biomolecular simulations we propose an adaptive multigrid method for evaluating the nonlocal long range interactions in biomolecular models the method provides o h 3 accuracy for the forces where h is the mesh size of the finest cubic grid even though the stencil size is minimal two new algorithmic components are developed an adaptive technique for spreading a three dimensional charge distribution onto a space mesh and a procedure for performing coarse to fine grid interpolation of the potential force components and their derivatives grid to particle interpolation at the finest level in a consistent fashion the adaptive mesh used in the spreading procedure contains two types of nodes regular positioned at nodes cubic grid and flexible one per cubic cell thus the charges located in each grid cell are spread to these nine nodes the method conserves charge dipole and quadrupole moments of cells the spreading and interpolation techniques are economical in computation time and storage due to the overlap design in the stencils implementation is also straightforward our initial tests demonstrate the favorable performance for low accuracy with respect to the fast multipole technique

comparison of the bergman and szego kernels abstract the quotient of the szego and bergman kernels for a smooth bounded pseudoconvex domains in c n is bounded from above by a constant multiple of log p for any p n where is the distance to the boundary for a class of domains that includes those of d angelo finite type and those with plurisubharmonic defining functions the quotient is also bounded from below by a constant multiple of log p for any p 1 moreover for convex domains the quotient is bounded from above and below by constant multiples of

a fast dual boundary element method for 3d anisotropic crack problems in the present paper a fast solver for dual boundary element analysis of 3d anisotropic crack problems is formulated implemented and tested the fast solver is based on the use of hierarchical matrices for the representation of the collocation matrix the admissible low rank blocks are computed by adaptive cross approximation aca the performance of aca against the accuracy of the adopted computational scheme for the evaluation of the anisotropic kernels is investigated focusing on the balance between the kernel representation accuracy and the accuracy required for aca the system solution is computed by a preconditioned gmres and the preconditioner is built exploiting the hierarchical arithmetic and taking full advantage of the hierarchical format the effectiveness of the proposed technique for anisotropic crack problems has been numerically demonstrated highlighting the accuracy as well as the significant reduction in memory storage and analysis time in particular it has been numerically shown that the computational cost grows almost linearly with the number of degrees of freedom obtaining up to solution speedups of order 10 for systems of order 104 moreover the sensitivity of the performance of the numerical scheme to materials with different degrees of anisotropy has been assessed copyright 2009 john wiley sons ltd

adaptive multiscale moment method ammm for analysis of scattering from three dimensional perfectly conducting structures the adaptive multiscale moment method ammm is presented for the analysis of scattering from three dimensional 3d perfectly conducting bodies this algorithm employs the conventional moment method mm using the subsectional triangular patch basis functions and a special matrix transformation which is derived from solving the fredholm equation of the first kind by the multiscale technique this methodology is more suitable for problems where the matrix solution time is much greater than the matrix fill time the widely used triangular patch vector basis functions developed by rao et al 1982 are used for expansion and testing functions in the conventional mm the objective here is to compress the unknowns in existing mm codes which solves for the currents crossing the edges of the triangular patch basis functions by use of a matrix transformation the currents source terms and impedance matrix can be arranged in the form of different scales from one scale to another scale the initial guess for the solution can be predicted according to the properties of the multiscale technique ammm can reduce automatically the size of the linear equations so as to improve the efficiency of the conventional mm the basic difference between this methodology and the other wavelet based techniques that have been presented so far is that we apply the compression not to the impedance matrix but to the solution itself directly in an iterative fashion even though it is an unknown two numerical results are presented which demonstrate that the ammm is a useful method for analysis of electromagnetic scattering from arbitrary shaped 3d perfectly conducting bodies

wavelet discretizations of parabolic integrodifferential equations we consider parabolic problems dot u au f in 0 t times omega t infty where omega subset mathbb r d is a bounded domain and a is a strongly elliptic classical pseudodifferential operator of order rho in 0 2 in tilde h rho 2 omega we use a theta scheme for time discretization and a galerkin method with n degrees of freedom for space discretization the full galerkin matrix for a can be replaced with a sparse matrix using a wavelet basis and the linear systems for each time step are solved approximatively with gmres we prove that the total cost of the algorithm for m time steps is bounded by o mn log n beta operations and o n log n beta memory we show that the algorithm gives optimal convergence rates up to logarithmic terms for the computed solution with respect to l2 in time and the energy norm in space

inexact data sparse boundary element tearing and interconnecting methods the boundary element tearing and interconnecting beti methods have recently been introduced as boundary element counterparts of the well established finite element tearing and interconnecting feti methods in this paper we present inexact data sparse versions of the beti methods which avoid the elimination of the primal unknowns and dense matrices however instead of symmetric and positive definite systems we finally have to solve twofold saddle point problems the proposed iterative solvers and preconditioners result in almost optimal solvers whose complexity is proportional to the number of unknowns on the skeleton up to some polylogarithmical factor moreover the solvers are robust with respect to large coefficient jumps

combining kronecker product approximation with discrete wavelet transforms to solve dense function related linear systems a new solution technique is proposed for linear systems with large dense matrices of a certain class including those that come from typical integral equations of potential theory this technique combines kronecker product approximation and wavelet sparsification for the kronecker product factors the user is only required to supply a procedure for computation of each entry of the given matrix the main sources of efficiency are the incomplete cross approximation procedure adapted from the mosaic skeleton method of the second author and data sparse preconditioners the incomplete lu decomposition with dynamic choice of the fill in structure with a prescribed threshold and the inverse kronecker product preconditioner constructed for the sum of kronecker products of sparsified finger like matrices computed by the discrete wavelet transform in some model but quite representative examples the new technique allowed us to solve dense systems with more than 1 million unknowns in a few minutes on a personal computer with 1 gbyte operative memory

multiscale approximation of the solution of weakly singular second kind fredholm integral equation in legendre multiwavelet basis numerical solution of fredholm integral equation of second kind with weakly singular kernel is obtained in this paper by employing legendre multi wavelet basis the low and high pass filters for two scale relations involving legendre multiwavelets having four or five vanishing moments of their wavelets have been derived and are used in the evaluation of integrals for the multiscale representation of the integral operator explicit expressions for the elements of the matrix associated with the multiscale representation are given an estimate for the holder exponent of the solution of the integral equation at any point in its domain is obtained a number of examples is provided to illustrate the efficiency of the method developed here

differential identities with automorphisms and antiautomorphisms i abstract let r be a prime ring with the extended centroid c an anti automorphism g of r is said to be frobenius if in the case of char r g for all c or in the case of char r p 0 g p n for all c where n is a fixed integer which may be zero positive or negative anti automorphisms g 1 g 2 are said to be strongly independent if for any i j g i g 1 j is not frobenius the following is proved main theorem let z ijk be a polynomial with frobenius anti automorphisms assume that x jgk i 0 holds on a nonzero two sided ideal of r where x i are distinct indeterminates j are distinct regular derivation words with respect to a linearly ordered basis of outer derivations and g k are strongly independent anti automorphisms of r then z ijk 0 holds on the two sided utumi quotient ring of r if z ijk does not involve antiautomorphisms then z ijk 0 also holds on the left utumi quotient ring of r

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

the cauchy szegő projection for domains in mathbb c n with minimal smoothness we prove the lp bd regularity of the cauchy szeg projection also known as the szeg projection for bounded domains d cn which are strongly pseudoconvex and whose boundary satisfies the minimal regularity condition of class c2

prewavelet analysis of the heat equation summary we consider the heat equation in a smooth domain of r 2 with dirichlet and neumann boundary conditions it is solved by using its integral formulation with double layer potentials where the unknown the jump of the solution through the boundary belongs to an anisotropic sobolev space we approximate by the galerkin method and use a prewavelet basis on t 0 t which characterizes the anisotropic space the use of prewavelets allows to compress the stiffness matrix from o n 2 to o n log n when n is the size of the matrix and the condition number of the compressed matrix is uniformly bounded as the initial one in the prewavelet basis finally we show that the compressed scheme converges as fast as the galerkin one even for the dirichlet problem which does not admit a coercive variational formulation

multilevel matrix multiplication and fast solution of integral equations abstract a fast multigrid approach is described for the task of calculating k x y u y dy for each x r d discretizing by an equidistant grid with n points and meshsize h and approximating the integrations to o h2s accuracy it is shown that the complexity of this calculation can be reduced from o n2 to o sn provided the kernel k is sufficiently smooth for potential type kernels the complexity is reduced to o sn log n corresponding integral equations can be solved to a similar accuracy in basically the same amount of work using a special kind of distributed relaxation in a multigrid algorithm one and two dimensional numerical tests and theoretical derivations of optimal strategies are reported the method is applicable to the task of multiplying by any matrix with appropriate smoothness properties including most types of many body interactions