wavelet like bases for the fast solutions of second kind integral equations a class of vector space bases is introduced for the sparse representation of discretizations of integral operators an operator with a smooth nonoscillatory kernel possessing a finite number of singularities in each row or column is represented in these bases as a sparse matrix to high precision a method is presented that employs these bases for the numerical solution of second kind integral equations in time bounded by o n log 2 n where n is the number of points in the discretization numerical results are given which demonstrate the effectiveness of the approach and several generalizations and applications of the method are discussed

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

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

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

default theories that always have extensions default logic is an important and influential formalization of commonsense reasoning determining whether a given default theory has an extension is the main computational problem pertinent to default logic the analog of testing for validity and finding deductions in more classical logics substantially generalizing a result by etherington 5 we show that all default theories that have no odd cycles in some precise sense have an extension which can be found efficiently we also give a proof that it is np complete to find extensions even for default theories with no prerequisites and at most two literals per default a case substantially simpler than the np completeness results in the literature

alternative foundations for reiter s default logic abstract we introduce an alternative conceptual basis for default reasoning in reiter s default logic in fact most formal or computational treatments of default logic suffer from the necessity of exhaustive consistency checks with respect to the finally resulting set of conclusions often this so called extension is just about being constructed on the theoretical side this exhaustive approach is reflected by the usual fixed point characterizations of extensions our goal is to reduce such global considerations to local and strictly necessary ones for this purpose we develop various techniques and instruments that draw on an analysis of interaction patterns between default rules embodied by their mutual blocking behavior these formal tools provide us with alternative means for addressing a variety of questions in default logic we demonstrate the utility of our approach by applying it to three traditional problems first we obtain a range of criteria guaranteeing the existence and non existence of extensions second we get alternative characterizations of extensions that avoid fixed point conditions finally we furnish a formal account of default proofs that was up to now neglected in the literature

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

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

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

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

hard problems for simple default logics abstract we investigate the complexity of reasoning with a number of limited default logics surprising negative results the high complexity of simple three literal default rules as well as positive results a fast algorithm for skeptical reasoning with binary defaults are reported and sources of complexity are discussed these results impact on work on defeasible inheritance hierarchies as well as default reasoning in general

a priori sparsity patterns for parallel sparse approximate inverse preconditioners parallel algorithms for computing sparse approximations to the inverse of a sparse matrix either use a prescribed sparsity pattern for the approximate inverse or attempt to generate a good pattern as part of the algorithm this paper demonstrates that for pde problems the patterns of powers of sparsified matrices psms can be used a priori as effective approximate inverse patterns and that the additional effort of adaptive sparsity pattern calculations may not be required psm patterns are related to various other approximate inverse sparsity patterns through matrix graph theory and heuristics concerning the pde s green s function a parallel implementation shows that psm patterned approximate inverses are significantly faster to construct than approximate inverses constructed adaptively while often giving preconditioners of comparable quality

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

multiresolution approximate inverse preconditioners we introduce a new preconditioner for elliptic pdes on unstructured meshes using a wavelet inspired basis we compress the inverse of the matrix allowing an effective sparse approximate inverse by solving the sparsity vs accuracy conflict the key issue in this compression is to use second generation wavelets which can be adapted to the unstructured mesh the true boundary conditions and even the pde coefficients we also show how this gives a new perspective on multiresolution algorithms such as multigrid interpreting the new preconditioner as a variation on node nested multigrid in particular we hope the new preconditioner will combine the best of both worlds fast convergence when multilevel methods can succeed but with robust performance for more difficult problems the rest of the paper discusses the core issues for the preconditioner ordering and construction of a factored approximate inverse in the multiresolution basis robust interpolation on unstructured meshes automatic mesh coarsening and purely algebraic alternatives some exploratory numerical experiments suggest the superiority of the new basis over the standard basis for several tough problems including discontinuous anisotropic coefficients strong convection and indefinite reaction problems on unstructured meshes with scalability like hierarchical basis methods achieved

a note on weak type l1 boundedness of czos with rough kernels abstract for f s r 2 and l 1 s 1 s 1 x d x 0 define t f x lim 0 x y y y y 2 f x y d y in this paper we shall prove that there is a class of functions in h 1 s 1 l ln l s 1 such that t is weak type l 1 bounded

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

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

singular integral operators with rough convolution kernels in all dimensions again under the assumption q e l log l it is conceivable that a variant of the arguments in 3 for the maximal operator could also work for the singular integral operator in fact in unpublished work the authors of 3 obtained a weak type 1 1 inequality in dimension d 7 however their arguments if applied to the singular integral operator lead to substantial technical difficulties and no

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

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

on two variants of an algebraic wavelet preconditioner a recursive method of constructing preconditioning matrices for the nonsymmetric stiffness matrix in a wavelet basis is proposed for solving a class of integral and differential equations it is based on a level by level application of the wavelet scales decoupling the different wavelet levels in a matrix form just as in the well known nonstandard form the result is a powerful iterative method with built in preconditioning leading to two specific algebraic multilevel iteration algorithms one with an exact schur preconditioning and the other with an approximate schur preconditioning numerical examples are presented to illustrate the efficiency of the new algorithms

adaptive pattern research for block fsai preconditioning an adaptive algorithm is presented to generate automatically the nonzero pattern of the block factored sparse approximate inverse bfsai preconditioner it is demonstrated that in symmetric positive definite spd problems bfsai minimizes an upper bound to the kaporin number of the preconditioned matrix the mathematical structure of this bound suggests an efficient and easily parallelizable strategy for improving the given nonzero pattern of bfsai thus providing a novel adaptive bfsai abf preconditioner numerical experiments performed on large sized finite element problems show that abf coupled with a block incomplete cholesky ic outperforms bfsai ic even by a factor of 4 preserving the same preconditioner density and exhibiting an excellent parallelization degree

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

orderings for incomplete factorization preconditioning of nonsymmetric problems numerical experiments are presented whereby the effect of reorderings on the convergence of preconditioned krylov subspace methods for the solution of nonsymmetric linear systems is shown the preconditioners used in this study are different variants of incomplete factorizations it is shown that certain reorderings for direct methods such as reverse cuthill mckee can be very beneficial the benefit can be seen in the reduction of the number of iterations and also in measuring the deviation of the preconditioned operator from the identity

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 methods for second order elliptic problems preconditioning and adaptivity wavelet methods allow us to combine high order accuracy efficient preconditioning techniques and adaptive approximations in order to solve efficiently elliptic operator equations many difficulties remain in particular related to the adaptation of wavelet decompositions to bounded domains with prescribed boundary conditions leading to possibly high constants in the cal o 1 preconditioning in this paper we consider the framework of conforming domain decomposition to generate our wavelet bases and second order operators we emphasize the choice of the wavelets near the boundary of the tensor product reference domain in order to optimize the efficiency of the diagonal preconditioning of elliptic operators in order to improve the constants obtained by such diagonal preconditionings we propose to take into account interactions between the scales through the computation of a sparse approximate inverse spai on a set of nonzero entries obtained from the compression of the operator itself in the wavelet basis the efficiency of these methods is illustrated by solving elliptic second order problems with variable or constant coefficients and homogeneous boundary conditions on a uniform discretization finally we propose a coupling of the iterative solver with an adaptive space refinement technique on the laplacian model problem our experiments show that this algorithm generates an optimal nonlinear approximation of the solution

approximate inverse preconditioners via sparse sparse iterations the standard incomplete lu ilu preconditioners often fail for general sparse indefinite matrices because they give rise to unstable factors l and u in such cases it may be attractive to approximate the inverse of the matrix directly this paper focuses on approximate inverse preconditioners based on minimizing i am f where am is the preconditioned matrix an iterative descent type method is used to approximate each column of the inverse for this approach to be efficient the iteration must be done in sparse mode i e with sparse matrix by sparse vector operations numerical dropping is applied to maintain sparsity compared to previous methods this is a natural way to determine the sparsity pattern of the approximate inverse this paper describes newton global and column oriented algorithms and discusses options for initial guesses self preconditioning and dropping strategies some limited theoretical results on the properties and convergence of approximate inverses are derived numerical tests on problems from the harwell boeing collection and the fidap fluid dynamics analysis package show the strengths and limitations of approximate inverses finally some ideas and experiments with practical variations and applications are presented

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

a logic for default reasoning the need to make default assumptions is frequently encountered in reasoning about incompletely specified worlds inferences sanctioned by default are best viewed as beliefs which may well be modified or rejected by subsequent observations it is this property which leads to the non monotonicity of any logic of defaults in this paper we propose a logic for default reasoning we then specialize our treatment to a very large class of commonly occuring defaults for this class we develop a complete proof theory and show how to interface it with a top down resolution theorem prover finally we provide criteria under which the revision of derived beliefs must be effected

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

sparse approximate inverse preconditioners using norm minimization techniques we investigate the use of sparse approximate inverse preconditioners for the iterative solution of unsymmetric linear systems of equations we consider the approximations proposed by cosgrove diaz and griewank internat j comput math 44 1992 pp 91 110 and huckle and grote a new approach to parallel preconditioning with sparse approximate inverses tech report sccm 94 03 stanford university 1994 which are based on norm minimization techniques such methods are of particular interest because of the considerable scope for parallelization we propose a number of enhancements which may improve their performance when run in a sequential environment these methods can perform unfavorably when compared with other techniques however they can be successful when other methods fail and simulations indicate that they can be competitive when considered in a parallel environment

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

an implicit wavelet sparse approximate inverse preconditioner wavelet based sparse approximate inverse preconditioners are considered for the linear system ax b the preconditioners are good sparse approximations to the inverse of a computed by taking advantage of the compression obtained by working in a wavelet basis when the representation of a in a single scale basis for example a finite element basis is available the formulation presented obviates computation of the representation of a in the wavelet basis and removes the associated costs efficient application for both sparse and dense a is considered

preconditioning techniques for nonsymmetric and indefinite linear systems abstract the standard preconditioning techniques for conjugate gradient methods often fail for matrices that are indefinite and or strongly nonsymmetric the most common alterative considered for these cases are either to use expensive direct solvers or to resort to one of many techniques based on the normal equations this paper examines several such alternatives and compares them in particular an incomplete lq factorization is proposed and some of its implementation details are described a number of experiments are reported to compare these methods

a graph theoretic approach to default logic a network representation of propositional seminormal disjunction free default theories is presented leading to a graph theoretic approach to their analysis the problem of finding an extension is proved to be equivalent to that of determining a kernel for a corresponding graph allowing stronger complexity results as well as new conditions for the existence of extensions

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

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

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

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

model preference default theories most formal theories of default inference have very poor computational properties and are easily shown to be intractable or worse undecidable we are therefore investigating limited but eciently computable theories of default reasoning this paper defines systems of propositional model preference defaults which provide a true modeltheoretic account of default inference with exceptions the most general system of model preference defaults is decidable but still intractable inspired by the very good linear complexity of propositional horn theories we consider systems of horn defaults surprisingly finding a most preferred model in even this very limited system is shown to be np hard tractability can be achieved in two ways by eliminating the specificity ordering among default rules thus limiting the system s expressive power and by restricting our attention to systems of acyclic horn defaults these acyclic theories can encode acyclic defeasible inheritance hierarchies but are strictly more general this analysis suggests several directions for future research finding other syntactic restrictions which permit ecient computation or more

autoepistemic logic autoepistemic logic is one of the principal modes of nonmonotonic reasoning it unifies several other modes of nonmonotonic reasoning and has important application in logic programming in the paper a theory of autoepistemic logic is developed this paper starts with a brief survey of some of the previously known results then the nature of nonmonotonicity is studied by investigating how membership of autoepistemic statements in autoepistemic theories depends on the underlying objective theory a notion similar to set theoretic forcing is introduced expansions of autoepistemic theories are also investigated expansions serve as sets of consequences of an autoepistemic theory and they can also be used to define semantics for logic programs with negation theories that have expansions are characterized and a normal form that allows the description of all expansions of a theory is introduced our results imply algorithms to determine whether a theory has a unique expansion sufficient conditions stratification that imply existence of a unique expansion are discussed the definition of stratified theories is extended and under some additional assumptions efficient algorithms for testing whether a theory is stratified are proposed the theorem characterizing expansions is applied to two classes of theories k1 theories and ae programs in each case simple hypergraph characterization of expansions of theories from each of these classes is given finally connections with stable model semantics for logic programs with negation is discussed in particular it is proven that the problem of existence of stable models is np complete authors abstract

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

weak type l1 boundedness of czos with odd rough kernels abstract for f s r d and l 1 s d 1 s d 1 x d x 0 the singular integral operator t is defined by t f x lim e 0 x y e y y y d f x y d y in this paper we prove some sufficient conditions for odd h 1 such that t is weak type l 1 bounded

a stability analysis of incomplete lu factorizations abstract the combination of iterative methods with preconditionings based on incomplete lu factorizations constitutes an effective class of methods for solving the sparse linear systems arising from the discretization of elliptic partial differential equations in this paper we show that there are some settings in which the incomplete lu preconditioners are not effective and we demonstrate that their poor performance is due to numerical instability our analysis consists of an analytic and numerical study of a sample two dimensional non self adjoint elliptic problem discretized by several finite difference schemes author

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

graph theoretical structures in logic programs and default theories abstract in this paper we present a graph representation of logic programs and default theories we show that many of the semantics proposed for logic programs with negation can be expressed in terms of notions emerging from graph theory establishing in this way a link between the fields namely the stable models the partial stable models and the well founded semantics correspond respectively to the kernels semikernels and the initial acyclic part of an associated graph this link allows us to consider both theoretical existence uniqueness and computational problems tractability algorithms approximations from a more abstract and rather combinatorial point of view it also provides a clear and intuitive understanding about how conflicts between rules are resolved within the different semantics furthermore we extend the basic framework developed for logic programs to the case of default logic by introducing the notions of partial deterministic and well founded extensions for default theories these semantics capture different ways of reasoning with a default theory

weak type 1 1 bounds for rough operators ii in this paper we consider four operators arising in harmonic analysis in r n 1 they share one feature each is known to be bounded on lp for all 1 p o but for each it has been an open question whether a weak type 1 1 bound also holds in two cases we show that the weak 1 1 bound does hold and for the others we establish weaker estimates which are still sharper than the lp boundedness although the four operators are quite distinct they all involve convolution kernels which are related to the singular integrals of calderon and zygmund but which lack the smoothness required for the usual analysis in each case this defect is mitigated by the presence of some form of curvature consider the bochner riesz means

parallel preconditioning with sparse approximate inverses a parallel preconditioner is presented for the solution of general sparse linear systems of equations a sparse approximate inverse is computed explicitly and then applied as a preconditioner to an iterative method the computation of the preconditioner is inherently parallel and its application only requires a matrix vector product the sparsity pattern of the approximate inverse is not imposed a priori but captured automatically this keeps the amount of work and the number of nonzero entries in the preconditioner to a minimum rigorous bounds on the clustering of the eigenvalues and the singular values are derived for the preconditioned system and the proximity of the approximate to the true inverse is estimated an extensive set of test problems from scientific and industrial applications provides convincing evidence of the effectiveness of this approach