imposing boundary and interface conditions in multi resolution wavelet galerkin method for numerical solution of helmholtz problems abstract in this paper we propose a new formulation for numerical solution of boundary value problems using mesh free multi resolution wavelet galerkin wg method a difficulty with the wg method is imposing boundary condition constraints single scale wavelet basis functions together with a jump function approach using cubic polynomial functions have been used in solving some boundary problems in the literature ramp and cubic polynomial functions have been also used in the jump function approach described in element free galerkin context and it is proven ramp functions exactly satisfy the desirable property for the derivative of jump but the cubic polynomial functions do not in this paper a multi resolution wg method is employed and both ramp and cubic polynomial jump functions are applied obtained results in one dimension are compared with analytical solutions results obtained with ramp jump functions in two dimensions are also reported the results obtained with the wg method manifest boundary and interface conditions are accurately imposed when the ramp jump functions are applied numerical results are in good agreement with accurate solutions

the existence and uniqueness of solution to wavelet collocation the study of the numerical solutions of pdes with wavelet collocation has yielded a number of substantial results however the existence and uniqueness of solution has not been discussed yet in this paper the existence and uniqueness of solution to wavelet collocation for elliptic equations is established and discussed moreover wavelet collocation is applied to numerical example to examine its appropriateness according to numerical example and analysis it is seen that the existence and uniqueness of solution of this paper is feasible and the new theory is meaningful for developing wavelet collocation

a fast adaptive diffusion wavelet method for burger s equation abstract a fast adaptive diffusion wavelet method is developed for solving the burger s equation the diffusion wavelet is developed in 2006 coifman and maggioni 2006 and its most important feature is that it can be constructed on any kind of manifold classes of operators which can be used for construction of the diffusion wavelet include second order finite difference differentiation matrices the efficiency of the method is that the same operator is used for the construction of the diffusion wavelet as well as for the discretization of the differential operator involved in the burger s equation the diffusion wavelet is used for the construction of an adaptive grid as well as for the fast computation of the dyadic powers of the finite difference matrices involved in the numerical solution of burger s equation in this paper we have considered one dimensional and two dimensional burger s equation with dirichlet and periodic boundary conditions for each test problem the cpu time taken by fast adaptive diffusion wavelet method is compared with the cpu time taken by finite difference method and observed that the proposed method takes lesser cpu time we have also verified the convergence of the given method

adaptive anisotropic wavelet collocation method on general curvilinear coordinate systems a new general framework for an adaptive anisotropic wavelet collocation method a awcm for the solution of partial differential equations is developed this proposed framework addresses two major shortcomings of existing wavelet based adaptive numerical methodologies namely the reliance on a rectangular domain and the curse of anisotropy i e drastic over resolution of sheet and filament like features arising from the inability of the wavelet refinement mechanism to distinguish highly correlated directional information in the solution the a awcm addresses both of these challenges by incorporating coordinate transforms into the adaptive wavelet collocation method for the solution of pdes the resulting integrated framework leverages the advantages of both the curvilinear anisotropic meshes and wavelet based adaptive refinement in a complimentary fashion resulting in greatly reduced cost of resolution for anisotropic features the proposed adaptive anisotropic wavelet collocation method retains the a priori error control of the solution and fully automated mesh refinement while offering new abilities through the flexible mesh geometry including body fitting the new a awcm is demonstrated for a variety of cases including parabolic diffusion acoustic scattering and unsteady external flow

an adaptive wavelet collocation method for the solution of partial differential equations on the sphere a dynamic adaptive numerical method for solving partial differential equations on the sphere is developed the method is based on second generation spherical wavelets on almost uniform nested spherical triangular grids and is an extension of the adaptive wavelet collocation method to curved manifolds wavelet decomposition is used for grid adaption and interpolation an o n hierarchical finite difference scheme based on the wavelet multilevel decomposition is used to approximate laplace beltrami jacobian and flux divergence operators the accuracy and efficiency of the method is demonstrated using linear and nonlinear examples relevant to geophysical flows although the present paper considers only the sphere the strength of this new method is that it can be extended easily to other curved manifolds by considering appropriate coarse approximations to the desired manifold here we used the icosahedral approximation to the sphere at the coarsest level

a review of geometric transformations for nonrigid body registration this paper provides a comprehensive and quantitative review of spatial transformations models for nonrigid image registration it explains the theoretical foundation of the models and classifies them according to this basis this results in two categories physically based models described by partial differential equations of continuum mechanics e g linear elasticity and fluid flow and basis function expansions derived from interpolation and approximation theory e g radial basis functions b splines and wavelets recent work on constraining the transformation so that it preserves the topology or is diffeomorphic is also described the final section reviews some recent evaluation studies the paper concludes by explaining under what conditions a particular transformation model is appropriate

a multilevel wavelet collocation method for solving partial differential equations in a finite domain abstract a multilevel wavelet collection method for the solution of partial differential equations is developed two different approaches of treating general boundary conditions are suggested both are based on the wavelet interpolation technique developed in the present research the first approach uses wavelets as a basis and results in a differential algebraic system of equations where the algebraic part arises from boundary conditions the second approach utilizes extended wavelets which satisfy boundary conditions exactly this approach results in a system of coupled ordinary differential equations the method is tested on the one dimensional burgers equation with small viscosity the solutions are compared with those resulting from the use of other numerical algorithms the present results indicate that the method is competitive with well established numerical algorithms

simultaneous space time adaptive wavelet solution of nonlinear parabolic differential equations dynamically adaptive numerical methods have been developed to efficiently solve differential equations whose solutions are intermittent in both space and time these methods combine an adjustable time step with a spatial grid that adapts to spatial intermittency at a fixed time the same time step is used for all spatial locations and all scales this approach clearly does not fully exploit space time intermittency we propose an adaptive wavelet collocation method for solving highly intermittent problems e g turbulence on a simultaneous space time computational domain which naturally adapts both the space and time resolution to match the solution besides generating a near optimal grid for the full space time solution this approach also allows the global time integration error to be controlled the efficiency and accuracy of the method is demonstrated by applying it to several highly intermittent 1d t dimensional and 2d t dimensional test problems in particular we found that the space time method uses roughly 18 times fewer space time grid points and is roughly 4 times faster than a dynamically adaptive explicit time marching method while achieving similar global accuracy

parallel adaptive wavelet collocation method for pdes a parallel adaptive wavelet collocation method for solving a large class of partial differential equations is presented the parallelization is achieved by developing an asynchronous parallel wavelet transform which allows one to perform parallel wavelet transform and derivative calculations with only one data synchronization at the highest level of resolution the data are stored using tree like structure with tree roots starting at a priori defined level of resolution both static and dynamic domain partitioning approaches are developed for the dynamic domain partitioning trees are considered to be the minimum quanta of data to be migrated between the processes this allows fully automated and efficient handling of non simply connected partitioning of a computational domain dynamic load balancing is achieved via domain repartitioning during the grid adaptation step and reassigning trees to the appropriate processes to ensure approximately the same number of grid points on each process the parallel efficiency of the approach is discussed based on parallel adaptive wavelet based coherent vortex simulations of homogeneous turbulence with linear forcing at effective non adaptive resolutions up to 20483 using as many as 2048 cpu cores

an adaptive multilevel wavelet solver for elliptic equations on an optimal spherical geodesic grid an adaptive multilevel wavelet solver for elliptic equations on an optimal spherical geodesic grid is developed the method is based on second generation spherical wavelets on almost uniform optimal spherical geodesic grids it is an extension of the adaptive multilevel wavelet solver o v vasilyev and n k r kevlahan j comput phys 206 2005 pp 412 431 to curved manifolds wavelet decomposition is used for grid adaption and interpolation a hierarchical finite difference scheme based on the wavelet multilevel decomposition is used to approximate the laplace beltrami operator the optimal spherical geodesic grid internat j comput geom appl 16 2006 pp 75 93 is convergent in terms of local mean curvature and has lower truncation error than conventional spherical geodesic grids the overall computational complexity of the solver is o mathcal n where mathcal n is the number of grid points after adaptivity the accuracy and efficiency of the method is demonstrated for the spherical poisson equation although the present paper considers the sphere the strength of this new method is that it can be extended easily to other curved manifolds by choosing an appropriate coarse approximation and using recursive surface subdivision

faster computation of the first factor of the class number of q x p we describe two fast methods for computing the first factor of the class number of the cyclotomic field q p in o p 2 log 5 p and o p 2 log p steps of elementary arithmetic operations on the numbers of size p respectively the first is deterministic while the second holds under the grh this is an improvement over the previous method of lehmer and masley which has complexity o p 3 81

a dynamically adaptive multilevel wavelet collocation method for solving partial differential equations in a finite domain a dynamically adaptive multilevel wavelet collocation method is developed for the solution of partial differential equations the multilevel structure of the algorithm provides a simple way to adapt computational refinements to local demands of the solution high resolution computations are performed only in regions where sharp transitions occur the scheme handles general boundary conditions the method is applied to the solution of the one dimensional burgers equation with small viscosity a moving shock problem and a nonlinear thermoacoustic wave problem the results indicate that the method is very accurate and efficient

wavelet optimized finite difference mesh for mhd flow in a circular duct abstract there is a large class of important problems in scientific and engineering applications where the solution shows very little variation in the core region but very rapid changes close to the boundary of the domain of interest magnetohydrodynamic flow problems belong to this category treatment of such problems numerically therefore requires a non uniform mesh which is very fine near the boundary the determination of the size of the grid in different parts of the domain is therefore a challenging problem some recent studies suggest that wavelet methods can be effectively applied to create an optimized mesh the method has mostly been used for one dimensional problems the present paper investigates the problem for fully developed mhd duct flow problems which are basically two dimensional the boundary layer character is more pronounced in one direction as compared to the other the governing equations comprise a coupled system of two partial differential equations with dirichlet boundary conditions wavelet based adaptive mesh is first created and then the system is discretized over this mesh computations have been carried out at different precision levels for various hartmann numbers for a pipe of circular section contours are also drawn to depict the core and the boundary layer region

wavelet optimized finite difference method with non static re gridding the main aim of wavelet based numerical methods for solving partial differential equations is to develop adaptive schemes in order to achieve accuracy and computational efficiency the wavelet optimized finite difference method wofd uses wavelets to generate appropriate grids to apply finite difference method its standard implementation carries out static re griddings after a fixed number of time steps we present an effective implementation of wofd method that reduces the number of static re griddings hence leading to reduction of flops without significant loss of accuracy numerical experiments are performed on different cases of heat and schrodinger equations

adaptive multiresolution collocation methods for initial boundary value problems of nonlinear pdes we have designed a cubic spline wavelet like decomposition for the sobolev space h 0 2 i where i is a bounded interval based on a special point value vanishing property of the wavelet basis functions a fast discrete wavelet transform dwt is constructed this dwt will map discrete samples of a function to its wavelet expansion coefficients in at most 7n log n operations using this transform we propose a collocation method for the initial boundary value problem of nonlinear partial differential equations pdes then we test the efficiency of the dwt and apply the collocation method to solve linear and nonlinear pdes

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

on the use of biorthogonal interpolating wavelets for large eddy simulation of turbulence recent work in the field of turbulence modelling has demonstrated the benefits of the wavelet based multiresolution analysis technique as a tool for the formulation of the large eddy simulation les equations in this formalism the les equations are obtained by projecting the navier stokes equations onto a hierarchy of wavelet spaces this paper investigates the use of biorthogonal interpolating wavelets as a basis for this projection placing special emphasis on the wavelet based differential operators that define this mapping a detailed analysis of their convergence properties is presented and compared to those of their orthogonal counterpart the daubechies wavelets based on this study we highlight the weaknesses of the unlifted interpolating wavelet representation for les sub grid modelling finally we establish a link between the unlifted framework and the sampling based les approach recently proposed in the literature

an adaptive spline wavelet adi sw adi method for two dimensional reaction diffusion equations we study a spline wavelet alternative direction implicit sw adi algorithm for solving two dimensional reaction diffusion equations this algorithm is based on a collocation method for pdes with a specially designed spline wavelet for the sobolev spaceh2 i on a closed intervali by using the tensor product nature of adaptive wavelet meshes we propose a sw adi method for two dimensional problems the proposed sw adi method is an efficient time dependent adaptive method with second order accuracy for solutions with localized phenomena such as in flame propagations issues like new boundary wavelets for more accurate boundary conditions adaptive strategy for two dimensional meshes data structure and storage and implementation details and numerical results will be discussed

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

second generation wavelet collocation method for the solution of partial differential equations an adaptive numerical method for solving partial differential equations is developed the method is based on the whole new class of second generation wavelets wavelet decomposition is used for grid adaptation and interpolation while a new o n hierarchical finite difference scheme which takes advantage of wavelet multilevel decomposition is used for derivative calculations the treatment of nonlinear terms and general boundary conditions is a straightforward task due to the collocation nature of the algorithm in this paper we demonstrate the algorithm for one particular choice of second generation wavelets namely lifted interpolating wavelets on an interval with uniform regular sampling the main advantage of using second generation wavelets is that wavelets can be custom designed for complex domains and irregular sampling thus the strength of the new method is that it can be easily extended to the whole class of second generation wavelets leaving the freedom and flexibility to choose the wavelet basis depending on the application

the applications of partial integro differential equations related to adaptive wavelet collocation methods for viscosity solutions to jump diffusion models this paper presents adaptive wavelet collocation methods for the numerical solutions to partial integro differential equations pides arising from option pricing in a market driven by jump diffusion process the first contribution of this paper lies in the formulation of the wavelet collocation schemes the integral and differential operators are formulated in the collocation setting exactly and efficiently in both adaptive and non adaptive wavelet settings the wavelet compression technique is employed to replace the full matrix corresponding to the nonlocal integral term by a sparse matrix an adaptive algorithm is developed which automatically obtains the solution on a near optimal grid the second contribution of this paper is the theoretical analysis of the wavelet collocation schemes due to the possible degeneracy of the parabolic operators classical solutions of the jump diffusion models may not exist in this paper we first prove the convergence and stability of the proposed numerical schemes under the framework of viscosity solution theory and then the numerical experiments demonstrate the accuracy and computational efficiency of the methods we developed

quadratic forms over c t1 t2 abstract we show that dimension 4 quadratic forms over c t1 t2 ti transcendental do not satisfy what might be naturally taken as a strong hasse principle though a weak hasse principle exists we reduced solvability to a question of determining the quadratic reciprocity group for the discriminant d the classes modulo d of elements x such that d is a quadratic residue modulo x we determine this group in terms of the class group of a function field we prove that the weak hasse principle does not hold for forms over c t1 d therefore witt and brauer groups of 2 dimensional function fields over c do not restrict monomorphically to their localizations

on the adaptive numerical solution of nonlinear partial differential equations in wavelet bases this work develops fast and adaptive algorithms for numerically solving nonlinear partial differential equations of the formut lu nf u where l and n are linear differential operators andf u is a nonlinear function these equations are adaptively solved by projecting the solutionuand the operators l and n into a wavelet basis vanishing moments of the basis functions permit a sparse representation of the solution and operators using these sparse representations fast and adaptive algorithms that apply operators to functions and evaluate nonlinear functions are developed for solving evolution equations for a wavelet representation of the solutionuthat containsnssignificant coefficients the algorithms update the solution usingo ns operations the approach is applied to a number of examples and numerical results are given

convergence criteria for aronszajn s method and for the bazley fox method on etablit des criteres de convergence pour les methodes d aronszajn et de bazley fox appliquees aux problemes autoadjoints aux valeurs propres de la forme bx x

diophantine undecidability of c t1 t2 we give a short proof that the diophantine theory of the complex function field c t 1 t 2 where t i are algebraically independent over the complex numbers in the language t 1 t 2 0 1 is undecidable we construct a model of z represented by isogenies of an elliptic curve

filter bank methods for hyperbolic pdes we use biorthogonal filter banks to solve hyperbolic pdes adaptively with a sparse multilevel representation of the solution the methods described are of finite difference type and the filter banks are used to give a sparse representation of signals and to transform between grids on different scales we derive bounds for the error and number of coefficients in the sparse representation these bounds also apply for filter banks that are not associated with any wavelets we develop algorithms for fast differentiation and multiplication in detail the strength of the method is shown in various test problems

convergence theorems for intermediate problems ii convergence theorems for the practical eigenvector free methods of gay and goerisch are obtained under a variety of hypotheses so that our theorems apply to both traditional boundary value problems and atomic problems in addition we prove convergence of the t t method of bazley and fox without an alignment of projections hypothesis required in previous literature

a fast adaptive wavelet collocation algorithm for multidimensional pdes a fast multilevel wavelet collocation method for the solution of partial differential equations in multiple dimensions is developed the computational cost of the algorithm is independent of the dimensionality of the problem and is o n where n is the total number of collocation points the method can handle general boundary conditions the multilevel structure of the algorithm provides a simple way to adapt computational refinements to local demands of the solution high resolution computations are performed only in regions where singularities or sharp transitions occur numerical results demonstrate the ability of the method to resolve localized structures such as shocks which change their location and steepness in space and time the present results indicate that the method has clear advantages in comparison with well established numerical algorithms

a hybrid wavelet based adaptive immersed boundary finite difference lattice boltzmann method for two dimensional fluid structure interaction abstract a second generation wavelet based adaptive finite difference lattice boltzmann method fd lbm is developed in this paper in this approach the adaptive wavelet collocation method awcm is firstly to the best of our knowledge incorporated into the fd lbm according to the grid refinement criterion based on the wavelet amplitudes of density distribution functions an adaptive sparse grid is generated by the omission and addition of collocation points on the sparse grid the finite differences are used to approximate the derivatives to eliminate the special treatments in using the fd based derivative approximation near boundaries the immersed boundary method ibm is also introduced into fd lbm by using the adaptive technique the adaptive code requires much less grid points as compared to the uniform mesh code as a consequence the computational efficiency can be improved to justify the proposed method a series of test cases including fixed boundary cases and moving boundary cases are invested a good agreement between the present results and the data in previous literatures is obtained which demonstrates the accuracy and effectiveness of the present awcm ib lbm

location of essential spectrum of intermediate hamiltonians restricted to symmetry subspaces a theorem is presented on the location of the essential spectrum of certain intermediate hamiltonians used to construct lower bounds to bound state energies of multiparticle atomic and molecular systems this result is an analog of the hunziker van winter zhislin theorem for exact hamiltonians which implies that the continuum of an n electron system begins at the ground state energy for the corresponding system with n 1 electrons the work presented here strengthens earlier results of beattie siam j math anal 16 492 1985 in that one may now consider hamiltonians restricted to the symmetry subspaces appropriate to the permutational symmetry required by the pauli exclusion principle or to other physically relevant symmetry subspaces the associated convergence theory is also given guaranteeing that all bound state energies can be approximated from below with arbitrary accuracy

image registration using wavelet based motion model an image registration algorithm is developed to estimate dense motion vectors between two images using the coarse to fine wavelet based motion model this motion model is described by a linear combination of hierarchical basis functions proposed by cai and wang siam numer anal 33 3 937 970 1996 the coarser scale basis function has larger support while the finer scale basis function has smaller support with these variable supports in full resolution the basis functions serve as large to small windows so that the global and local information can be incorporated concurrently for image matching especially for recovering motion vectors containing large displacements to evaluate the accuracy of the wavelet based method two sets of test images were experimented using both the wavelet based method and a leading pyramid spline based method by szeliski et al international journal of computer vision 22 3 199 218 1996 one set of test images taken from barron et al international journal of computer vision 12 43 77 1994 contains small displacements the other set exhibits low texture or spatial aliasing after image blurring and contains large displacements the experimental results showed that our wavelet based method produced better motion estimates with error distributions having a smaller mean and smaller standard deviation

relative class number of imaginary abelian fields of prime conductor below 10000 in this paper we compute the relative class number of all imaginary abelian fields of prime conductor below 10000 our approach is based on a novel multiple evaluation technique and assuming the erh it has a running time of o p2 log 2 p log log p where p is the conductor of the field

using divergence free wavelets for the numerical solution of the 2 d stationary navier stokes equations a galerkin wavelet scheme is presented for solving the 2 d stationary navier stokes equations using the scaling generator of the divergence free wavelets moreover some boundary generators are constructed to improve the approximation accuracy finally the optimal error estimates and the numerical results are reported

an adaptive multilevel wavelet collocation method for elliptic problems an adaptive multilevel wavelet collocation method for solving multi dimensional elliptic problems with localized structures is described the method is based on multi dimensional second generation wavelets and is an extension of the dynamically adaptive second generation wavelet collocation method for evolution problems int j comp fluid dyn 17 2003 151 wavelet decomposition is used for grid adaptation and interpolation while a hierarchical finite difference scheme which takes advantage of wavelet multilevel decomposition is used for derivative calculations the multilevel structure of the wavelet approximation provides a natural way to obtain the solution on a near optimal grid in order to accelerate the convergence of the solver an iterative procedure analogous to the multigrid algorithm is developed the overall computational complexity of the solver is o n where n is the number of adapted grid points the accuracy and computational efficiency of the method are demonstrated for the solution of two and three dimensional elliptic test problems

a brinkman penalization method for compressible flows in complex geometries to simulate flows around solid obstacles of complex geometries various immersed boundary methods had been developed their main advantage is the efficient implementation for stationary or moving solid boundaries of arbitrary complexity on fixed non body conformal cartesian grids the brinkman penalization method was proposed for incompressible viscous flows by penalizing the momentum equations its main idea is to model solid obstacles as porous media with porosity f and viscous permeability approaching zero it has the pronounced advantages of mathematical proof of error bound strong convergence and ease of numerical implementation with the volume penalization technique in this paper it is extended to compressible flows the straightforward extension of penalizing momentum and energy equations using brinkman penalization with respective normalized viscous h and thermal h t permeabilities produces unsatisfactory results mostly due to nonphysical wave transmissions into obstacles resulting in considerable energy and mass losses in reflected waves the objective of this paper is to extend the brinkman penalization technique to compressible flows based on a physically sound mathematical model for compressible flows through porous media in addition to penalizing momentum and energy equations the continuity equation for porous media is considered inside obstacles in this model the penalized porous region acts as a high impedance medium resulting in negligible wave transmissions the asymptotic analysis reveals that the proposed brinkman penalization technique results in the amplitude and phase errors of order o email protected 1 2 and o h h t 1 email protected 3 4 when the boundary layer within the porous media is respectively resolved or unresolved the proposed method is tested using 1 and 2 d benchmark problems the results of direct numerical simulation are in excellent agreement with the analytical solutions the numerical simulations verify the accuracy and convergence rates

a wavelet optimized very high order adaptive grid and order numerical method wavelets detect information at different scales and at different locations throughout a computational domain furthermore wavelets can detect the local polynomial content of computational data numerical methods are most efficient when the basis functions of the method are similar to the data present by designing a numerical scheme in a completely adaptive manner around the data present in a computational domain one can obtain optimal computational efficiency this paper extends the numerical wavelet optimized finite difference wofd method to arbitrarily high order so that one obtains in effect an adaptive grid and adaptive order numerical method which can achieve errors equivalent to errors obtained with a spectrally accurate numerical method

adaptive multiresolution schemes for shock computations in this paper we present adaptive multiresolution schemes for the computation of discontinuous solutions of hyperbolic conservation laws starting with the given grid we consider the grid averages of the numerical solution for a hierarchy of nested grids which is obtained by diadic coarsening and we compute its equivalent multiresolution representation this representation of the numerical solution consists of the grid averages of the coarsest grid and the set of errors in predicting the grid averages of each level of resolution from the next coarser one these errors depend on the size of the grid and the local regularity of the solution at a jump discontinuity they remain the size of the jump independent of the level of resolution this observation enables us to identify the location of discontinuities in the numerical solution in a region of smoothness once the numerical solution is resolved to our satisfaction at a certain locality of some grid then the prediction errors there for this grid and all finer ones are small this enables us to obtain data compression by setting to zero the terms of the multiresolution representation that fall below a specified tolerance the numerical flux of the adaptive scheme is taken to be that of a standard centered scheme unless it corresponds to an identified discontinuity in which case it is taken to be the flux of an eno scheme we use the data compression of the numerical solution in order to reduce the number of numerical flux evaluations as follows we start with the computation of the exact numerical fluxes at the few grid points of the coarsest grid and then we proceed through diadic refinement to the given grid at each step of refinement we add values for the numerical flux at the new grid points which are the centers of the coarser cells wherever the solution is locally well resolved i e the corresponding prediction error is below the specified tolerance the costly exact value of the numerical flux function is replaced by an accurate enough approximate value which is obtained by an inexpensive interpolation from the values of the coarser grid

adaptive wavelet enriched hierarchical finite element model for polycrystalline microstructures abstract this paper proposes an adaptive wavelet enriched hierarchical finite element model fem to enhance computational efficiency and solution accuracy for heterogeneous domains of anisotropic elastic materials it is motivated by the need to overcome shortcomings of conventional methods like the standard fem or fast fourier transformation fft based method for analyzing deformation in large polycrystalline microstructures the multi resolution wavelet functions are advantageous for selection of an optimal set of functions that can adaptively enrich the solution space with a prescribed level of accuracy the proposed method introduces a discretization space that conforms to the profile of the solution sought the method engages a second generation family of wavelets with a lifting scheme to generate hierarchical interpolation functions an iterative algorithm efficiently calculates estimates of the solution from the previous iterate using a modified jacobi method the adaptive fe method performs significantly better than uniformly refined fem in validation tests with focus on the convergence rate and error mitigation a polycrystalline microstructure with elastically anisotropic grains is simulated by the adaptive wavelet enriched hierarchical method showing high convergence rates

a parallel fast algorithm for computing the helmholtz integral operator in 3 d layered media in this paper we propose a parallel fast algorithm for computing the product of the discretized helmholtz integral operator in layered media and a vector in o n qn z 2n xn ylogn xn y operations here n xn yn z is the number of sources and n q is the number of quadrature points used in the evaluation of the sommerfeld integral in the definition of layered media green s function for problems in thin layer media n z o 1 such a product forms the key step of many iterative solvers such as the krylov subspace based gmres and bicgstab for linear systems arising from the integral equation methods for the helmholtz equations the fast solver is based on two important techniques which reduce the cost of quadrature summations in the sommerfeld contour integral for green s functions in 3 d layered media the first technique is the removal of surface pole effects along the real axis integration contour by identifying the pole locations with a discrete wavelet transform in the second technique we apply a window based high frequency filter to shorten the contour length as a result the integral operator for the 3 d layered media can be efficiently written as a sum of 2 d hankel cylindrical integral operators and the latter can be calculated by either a tree code or a 2 d wideband fast multipole method in a fast manner numerical results show the efficiency and parallelism of the proposed fast algorithm

multiresolution algorithms for the numerical solution of hyperbolic conservation laws given any scheme in conservation form and an appropriate uniform grid for the numerical solution of the initial value problem for one dimensional hyperbolic conservation laws we describe a multiresolution algorithm that approximates this numerical solution to a prescribed tolerance in an efficient manner to do so we consider the grid averages of the numerical solution for a hierarchy of nested diadic grids in which the given grid is the finest and introduce an equivalent multiresolution representation the multiresolution representation of the numerical solution consists of its grid averages for the coarsest grid and the set of errors in predicting the grid averages of each level of resolution in this hierarchy from those of the next coarser one once the numerical solution is resolved to our satisfaction in a certain locality of some grid then the prediction errors there are small for this particular grid and all finer ones this enables us to compress data by setting to zero small components of the representation which fall below a prescribed tolerance therefore instead of computing the time evolution of the numerical solution on the given grid we compute the time evolution of its compressed multiresolution representation algorithmically this amounts to computing the numerical fluxes of the given scheme at the points of the given grid by a hierarchical algorithm which starts with the computation of these numerical fluxes at the points of the coarsest grid and then proceeds through diadic refinements to the given grid at each step of refinement we add the values of the numerical flux at the center of the coarser cells the information in the multiresolution representation of the numerical solution is used to determine whether the solution is locally well resolved when this is the case we replace the costly exact value of the numerical flux with an accurate enough approximate value which is obtained by an inexpensive interpolation from the coarser grid the computational efficiency of this multiresolution algorithm is proportional to the rate of data compression for a prescribed level of tolerance that can be achieved for the numerical solution of the given scheme

elastic full waveform inversion using the second generation wavelet and an adaptive operator length scheme abstractelastic full waveform inversion fwi can generally yield subsurface model parameters of high spatial resolution however the computational and storage burden is tremendous to mitigate this problem we developed the following three strategies first we derived a time space domain adaptive staggered grid finite difference method the size of the calculation was reduced by using smaller operator lengths for high s wave velocities second we evaluated a multiscale elastic fwi scheme based on the second generation wavelet by which scale decompositions of data and model space were implemented with the scale increasing spatial sampling intervals and time steps became larger therefore the minimum computational cost could be achieved by combining the variable operator length scheme with the multiscale scheme third to economize the memory consumption we extended the efficient boundary storage and checkpointing schemes into elastic fwi only wavefields of several parts in the boundary area betwe

interpolating wavelet collocation method of time dependent maxwell s equations characterization of electrically large optical waveguide discontinuities biorthogonal interpolating wavelets have been applied to electromagnetic field modeling through the wavelet collocation method in time domain yielding a versatile first principle algorithm for the solution of time dependent maxwell s equations with inhomogeneous media the resulting scheme maintains high accuracy while by virtue of its sub gridding capability significant reduction of the computational expenditure has been obtained the proposed method has been applied to the analysis of two dimensional dielectric waveguide discontinuities particularly for the modeling of electrically large optical waveguides where the dimension of the analyzed structure is much larger than the wavelength of the highest frequency content of the transmitted signal the proposed method has been proven to be highly efficient compared to the standard finite difference method

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

monotone convergence theorems for variational triples with applications to intermediate problems the variational eigenvalue problem for a real quadratic form b with respect to a positive definite form a on a vector space v may be represented by the triple b a v methods of intermediate problems provide approximations from below to the lower eigenvalues of b a v using monotone increasing sequences of such triples it is shown that every such approximation method is canonically equivalent to weinstein s method general convergence theorems are proved for such methods these results generalise known convergence results for increasing sequences of quadratic forms the results are applied to some specific approximation methods and are illustrated using a differential eigenvalue problem

a new finite difference diffusion scheme a new second order accurate explicit diffusion scheme is presented and discussed the scheme is derived as a weighted average of the conventional forward in time explicit diffusion scheme over one grid length and the same scheme but over two grid lengths varying the weighting factors produces a family of schemes for optimum use a new scheme with the weighting factor dependent on the viscous stability number is proposed it is slightly more computationally expensive than the conventional explicit scheme typically by 25 but is numerically stable at viscous stability numbers four times as large further it is about 20 computationally less expensive than the fully implicit scheme even in the simplest one dimensional model this three level locally implicit scheme has been implemented in both a simple one dimensional diffusion model and also in a complex three dimensional large eddy simulation model it has been found to behave well and is profitable in both models

solving hyperbolic pdes using interpolating wavelets a method is presented for adaptively solving hyperbolic pdes the method is based on an interpolating wavelet transform using polynomial interpolation on dyadic grids the adaptability is performed automatically by thresholding the wavelet coefficients operations such as differentiation and multiplication are fast and simple due to the one to one correspondence between point values and wavelet coefficients in the interpolating basis treatment of boundary conditions is simplified in this sparse point representation spr numerical examples are presented for one and two dimensional problems it is found that the proposed method outperforms a finite difference method on a uniform grid for certain problems in terms of flops

an adaptive wavelet collocation method for fluid structure interaction at high reynolds numbers two mathematical approaches are combined to calculate high rey nolds number incompressible fluid structure interaction a wavelet method to dynamically adapt the computational grid to flow intermittency and obstacle motion and brinkman penalization to enforce solid boundaries of arbitrary complexity we also implement a wavelet based multilevel solver for the poisson problem for the pressure at each time step the method is applied to two dimensional flow around fixed and moving cylinders for reynolds numbers in the range 3 times 10 1 le re le 10 5 the compression ratios of up to 1000 are achieved for the first time it is demonstrated in actual dynamic simulations that the compression scales like re 1 2 over five orders of magnitude while computational complexity scales like re this represents a significant improvement over the classical complexity estimate of re 9 4 for two dimensional turbulence

on a wavelet based method for the numerical simulation of wave propagation a wavelet based method for the numerical simulation of acoustic and elastic wave propagation is developed using a displacement velocity formulation and treating spatial derivatives with linear operators the wave equations are rewritten as a system of equations whose evolution in time is controlled by first order derivatives the linear operators for spatial derivatives are implemented in wavelet bases using an operator projection technique with nonstandard forms of wavelet transform using a semigroup approach the discretized solution in time can be represented in an explicit recursive form based on taylor expansion of exponential functions of operator matrices the boundary conditions are implemented by augmenting the system of equations with equivalent force terms at the boundaries the wavelet based method is applied to the acoustic wave equation with rigid boundary conditions at both ends in 1 d domain and to the elastic wave equation with a traction free boundary conditions at a free surface in 2 d spatial media the method can be applied directly to media with plane surfaces and surface topography can be included with the aid of distortion of the grid describing the properties of the medium the numerical results are compared with analytic solutions based on the cagniard technique and show high accuracy the wavelet based approach is also demonstrated for complex media including highly varying topography or stochastic heterogeneity with rapid variations in physical parameters these examples indicate the value of the approach as an accurate and stable tool for the simulation of wave propagation in general complex media

an adaptive meshfree diffusion wavelet method for partial differential equations on the sphere abstract an adaptive meshfree diffusion wavelet method for solving partial differential equations pdes on the sphere is developed approximation formulae for laplacian beltrami 2 and gradient operators are derived using radial basis functions rbfs and the convergence of these approximations to 2 and is verified for two test functions the matrix approximating the laplace beltrami operator is used for the construction of the diffusion wavelet the diffusion wavelet is used for the adaptation of node arrangement as well as for the fast computation of dyadic powers of the matrices involved in the numerical solution of the pde the efficiency of the method is that the same operator is used for the construction of the diffusion wavelet and for the approximation of the differential operators as a part of the wavelet method the behaviour of the compression error with respect to different parameters involved in the construction of the diffusion wavelet is tested the cpu time taken by the proposed method is compared with the cpu time taken by the rbf based collocation method and it is observed that the proposed method performs better the method is tested on three test problems namely spherical diffusion equation linear problem of computing a moving steep front nonlinear and problem of turing patterns system of nonlinear reaction diffusion equations

an adaptive wavelet method for nonlinear differential algebraic equations abstract a multi resolution collocation method with specially designed spline wavelet is presented to numerically solve a system of nonlinear differential algebraic equations of index one in circuit simulation as the wavelet collocation method is applied to such system we will have to solve large nonlinear algebraic equations for the algebraic equations we give a modified newton type iterative method which bases on two wavelet adaptive techniques we prove the convergence the stability and the complexity on the adaptive method reported here the new approach could effectively reduce computational costs and storage requirements to large extent numerical experiments are given to further illustrate the method s principle

an adaptive wavelet vaguelette algorithm for the solution of pdes the paper first describes a fast algorithm for the discrete orthonormal wavelet transform and its inverse without using the scaling function this approach permits to compute the decomposition of a function into a lacunary wavelet basis i e a basis constituted of a subset of all basis functions up to a certain scale without modification the construction is then extended to operator adapted biorthogonal wavelets this is relevant for the solution of certain nonlinear evolutionary pdes where a priori information about the significant coefficients is available we pursue the approach described in j frohlich and k schneider europ j mech b fluids13 439 1994 which is based on the explicit computation of the scalewise contributions of the approximated function to the values at points of hierarchical grids here we present an improved construction employing the cardinal function of the multiresolution the new method is applied to the helmholtz equation and illustrated by comparative numerical results it is then extended for the solution of a nonlinear parabolic pde with semi implicit discretization in time and self adaptive wavelet discretization in space results with full adaptivity of the spatial wavelet discretization are presented for a one dimensional flame front as well as for a two dimensional problem

class number parity for the p th cyclotomic field we study the parity of the class number of the pth cyclotomic field for p prime by analytic methods we derive a parity criterion in terms of polynomials over the field of 2 elements the conjecture that the class number is odd for p a prime of the form 2q 1 with q prime is proved in special cases and a heuristic argument is given in favor of the conjecture an implementation of the criterion on a computer shows that no small counterexamples to the conjecture exist

wavelet based adaptive large eddy simulation with explicit filtering stochastic coherent adaptive large eddy simulation is a novel approach to the numerical simulation of turbulence where the coherent energetic eddies are solved for while modeling the influence of the less energetic coherent incoherent background flow the formal separation between resolved and unresolved field is obtained by wavelet threshold filtering that is inherent to the adaptive wavelet collocation numerical method a new explicit wavelet filtering strategy is introduced and tested by considering two different filtering levels the physical level which controls the turbulence model and the numerical level that is responsible for the accuracy of numerical simulations the theoretical basis for wavelet based adaptive large eddy simulation with explicit filtering and consistent dynamic modeling is given numerical experiments are presented for unsteady homogeneous turbulence demonstrating the existence of grid independent solutions

improved convergence rates for intermediate problems improved convergence rate estimates are derived for a variant of aronszajn type intermediate problems that is both computationally feasible and convergent for problems with nontrivial essential spectra in a previous paper the authors obtained rate of convergence estimates for this method in terms of containment gaps between subspaces in the present work techniques for estimating relatively unbounded perturbations are refined in order to apply the kato temple inequalities this yields convergence rates for the intermediate operator eigenvalues in terms of squares of containment gaps between subspaces convergence rate estimates are also obtained for the intermediate problem eigenvectors and comparisons are made with previously known results for the method of special choice

adaptive wavelet collocation method on the shallow water model this paper presents an integrated approach for modeling several ocean test problems on adaptive grids using novel boundary techniques the adaptive wavelet collocation method solves the governing equations on temporally and spatially varying meshes which allows higher effective resolution to be obtained with less computational cost it is a general method for the solving a large class of partial differential equations but is applied to the shallow water equations here in addition to developing wavelet based computational models this work also uses an extension of the brinkman penalization method to represent irregular and non uniform continental boundaries this technique is used to enforce no slip boundary conditions through the addition of a term to the field equations when coupled with the adaptive wavelet collocation method the flow near the boundary can be well resolved it is especially useful for simulations of boundary currents and tsunamis where flow and the boundary is important thus those are the test cases presented here

a numerical strategy to discretize and solve the poisson equation on dynamically adapted multiresolution grids for time dependent streamer discharge simulations we develop a numerical strategy to solve multi dimensional poisson equations on dynamically adapted grids for evolutionary problems disclosing propagating fronts the method is an extension of the multiresolution finite volume scheme used to solve hyperbolic and parabolic time dependent pdes such an approach guarantees a numerical solution of the poisson equation within a user defined accuracy tolerance most adaptive meshing approaches in the literature solve elliptic pdes level wise and hence at uniform resolution throughout the set of adapted grids here we introduce a numerical procedure to represent the elliptic operators on the adapted grid strongly coupling inter grid relations that guarantee the conservation and accuracy properties of multiresolution finite volume schemes the discrete poisson equation is solved at once over the entire computational domain as a completely separate process the accuracy and numerical performance of the method are assessed in the context of streamer discharge simulations