Showing papers on "Basis function published in 2005"
TL;DR: In this article, the concept of isogeometric analysis is proposed and the basis functions generated from NURBS (Non-Uniform Rational B-Splines) are employed to construct an exact geometric model.
5,137 citations
TL;DR: In this article, a general approach for obtaining systematic sequences of atomic single-particle basis sets for use in correlated electronic structure calculations of molecules was developed, and all the constituent functions are defined as the solutions of variational problems.
543 citations
TL;DR: A revised procedure is proposed, which uses grid-based quadrature for partitioning the contributions to the charge density from diffuse basis functions and the resulting procedure is very stable, and the calculated multipole moments converge rapidly to stable values as the size of the basis is increased.
Abstract: The distributed multipole analysis procedure, for describing a molecular charge distribution in terms of multipole moments on the individual atoms (or other sites) of the molecule, is not stable with respect to a change of basis set, and indeed, the calculated moments change substantially and unpredictably when the basis set is improved, even though the resulting electrostatic potential changes very little. A revised procedure is proposed, which uses grid-based quadrature for partitioning the contributions to the charge density from diffuse basis functions. The resulting procedure is very stable, and the calculated multipole moments converge rapidly to stable values as the size of the basis is increased.
529 citations
TL;DR: In this article, a low-complexity channel estimator for a multiuser multicarrier code division multiple access (MC-CDMA) downlink in a time-variant frequency-selective channel is proposed and analyzed.
Abstract: We propose and analyze a low-complexity channel estimator for a multiuser multicarrier code division multiple access (MC-CDMA) downlink in a time-variant frequency-selective channel. MC-CDMA is based on orthogonal frequency division multiplexing (OFDM). The time-variant channel is estimated individually for every flat-fading subcarrier, assuming small intercarrier interference. The temporal variation of every subcarrier over the duration of a data block is upper bounded by the Doppler bandwidth determined by the maximum velocity of the users. Slepian showed that time-limited snapshots of bandlimited sequences span a low-dimensional subspace. This subspace is also spanned by discrete prolate spheroidal (DPS) sequences. We expand the time-variant subcarrier coefficients in terms of orthogonal DPS sequences we call Slepian basis expansion. This enables a time-variant channel description that avoids the frequency leakage effect of the Fourier basis expansion. The square bias of the Slepian basis expansion per subcarrier is three magnitudes smaller than the square bias of the Fourier basis expansion. We show simulation results for a fully loaded MC-CDMA downlink with classic linear minimum mean square error multiuser detection. The users are moving with 19.4 m/s. Using the Slepian basis expansion channel estimator and a pilot ratio of only 2%, we achieve a bit error rate performance as with perfect channel knowledge.
501 citations
TL;DR: Convergence is proved for Discontinuous elements on tetrahedral meshes, as well as a discrete divergence preservation property, and a discrete analog of the electromagnetic energy is conserved for metallic cavities.
Abstract: A Discontinuous Galerkin method is used for to the numerical solution of the time-domain Maxwell equations on unstructured meshes. The method relies on the choice of local basis functions, a centered mean approximation for the surface integrals and a second-order leap-frog scheme for advancing in time. The method is proved to be stable for cases with either metallic or absorbing boundary conditions, for a large class of basis functions. A discrete analog of the electromagnetic energy is conserved for metallic cavities. Convergence is proved for Discontinuous elements on tetrahedral meshes, as well as a discrete divergence preservation property. Promising numerical examples with low-order elements show the potential of the method.
284 citations
TL;DR: It is shown that the exponential B- Spline framework allows an exact implementation of continuous-time signal processing operators including convolution, differential operators, and modulation, by simple processing in the discrete B-spline domain.
Abstract: Causal exponentials play a fundamental role in classical system theory. Starting from those elementary building blocks, we propose a complete and self-contained signal processing formulation of exponential splines defined on a uniform grid. We specify the corresponding B-spline basis functions and investigate their reproduction properties (Green function and exponential polynomials); we also characterize their stability (Riesz bounds). We show that the exponential B-spline framework allows an exact implementation of continuous-time signal processing operators including convolution, differential operators, and modulation, by simple processing in the discrete B-spline domain. We derive efficient filtering algorithms for multiresolution signal extrapolation and approximation, extending earlier results for polynomial splines. Finally, we present a new asymptotic error formula that predicts the magnitude and the Nth-order decay of the L/sub 2/-approximation error as a function of the knot spacing T.
252 citations
TL;DR: A multiscale finite-volume (MSFV) method for multiphase flow and transport in heterogeneous porous media and shows that at a given time step, only a small fraction of the basis functions needs to be recomputed.
Abstract: We present a multiscale finite-volume (MSFV) method for multiphase flow and transport in heterogeneous porous media. The approach extends our recently developed MSFV method for single-phase flow. We use a sequential scheme that deals with flow (i.e., pressure and total velocity) and transport (i.e., saturation) separately and differently. For the flow problem, we employ two different sets of basis functions for the reconstruction of a conservative fine-scale total velocity field. Our basis functions are designed to have local support, and that allows for adaptive computation of the flow field. We use a criterion based on the time change of the total mobility field to decide when and where to recompute our basis functions. We show that at a given time step, only a small fraction of the basis functions needs to be recomputed. Numerical experiments of difficult two-dimensional and three-dimensional test cases demonstrate the accuracy, computational efficiency, and overall scalability of the method.
242 citations
TL;DR: In this paper, the behavior of the interpolants in the limit of nearly flat radial basis functions is studied both theoretically and numerically, and explicit criteria for different types of limits are given.
Abstract: Multivariate interpolation of smooth data using smooth radial basis functions is considered. The behavior of the interpolants in the limit of nearly flat radial basis functions is studied both theoretically and numerically. Explicit criteria for different types of limits are given. Using the results for the limits, the dependence of the error on the shape parameter of the radial basis function is investigated. The mechanisms that determine the optimal shape parameter value are studied and explained through approximate expansions of the interpolation error.
217 citations
TL;DR: This paper describes two different ways to numerically calculate the modes, shows how symmetry considerations can be exploited to simplify and understand them, and describes a generalization of the procedure involving projection onto uncoupled modes that allow streamwise and cross-stream components to evolve independently.
Abstract: The proper orthogonal decomposition identifies basis functions or modes which optimally capture the average energy content from numerical or experimental data. By projecting the Navier–Stokes equations onto these modes and truncating, one can obtain low-dimensional ordinary differential equation models for fluid flows. In this paper we present a tutorial on the construction of such models. In addition to providing a general overview of the procedure, we describe two different ways to numerically calculate the modes, show how symmetry considerations can be exploited to simplify and understand them, comment on how parameter variations are captured naturally in such models, and describe a generalization of the procedure involving projection onto uncoupled modes that allow streamwise and cross-stream components to evolve independently. We illustrate for the example of plane Couette flow in a minimal flow unit – a domain whose spanwise and streamwise extent is just sufficient to maintain turbulence.
212 citations
TL;DR: Two- and three-body approximations based on interfragment distances were developed and the values appropriate for their accurate application carefully determined, and the computational scaling of the two-body FMO-CC method was demonstrated to be nearly linear.
Abstract: The fragment molecular-orbital (FMO) method was combined with the single-reference coupled-cluster (CC) theory. The developed method (FMO-CC) was applied at the CCSD and CCSD(T) levels of theory, for the cc-pVnZ family of basis sets (n=D,T,Q) to water clusters and glycine oligomers (up to 32 molecules/residues using as large basis sets as possible for the given system). The two- and three-body FMO-CC results are discussed at length, with emphasis on the basis-set dependence and three-body effects. Two- and three-body approximations based on interfragment distances were developed and the values appropriate for their accurate application carefully determined. The error in recovering the correlation energy was several millihartree for the two-body FMO-CC method and in the submillihartree range for the three-body FMO-CC method. In the largest calculations, we were able to perform the CCSD(T) calculations of (H2O)32 with the cc-pVQZ basis set (3680 basis functions) and (GLY)32 with the cc-VDZ basis set (712 co...
209 citations
TL;DR: This paper examines methods for adapting the basis function during the learning process in the context of evaluating the value function under a fixed control policy using the Bellman approximation error as an optimization criterion.
Abstract: Reinforcement Learning (RL) is an approach for solving complex multi-stage decision problems that fall under the general framework of Markov Decision Problems (MDPs), with possibly unknown parameters. Function approximation is essential for problems with a large state space, as it facilitates compact representation and enables generalization. Linear approximation architectures (where the adjustable parameters are the weights of pre-fixed basis functions) have recently gained prominence due to efficient algorithms and convergence guarantees. Nonetheless, an appropriate choice of basis function is important for the success of the algorithm. In the present paper we examine methods for adapting the basis function during the learning process in the context of evaluating the value function under a fixed control policy. Using the Bellman approximation error as an optimization criterion, we optimize the weights of the basis function while simultaneously adapting the (non-linear) basis function parameters. We present two algorithms for this problem. The first uses a gradient-based approach and the second applies the Cross Entropy method. The performance of the proposed algorithms is evaluated and compared in simulations.
TL;DR: Techniques which allow us to use triharmonic radial basis functions for real-time freeform shape editing are presented and an incremental least-squares method enables us to approximately solve the involved linear systems in a robust and efficient manner.
Abstract: Current surface-based methods for interactive freeform editing of high resolution 3D models are very powerful, but at the same time require a certain minimum tessellation or sampling quality in order to guarantee sufficient robustness. In contrast to this, space deformation techniques do not depend on the underlying surface representation and hence are affected neither by its complexity nor by its quality aspects. However, while analogously to surfacebased methods high quality deformations can be derived from variational optimization, the major drawback lies in the computation and evaluation, which is considerably more expensive for volumetric space deformations. In this paper we present techniques which allow us to use triharmonic radial basis functions for real-time freeform shape editing. An incremental least-squares method enables us to approximately solve the involved linear systems in a robust and efficient manner and by precomputing a special set of deformation basis functions we are able to significantly reduce the per-frame costs. Moreover, evaluating these linear basis functions on the GPU finally allows us to deform highly complex polygon meshes or point-based models at a rate of 30M vertices or 13M splats per second, respectively.
TL;DR: An adaptive algorithm is used to choose the location of the collocation points of the radial basis function methods and produces results similar to the more well-known and analyzed spectral methods, but while allowing greater flexibility in the choice of grid point locations.
TL;DR: In this paper, the authors show that the low-frequency breakdown associated to the MoM solution of scattering by dielectric objects can be avoided by the classical Muller formulation without the loop-tree or loop-star basis functions.
Abstract: Numerical solution of electromagnetic scattering by homogeneous dielectric objects with the method of moments (MoM) and Rao-Wilton-Glisson (RWG) basis functions is discussed. It is shown that the low-frequency breakdown associated to the MoM solution of scattering by dielectric objects can be avoided by the classical Muller formulation without the loop-tree or loop-star basis functions. Two variations of the Muller formulation, T-Mu/spl uml/ller and N-Muller, are considered. It is demonstrated that only the N-Muller formulation with the Galerkin method and RWG functions gives stable solution. Discretization of the N-Muller formulation leads to a well-conditioned matrix equation and rapidly converging iterative solutions on a wide frequency range from very low frequencies to microwave frequencies. At zero frequency, the N-Muller formulation decouples into the electrostatic and magnetostatic integral equations.
TL;DR: In this article, the surface integral equations for solving electromagnetic scattering by dielectric and composite metallic and dielectoric objects with iterative methods are studied and four types of formulations are considered: T formulations, N formulations, the combined field integral equation formulation, and the Muller formulation.
Abstract: [1] In this paper, formulation of the surface integral equations for solving electromagnetic scattering by dielectric and composite metallic and dielectric objects with iterative methods is studied Four types of formulations are considered: T formulations, N formulations, the combined field integral equation formulation, and the Muller formulation By studying properties of the integral equations and their testing in the Galerkin method, “optimal” forms for each formulation type are derived Numerical examples demonstrate that the developed new formulations lead to clear improvements in the convergence rates when the matrix equations are solved iteratively with the generalized minimal residual method Both the Rao-Wilton-Glisson and Trintinalia-Ling (TL) basis functions are used in expanding the unknown electric and magnetic surface current densities In particular, the first-order TL basis functions are required in the N formulations to maintain the solution accuracy when the surfaces include sharp edges
01 Jul 2005
TL;DR: This paper describes the construction of second generation bandelet bases and their application to 3D geometry compression and has an error decay that is asymptotically optimal for geometrically regular surfaces.
Abstract: This paper describes the construction of second generation bandelet bases and their application to 3D geometry compression. This new coding scheme is orthogonal and the corresponding basis functions are regular. In our method, surfaces are decomposed in a bandelet basis with a fast bandeletization algorithm that removes the geometric redundancy of orthogonal wavelet coefficients. The resulting transform coding scheme has an error decay that is asymptotically optimal for geometrically regular surfaces. We then use these bandelet bases to perform geometry image and normal map compression. Numerical tests show that for complex surfaces bandelets bring an improvement of 1.5dB to 2dB over state of the art compression schemes.
01 Jul 2005
TL;DR: This work applies zonal harmonics (ZH) which approximate spherical functions as sums of circularly symmetric Legendre polynomials around different axes to PRT, and shows how shading with ZH transfer can be significantly accelerated by specializing to a given lighting environment.
Abstract: Precomputed radiance transfer (PRT) captures realistic lighting effects from distant, low-frequency environmental lighting but has been limited to static models or precomputed sequences. We focus on PRT for local effects such as bumps, wrinkles, or other detailed features, but extend it to arbitrarily deformable models. Our approach applies zonal harmonics (ZH) which approximate spherical functions as sums of circularly symmetric Legendre polynomials around different axes. By spatially varying both the axes and coefficients of these basis functions, we can fit to spatially varying transfer signals. Compared to the spherical harmonic (SH) basis, the ZH basis yields a more compact approximation. More important, it can be trivially rotated whereas SH rotation is expensive and unsuited for dense per-vertex or per-pixel evaluation. This property allows, for the first time, PRT to be mapped onto deforming models which re-orient the local coordinate frame. We generate ZH transfer models by fitting to PRT signals simulated on meshes or simple parametric models for thin membranes and wrinkles. We show how shading with ZH transfer can be significantly accelerated by specializing to a given lighting environment. Finally, we demonstrate real-time rendering results with soft shadows, inter-reflections, and subsurface scatter on deforming models.
16 May 2005
TL;DR: This paper presents a novel fast integral equation method, termed IE-FFT, for solving large electromagnetic scattering problems, which utilizes the Toeplitz property of the coefficient matrix and is therefore applicable to both static and wave propagation problems.
Abstract: This paper presents a novel fast integral equation method, termed IE-FFT, for solving large electromagnetic scattering problems Similar to other fast integral equation methods, the IE-FFT algorithm starts by partitioning the basis functions into multilevel clustering groups Subsequently, the entire impedance matrix is decomposed into two parts: one for the self and/or near field couplings, and one for well-separated group couplings The IE-FFT algorithm employs two discretizations one is for the unknown current on an unstructured triangular mesh, and the other is a uniform Cartesian grid for interpolating the Green's function By interpolating the Green's function on a regular Cartesian grid, the couplings between two well-separated groups can be computed using the fast Fourier transform (FFT) Consequently, the IE-FFT algorithm does not require the knowledge of addition theorem It simply utilizes the Toeplitz property of the coefficient matrix and is therefore applicable to both static and wave propagation problems
07 Aug 2005
TL;DR: This paper presents a novel framework called proto-reinforcement learning (PRL), based on a mathematical model of a proto-value function: these are task-independent basis functions that form the building blocks of all value functions on a given state space manifold.
Abstract: This paper presents a novel framework called proto-reinforcement learning (PRL), based on a mathematical model of a proto-value function: these are task-independent basis functions that form the building blocks of all value functions on a given state space manifold. Proto-value functions are learned not from rewards, but instead from analyzing the topology of the state space. Formally, proto-value functions are Fourier eigenfunctions of the Laplace-Beltrami diffusion operator on the state space manifold. Proto-value functions facilitate structural decomposition of large state spaces, and form geodesically smooth orthonormal basis functions for approximating any value function. The theoretical basis for proto-value functions combines insights from spectral graph theory, harmonic analysis, and Riemannian manifolds. Proto-value functions enable a novel generation of algorithms called representation policy iteration, unifying the learning of representation and behavior.
TL;DR: A simple preconditioning scheme that is based upon constructing least-squares approximate cardinal basis functions (ACBFs) from linear combinations of the RBF-PDE matrix elements that transforms a badly conditioned linear system into one that is very well conditioned, allowing us to solve for the expansion coefficients iteratively so the authors can reconstruct the unknown solution everywhere on the domain.
Abstract: Although meshless radial basis function (RBF) methods applied to partial differential equations (PDEs) are not only simple to implement and enjoy exponential convergence rates as compared to standard mesh-based schemes, the system of equations required to find the expansion coefficients are typically badly conditioned and expensive using the global Gaussian elimination (G-GE) method requiring \(\mathcal{O}(N^{3})\) flops We present a simple preconditioning scheme that is based upon constructing least-squares approximate cardinal basis functions (ACBFs) from linear combinations of the RBF-PDE matrix elements The ACBFs transforms a badly conditioned linear system into one that is very well conditioned, allowing us to solve for the expansion coefficients iteratively so we can reconstruct the unknown solution everywhere on the domain Our preconditioner requires \(\mathcal{O}(mN^{2})\) flops to set up, and \(\mathcal{O}(mN)\) storage locations where m is a user define parameter of order of 10 For the 2D MQ-RBF with the shape parameter \(c\sim1/\sqrt{N}\) , the number of iterations required for convergence is of order of 10 for large values of N, making this a very attractive approach computationally As the shape parameter increases, our preconditioner will eventually be affected by the ill conditioning and round-off errors, and thus becomes less effective We tested our preconditioners on increasingly larger c and N A more stable construction scheme is available with a higher set up cost
TL;DR: A procedural object distribution function, a new texture basis function that distributes Procedurally generated objects over a procedurally generated texture, and a new texturing primitive that extends the range of textures that can be generated procedurally.
Abstract: In this article, we present a procedural object distribution function, a new texture basis function that distributes procedurally generated objects over a procedurally generated texture. The objects are distributed uniformly over the texture, and are guaranteed not to overlap. The scale, size, and orientation of the objects can be easily manipulated. The texture basis function is efficient to evaluate, and is suited for real-time applications. The new texturing primitive we present extends the range of textures that can be generated procedurally.The procedural object distribution function we propose is based on Poisson disk tiles and a direct stochastic tiling algorithm for Wang tiles. Poisson disk tiles are square tiles filled with a precomputed set of Poisson disk distributed points, inspired by Wang tiles. A single set of Poisson disk tiles enables the real-time generation of an infinite amount of Poisson disk distributions of arbitrary size. With the direct stochastic tiling algorithm, these Poisson disk distributions can be evaluated locally, at any position in the Euclidean plane.Poisson disk tiles and the direct stochastic tiling algorithm have many other applications in computer graphics. We briefly explore applications in object distribution, primitive distribution for illustration, and environment map sampling.
TL;DR: In this article, an adaptive sliding controller is proposed for controlling a non-autonomous quarter-car suspension system with time-varying loadings, where the bound of the car-body loading is assumed to be available.
Proceedings Article•
05 Dec 2005TL;DR: Two novel approaches to value function approximation are explored based on automatically constructing basis functions on state spaces that can be represented as graphs or manifolds, one of which uses the eigenfunctions of the Laplacian, in effect performing a global Fourier analysis on the graph.
Abstract: We investigate the problem of automatically constructing efficient representations or basis functions for approximating value functions based on analyzing the structure and topology of the state space. In particular, two novel approaches to value function approximation are explored based on automatically constructing basis functions on state spaces that can be represented as graphs or manifolds: one approach uses the eigenfunctions of the Laplacian, in effect performing a global Fourier analysis on the graph; the second approach is based on diffusion wavelets, which generalize classical wavelets to graphs using multiscale dilations induced by powers of a diffusion operator or random walk on the graph. Together, these approaches form the foundation of a new generation of methods for solving large Markov decision processes, in which the underlying representation and policies are simultaneously learned.
TL;DR: This work presents a Galerkin-free, proper orthogonal decomposition (POD)-assisted computational methodology for numerical simulations of the long-term dynamics of the incompressible Navier-Stokes equations, and demonstrates that it can successfully resolve complex flow dynamics at a reduced computational cost and capture theLong-term asymptotic state of the flow in cases where traditional Galerkins-POD models fail.
TL;DR: In this article, the authors studied the progressive iteration approximation property of a curve (tensor product surface) generated by blending a given data point set and a set of basis functions, and they showed that the curve has the same property as the B-spline and NURBS curve.
Abstract: In this paper, we study the progressive iteration approximation property of a curve (tensor product surface) generated by blending a given data point set and a set of basis functions. The curve (tensor product surface) has the progressive iteration approximation property as long as the basis is totally positive and the corresponding collocation matrix is nonsingular. Thus, the B-spline and NURBS curve (surface) have the progressive iteration approximation property, and Bezier curve (surface) also has the property if the corresponding collocation matrix is nonsingular.
01 May 2005-Graphical Models \/graphical Models and Image Processing \/computer Vision, Graphics, and Image Processing
TL;DR: This paper proposes a hierarchical approach to 3D scattered data interpolation and approximation with compactly supported radial basis functions that integrates the best aspects of scattered data fitting with locally and globally supported basis functions.
Abstract: In this paper, we propose a hierarchical approach to 3D scattered data interpolation and approximation with compactly supported radial basis functions. Our numerical experiments suggest that the approach integrates the best aspects of scattered data fitting with locally and globally supported basis functions. Employing locally supported functions leads to an efficient computational procedure, while a coarse-to-fine hierarchy makes our method insensitive to the density of scattered data and allows us to restore large parts of missed data. Given a point cloud distributed over a surface, we first use spatial down sampling to construct a coarse-to-fine hierarchy of point sets. Then we interpolate (approximate) the sets starting from the coarsest level. We interpolate (approximate) a point set of the hierarchy, as an offsetting of the interpolating function computed at the previous level. The resulting fitting procedure is fast, memory efficient, and easy to implement.
TL;DR: The modal basis is the real Fourier basis, which allows direct control of specific regions of the point-spread function and is computationally efficient enough to be implemented with off-the-shelf technology for a 2.5 kHz, 64 x 64 adaptive optics system.
Abstract: Optimal modal Fourier-transform wavefront control combines the speed of Fourier-transform reconstruction (FTR) with real-time optimization of modal gains to form a fast, adaptive wavefront control scheme. Our modal basis is the real Fourier basis, which allows direct control of specific regions of the point-spread function. We formulate FTR as modal control and show how to measure custom filters. Because the Fourier basis is a tight frame, we can use it on a circular aperture for modal control even though it is not an orthonormal basis. The modal coefficients are available during reconstruction, greatly reducing computational overhead for gain optimization. Simulation results show significant improvements in performance in low-signal-to-noise-ratio situations compared with nonadaptive control. This scheme is computationally efficient enough to be implemented with off-the-shelf technology for a 2.5 kHz, 64×64 adaptive optics system.
TL;DR: In this paper, the DSC-Ritz method is proposed for vibration analysis of Mindlin plates, where two basis functions are constructed by using DSC delta sequence kernels of the positive type.
Abstract: This paper introduces a novel method for the free vibration analysis of Mindlin plates. The proposed method takes the advantage of both the local bases of the discrete singular convolution (DSC) algorithm and the pb-2 Ritz boundary functions to arrive at a new approach, called DSC-Ritz method. Two basis functions are constructed by using DSC delta sequence kernels of the positive type. The energy functional of the Mindlin plate is represented by the newly constructed basis functions and is minimized under the Ritz variational principle. Extensive numerical experiments are considered by different combinations of boundary conditions of Mindlin plates of rectangular and triangular shapes. The performance of the proposed method is carefully validated by convergence analysis. The frequency parameters agree very well with those in the literature. Numerical experiments indicate that the proposed DSC-Ritz method is a very promising new method for vibration analysis of Mindlin plates. Copyright © 2004 John Wiley & Sons, Ltd.
TL;DR: The ACBF preconditioners previously used for interpolation problems are reviewed and a class of preconditionsers based on the one proposed in [Ling L, Kansa EJ] when a cardinality condition is enforced on different subsets are investigated.
Abstract: The approximate cardinal basis function (ACBF) preconditioning technique has been used to solve partial differential equations (PDEs) with radial basis functions (RBFs). In [Ling L, Kansa EJ. A least-squares preconditioner for radial basis functions collocation methods. Adv Comput Math; in press], a preconditioning scheme that is based upon constructing the least-squares approximate cardinal basis function from linear combinations of the RBF–PDE matrix elements has shown very attractive numerical results. This preconditioning technique is sufficiently general that it can be easily applied to many differential operators. In this paper, we review the ACBF preconditioning techniques previously used for interpolation problems and investigate a class of preconditioners based on the one proposed in [Ling L, Kansa EJ. A least-squares preconditioner for radial basis functions collocation methods. Adv Comput Math; in press] when a cardinality condition is enforced on different subsets. We numerically compare the ACBF preconditioners on several numerical examples of Poisson's, modified Helmholtz and Helmholtz equations, as well as a diffusion equation and discuss their performance.
TL;DR: The proposed basis is organized in hierarchical levels, and keeps the different scales of the problem directly into the basis functions representation; the current is divided into a solenoidal and a quasi-irrotational part, which allows mapping these two vector parts onto fully scalar quantities, where the wavelets are defined.
Abstract: This paper presents the construction, use, and properties of a multiresolution (wavelet) basis for the method of moments (MoM) analysis of metal antennas, scatterers, and microwave circuits discretized by triangular meshes. Several application examples show fast convergence of iterative solvers and accurate solutions with highly sparse MoM matrices. The proposed basis is organized in hierarchical levels, and keeps the different scales of the problem directly into the basis functions representation; the current is divided into a solenoidal and a quasi-irrotational part, which allows mapping these two vector parts onto fully scalar quantities, where the wavelets are defined. As a byproduct, this paper also presents a way to construct hierarchical sets of Rao-Wilton-Glisson (RWG) functions on a family of meshes obtained by subsequent refinement, i.e., with the RWG of coarser meshes expressed as linear combinations of those of finer meshes.