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170
Exponential integrators
, 2010
"... In this paper we consider the construction, analysis, implementation and application of exponential integrators. The focus will be on two types of stiff problems. The first one is characterized by a Jacobian that possesses eigenvalues with large negative real parts. Parabolic partial differential eq ..."
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Cited by 67 (5 self)
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In this paper we consider the construction, analysis, implementation and application of exponential integrators. The focus will be on two types of stiff problems. The first one is characterized by a Jacobian that possesses eigenvalues with large negative real parts. Parabolic partial differential equations and their spatial discretization are typical examples. The second class consists of highly oscillatory problems with purely imaginary eigenvalues of large modulus. Apart from motivating the construction of exponential integrators for various classes of problems, our main intention in this article is to present the mathematics behind these methods. We will derive error bounds that are independent of stiffness or highest frequencies in the system. Since the implementation of exponential integrators requires the evaluation of the product of a matrix function with a vector, we will briefly discuss some possible approaches as well. The paper concludes with some applications, in
A review of geometric transformations for nonrigid body registration
 IEEE TRANSACTIONS ON MEDICAL IMAGING
, 2007
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Semiparametric Regression of MultiDimensional Genetic Pathway Data: Least Squares Kernel Machines and Linear Mixed Models
"... SUMMARY. We consider a semiparametric regression model that relates a normal outcome to covariates and a genetic pathway, where the covariate effects are modeled parametrically and the pathway effect of multiple gene expressions is modeled parametrically or nonparametrically using least squares kern ..."
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Cited by 52 (10 self)
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SUMMARY. We consider a semiparametric regression model that relates a normal outcome to covariates and a genetic pathway, where the covariate effects are modeled parametrically and the pathway effect of multiple gene expressions is modeled parametrically or nonparametrically using least squares kernel machines (LSKMs). This unified framework allows a flexible function for the joint effect of multiple genes within a pathway by specifying a kernel function and allows for the possibility that each gene expression effect might be nonlinear and the genes within the same pathway are likely to interact with each other in a complicated way. This semiparametric model also makes it possible to test for the overall genetic pathway effect. We show that the LSKM semiparametric regression can be formulated using a linear mixed model. Estimation and inference hence can proceed within the linear mixed model framework using standard mixed model software. Both the regression coefficients of the covariate effects and the LSKM estimator of the genetic pathway effect can be obtained using the Best Linear Unbiased Predictor (BLUP) in the corresponding linear mixed model formulation. The smoothing parameter and the kernel parameter can be estimated as variance components using Restricted Maximum Likelihood (REML). A score test is developed to test for the genetic pathway effect. Model/variable selection within the LSKM framework is discussed.The methods are illustrated using a prostate cancer data set and evaluated using simulations.
Taylor models and other validated functional inclusion methods
 Int. J. Pure Appl. Math
"... Abstract: A detailed comparison between Taylor model methods and other tools for validated computations is provided. Basic elements of the Taylor model (TM) methods are reviewed, beginning with the arithmetic for elementary operations and intrinsic functions. We discuss some of the fundamental prope ..."
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Cited by 37 (3 self)
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Abstract: A detailed comparison between Taylor model methods and other tools for validated computations is provided. Basic elements of the Taylor model (TM) methods are reviewed, beginning with the arithmetic for elementary operations and intrinsic functions. We discuss some of the fundamental properties, including high approximation order and the ability to control the dependency problem, and pointers to many of the more advanced TM tools are provided. Aspects of the current implementation, and in particular the issue of floating point error control, are discussed. For the purpose of providing range enclosures, we compare with modern versions of centered forms and mean value forms, as well as the direct computation of remainder bounds by highorder interval automatic differentiation and show the advantages of the TM methods. We also compare with the socalled boundary arithmetic (BA) of Lanford, Eckmann, Wittwer, Koch et al., which was developed to prove existence of fixed points in several comparatively small systems, and the ultraarithmetic (UA) developed by Kaucher, Miranker et al. which
N.: Reflectance sharing: Predicting appearance from a sparse set of images of a known shape
 Montes & Ureña. University of Granada
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Polynomials and potential theory for Gaussian radial basis function interpolation
 SIAM J. Numer. Anal
"... Abstract. We explore a connection between Gaussian radial basis functions and polynomials. Using standard tools of potential theory, we find that these radial functions are susceptible to the Runge phenomenon, not only in the limit of increasingly flat functions, but also in the finite shape paramet ..."
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Cited by 25 (8 self)
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Abstract. We explore a connection between Gaussian radial basis functions and polynomials. Using standard tools of potential theory, we find that these radial functions are susceptible to the Runge phenomenon, not only in the limit of increasingly flat functions, but also in the finite shape parameter case. We show that there exist interpolation node distributions that prevent such phenomena and allow stable approximations. Using polynomials also provides an explicit interpolation formula that avoids the difficulties of inverting interpolation matrices, while not imposing restrictions on the shape parameter or number of points. Key words. Gaussian radial basis functions, RBF, potential theory, Runge phenomenon, convergence, stability
Eulerian Gaussian beams for Schrödinger equations in the semiclassical regime
 J. Comput. Phys
, 2009
"... We propose Gaussianbeam based Eulerian methods to compute semiclassical solutions of the Schrodinger equation. Traditional Gaussian beam type methods for the Schrodinger equation are based on the Lagrangian ray tracing. We develop a new Eulerian framework which uses global Cartesian coordinates, ..."
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Cited by 24 (0 self)
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We propose Gaussianbeam based Eulerian methods to compute semiclassical solutions of the Schrodinger equation. Traditional Gaussian beam type methods for the Schrodinger equation are based on the Lagrangian ray tracing. We develop a new Eulerian framework which uses global Cartesian coordinates, levelset based implicit representation and Liouville equations. The resulting method gives uniformly distributed phases and amplitudes in phase space simultaneously. To obtain semiclassical solutions to the Schrodinger equation with dierent initial wave functions, we only need to slightly modify the summation formula. This yields a very ecient method for computing semiclassical solutions to the Schrodinger equation. For instance, the proposed algorithm requires only O(sNn2) operations to compute s dierent solutions with s dierent initial wave functions under the in
uence of the same potential, where N = O(1=~), ~ is the Planck constant, and n N is the number of computed beams. Numerical experiments indicate that this Eulerian Gaussian beam approach yields accurate semiclassical solutions even at caustics. 1
GridFree Adaptive SemiLagrangian Advection Using Radial Basis Functions
 COMPUTERS AND MATHEMATICS WITH APPLICATIONS
, 2000
"... This paper proposes a new gridfree adaptive advection scheme. The resulting algorithm is a combination of the semiLagrangian method (SLM) and the gridfree radial basis function interpolation (RBF). The set of scattered interpolation nodes is subject to dynamic changes at run time: Based on a post ..."
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Cited by 19 (5 self)
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This paper proposes a new gridfree adaptive advection scheme. The resulting algorithm is a combination of the semiLagrangian method (SLM) and the gridfree radial basis function interpolation (RBF). The set of scattered interpolation nodes is subject to dynamic changes at run time: Based on a posteriori local error estimates, a selfadaptive local refinement and coarsening of the nodes serves to obtain enhanced accuracy at reasonable computational costs. Due to wellknown features of SLM and RBF, the method is guaranteed to be stable, it has good approximation behaviour, and it works for arbitrary space dimension. Numerical examples in two dimensions illustrate the performance of the method in comparison with existing gridbased advection schemes.
Accuracy of Radial Basis Function Interpolation and Derivative Approximations on 1D Infinite Grids
 Adv. Comput. Math
, 2005
"... Radial basis function (RBF) interpolation can be very e#ective for scattered data in any number of dimensions. As one of their many applications, RBFs can provide highly accurate collocationtype numerical solutions to several classes of PDEs. To better understand the accuracy that can be obtained, ..."
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Cited by 19 (6 self)
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Radial basis function (RBF) interpolation can be very e#ective for scattered data in any number of dimensions. As one of their many applications, RBFs can provide highly accurate collocationtype numerical solutions to several classes of PDEs. To better understand the accuracy that can be obtained, we survey here derivative approximations based on RBFs using a similar Fourier analysis approach that has become the standard way for assessing the accuracy of finite di#erence schemes. We find that the accuracy is directly linked to the decay rate, at large arguments, of the (generalized) Fourier transform of the radial function. Three di#erent types of convergence rates can be distinguished as the node density increases  polynomial, spectral, and superspectral, as exemplified for example by thin plate splines, multiquadrics, and Gaussians respectively.
Smooth approximation and rendering of large scattered data sets
 In Proceedings of the conference on Visualization ’01
, 2001
"... We present an efficient method to automatically compute a smooth approximation of large functional scattered data sets given over arbitrarily shaped planar domains. Our approach is based on the construction of a ¢¤ £continuous bivariate cubic spline and our method offers optimal approximation order ..."
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Cited by 18 (6 self)
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We present an efficient method to automatically compute a smooth approximation of large functional scattered data sets given over arbitrarily shaped planar domains. Our approach is based on the construction of a ¢¤ £continuous bivariate cubic spline and our method offers optimal approximation order. Both local variation and nonuniform distribution of the data are taken into account by using local polynomial least squares approximations of varying degree. Since we only need to solve small linear systems and no triangulation of the scattered data points is required, the overall complexity of the algorithm is linear in the total number of points. Numerical examples dealing with several real world scattered data sets with up to millions of points demonstrate the efficiency of our method. The resulting spline surface is of high visual quality and can be efficiently evaluated for rendering and modeling. In our implementation we achieve realtime frame rates for typical flythrough sequences and interactive frame rates for recomputing and rendering a locally modified spline surface. CR Categories: G.1.2 [Numerical Analysis]: Approximation— approximation of surfaces, least squares approximation, spline and piecewise polynomial approximation; I.3.3 [Computer Graphics]: Picture/Image Generation—display algorithms, viewing algorithms; I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling—surface representation, splines; E.4 [Coding