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Beylkin G.,University of Colorado at Boulder | Sandberg K.,Computational Solutions LLC
Journal of Computational Physics | Year: 2014

We use generalized Gaussian quadratures for exponentials to develop a new ODE solver. Nodes and weights of these quadratures are computed for a given bandlimit c and user selected accuracy ε, so that they integrate functions e i b x, for all |b| ≤ c, with accuracy ε. Nodes of these quadratures do not concentrate excessively near the end points of an interval as those of the standard, polynomial-based Gaussian quadratures. Due to this property, the usual implicit Runge-Kutta (IRK) collocation method may be used with a large number of nodes, as long as the method chosen for solving the nonlinear system of equations converges. We show that the resulting ODE solver is symplectic and demonstrate (numerically) that it is A-stable. We use this solver, dubbed Band-limited Collocation (BLC-IRK), for orbit computations in astrodynamics. Since BLC-IRK minimizes the number of nodes needed to obtain the solution, in this problem we achieve speed close to that of the traditional explicit multistep methods. © 2014 Elsevier Inc. Source


Sandberg K.,Computational Solutions LLC | Wojciechowski K.J.,University of Colorado at Denver
Journal of Computational Physics | Year: 2011

We develop a new type of derivative matrix for pseudospectral methods. The norm of these matrices grows at the optimal rate O(N2) for N-by-N matrices, in contrast to standard pseudospectral constructions that result in O(N4) growth of the norm. The smaller norm has a big advantage when using the derivative matrix for solving time dependent problems such as wave propagation. The construction is based on representing the derivative operator as an integral kernel, and does not rely on the interpolating polynomials. In particular, we construct second derivative matrices that incorporate Dirichlet or Neumann boundary conditions on an interval and on the disk, but the method can be used to construct a wide variety of commonly used operators for solving PDEs and integral equations. The construction can be used with any quadrature, including traditional Gauss-Legendre quadratures, but we have found that by using quadratures based on prolate spheroidal wave functions, we can achieve a near optimal sampling rate close to two points per wavelength, even for non-periodic problems. We provide numerical results for the new construction and demonstrate that the construction achieves similar or better accuracy than traditional pseudospectral derivative matrices, while resulting in a norm that is orders of magnitude smaller than the standard construction. To demonstrate the advantage of the new construction, we apply the method for solving the wave equation in constant and discontinuous media and for solving PDEs on the unit disk. We also present two compression algorithms for applying the derivative matrices in O(NlogN) operations. © 2011 Elsevier Inc. Source


Sandberg K.,Computational Solutions LLC
Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) | Year: 2010

In this paper we introduce the Curve Filter Transform, a powerful tool for enhancing curve-like structures in images. The method extends earlier works on orientation fields and the Orientation Field Transform. The result is a robust method that is less sensitive to noise and produce sharper images than the Orientation Field Transform. We describe the method and demonstrate its performance on several examples where we compare the result to the Canny edge detector and the Orientation Field Transform. The examples include a tomogram from a biological cell and we also demonstrate how the method can be used to enhance handwritten text. © 2010 Springer-Verlag. Source


Bradley B.K.,University of Colorado at Boulder | Jones B.A.,University of Colorado at Boulder | Beylkin G.,University of Colorado at Boulder | Sandberg K.,Computational Solutions LLC | Axelrad P.,University of Colorado at Boulder
Celestial Mechanics and Dynamical Astronomy | Year: 2014

We describe a new method for numerical integration, dubbed bandlimited collocation implicit Runge-Kutta (BLC-IRK), and compare its efficiency in propagating orbits to existing techniques commonly used in Astrodynamics. The BLC-IRK scheme uses generalized Gaussian quadratures for bandlimited functions. This new method allows us to use significantly fewer force function evaluations than explicit Runge-Kutta schemes. In particular, we use a low-fidelity force model for most of the iterations, thus minimizing the number of high-fidelity force model evaluations. We also investigate the dense output capability of the new scheme, quantifying its accuracy for Earth orbits. We demonstrate that this numerical integration technique is faster than explicit methods of Dormand and Prince 5(4) and 8(7), Runge-Kutta-Fehlberg 7(8), and approaches the efficiency of the 8th-order Gauss-Jackson multistep method. We anticipate a significant acceleration of the scheme in a multiprocessor environment. © 2014 Springer Science+Business Media Dordrecht. Source


Wojciechowski K.J.,University of Wisconsin - Stout | Chen J.,University of Colorado at Denver | Schreyer-Bennethum L.,University of Colorado at Denver | Sandberg K.,Computational Solutions LLC
Journal of Porous Media | Year: 2014

We mathematically analyze an initial-boundary value problem that involves a nonlinear Volterra partial integrodifferential equation derived using hybrid mixture theory and used to model swelling porous materials where the application is an immersed, porous cylindrical material imbibing fluid through its exterior boundary. The model is written as an initial-boundary value problem and we establish well-posedness and numerically solve it using a novel approach to constructing pseudospectral differentiation matrices in a polar geometry. Numerical results are obtained and interpretations are provided for a small variety of diffusion and permeability coefficients and parameters to simulate the model's behavior and to demonstrate its viability as a model for swelling porous materials exhibiting viscoelastic behavior. © 2014 by Begell House, Inc. Source

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