Numerical computation of the conformal map onto lemniscatic domains

We present a numerical method for the computation of the conformal map from unbounded multiply-connected domains onto lemniscatic domains. For $\ell$-times connected domains the method requires solving $\ell$ boundary integral equations with the Neumann kernel. This can be done in $O(\ell^2 n \log n)$ operations, where $n$ is the number of nodes in the discretization of each boundary component of the multiply connected domain. As demonstrated by numerical examples, the method works for domains with close-to-touching boundaries, non-convex boundaries, piecewise smooth boundaries, and for domains of high connectivity.Comment: Minor revision; simplified Example 6.1, and changed Example 6.2 to a set without symmetr

A Numerical Slow Manifold Approach to Model Reduction for Optimal Control of Multiple Time Scale ODE

Time scale separation is a natural property of many control systems that can be ex- ploited, theoretically and numerically. We present a numerical scheme to solve optimal control problems with considerable time scale separation that is based on a model reduction approach that does not need the system to be explicitly stated in singularly perturbed form. We present examples that highlight the advantages and disadvantages of the method

Hierarchical Bin Buffering: Online Local Moments for Dynamic External Memory Arrays

Local moments are used for local regression, to compute statistical measures such as sums, averages, and standard deviations, and to approximate probability distributions. We consider the case where the data source is a very large I/O array of size n and we want to compute the first N local moments, for some constant N. Without precomputation, this requires O(n) time. We develop a sequence of algorithms of increasing sophistication that use precomputation and additional buffer space to speed up queries. The simpler algorithms partition the I/O array into consecutive ranges called bins, and they are applicable not only to local-moment queries, but also to algebraic queries (MAX, AVERAGE, SUM, etc.). With N buffers of size sqrt{n}, time complexity drops to O(sqrt n). A more sophisticated approach uses hierarchical buffering and has a logarithmic time complexity (O(b log_b n)), when using N hierarchical buffers of size n/b. Using Overlapped Bin Buffering, we show that only a single buffer is needed, as with wavelet-based algorithms, but using much less storage. Applications exist in multidimensional and statistical databases over massive data sets, interactive image processing, and visualization

A multi-level preconditioned Krylov method for the efficient solution of algebraic tomographic reconstruction problems

Classical iterative methods for tomographic reconstruction include the class of Algebraic Reconstruction Techniques (ART). Convergence of these stationary linear iterative methods is however notably slow. In this paper we propose the use of Krylov solvers for tomographic linear inversion problems. These advanced iterative methods feature fast convergence at the expense of a higher computational cost per iteration, causing them to be generally uncompetitive without the inclusion of a suitable preconditioner. Combining elements from standard multigrid (MG) solvers and the theory of wavelets, a novel wavelet-based multi-level (WMG) preconditioner is introduced, which is shown to significantly speed-up Krylov convergence. The performance of the WMG-preconditioned Krylov method is analyzed through a spectral analysis, and the approach is compared to existing methods like the classical Simultaneous Iterative Reconstruction Technique (SIRT) and unpreconditioned Krylov methods on a 2D tomographic benchmark problem. Numerical experiments are promising, showing the method to be competitive with the classical Algebraic Reconstruction Techniques in terms of convergence speed and overall performance (CPU time) as well as precision of the reconstruction.Comment: Journal of Computational and Applied Mathematics (2014), 26 pages, 13 figures, 3 table

NumGfun: a Package for Numerical and Analytic Computation with D-finite Functions

This article describes the implementation in the software package NumGfun of classical algorithms that operate on solutions of linear differential equations or recurrence relations with polynomial coefficients, including what seems to be the first general implementation of the fast high-precision numerical evaluation algorithms of Chudnovsky & Chudnovsky. In some cases, our descriptions contain improvements over existing algorithms. We also provide references to relevant ideas not currently used in NumGfun

Approximate computations with modular curves

This article gives an introduction for mathematicians interested in numerical computations in algebraic geometry and number theory to some recent progress in algorithmic number theory, emphasising the key role of approximate computations with modular curves and their Jacobians. These approximations are done in polynomial time in the dimension and the required number of significant digits. We explain the main ideas of how the approximations are done, illustrating them with examples, and we sketch some applications in number theory

Fast multi-dimensional scattered data approximation with Neumann boundary conditions

An important problem in applications is the approximation of a function $f$ from a finite set of randomly scattered data $f(x_j)$. A common and powerful approach is to construct a trigonometric least squares approximation based on the set of exponentials $\{e^{2\pi i kx}\}$. This leads to fast numerical algorithms, but suffers from disturbing boundary effects due to the underlying periodicity assumption on the data, an assumption that is rarely satisfied in practice. To overcome this drawback we impose Neumann boundary conditions on the data. This implies the use of cosine polynomials $\cos (\pi kx)$ as basis functions. We show that scattered data approximation using cosine polynomials leads to a least squares problem involving certain Toeplitz+Hankel matrices. We derive estimates on the condition number of these matrices. Unlike other Toeplitz+Hankel matrices, the Toeplitz+Hankel matrices arising in our context cannot be diagonalized by the discrete cosine transform, but they still allow a fast matrix-vector multiplication via DCT which gives rise to fast conjugate gradient type algorithms. We show how the results can be generalized to higher dimensions. Finally we demonstrate the performance of the proposed method by applying it to a two-dimensional geophysical scattered data problem