Non-equispaced B-spline wavelets

This paper has three main contributions. The first is the construction of wavelet transforms from B-spline scaling functions defined on a grid of non-equispaced knots. The new construction extends the equispaced, biorthogonal, compactly supported Cohen-Daubechies-Feauveau wavelets. The new construction is based on the factorisation of wavelet transforms into lifting steps. The second and third contributions are new insights on how to use these and other wavelets in statistical applications. The second contribution is related to the bias of a wavelet representation. It is investigated how the fine scaling coefficients should be derived from the observations. In the context of equispaced data, it is common practice to simply take the observations as fine scale coefficients. It is argued in this paper that this is not acceptable for non-interpolating wavelets on non-equidistant data. Finally, the third contribution is the study of the variance in a non-orthogonal wavelet transform in a new framework, replacing the numerical condition as a measure for non-orthogonality. By controlling the variances of the reconstruction from the wavelet coefficients, the new framework allows us to design wavelet transforms on irregular point sets with a focus on their use for smoothing or other applications in statistics.Comment: 42 pages, 2 figure

Novel Approach to Real Polynomial Root-finding and Matrix Eigen-solving

Univariate polynomial root-finding is both classical and important for modern computing. Frequently one seeks just the real roots of a polynomial with real coefficients. They can be approximated at a low computational cost if the polynomial has no nonreal roots, but typically nonreal roots are much more numerous than the real ones. We dramatically accelerate the known algorithms in this case by exploiting the correlation between the computations with matrices and polynomials, extending the techniques of the matrix sign iteration, and exploiting the structure of the companion matrix of the input polynomial. We extend some of the proposed techniques to the approximation of the real eigenvalues of a real nonsymmetric matrix.Comment: 17 pages, added algorithm

New Acceleration of Nearly Optimal Univariate Polynomial Root-findERS

Univariate polynomial root-finding has been studied for four millennia and is still the subject of intensive research. Hundreds of efficient algorithms for this task have been proposed. Two of them are nearly optimal. The first one, proposed in 1995, relies on recursive factorization of a polynomial, is quite involved, and has never been implemented. The second one, proposed in 2016, relies on subdivision iterations, was implemented in 2018, and promises to be practically competitive, although user's current choice for univariate polynomial root-finding is the package MPSolve, proposed in 2000, revised in 2014, and based on Ehrlich's functional iterations. By proposing and incorporating some novel techniques we significantly accelerate both subdivision and Ehrlich's iterations. Moreover our acceleration of the known subdivision root-finders is dramatic in the case of sparse input polynomials. Our techniques can be of some independent interest for the design and analysis of polynomial root-finders.Comment: 89 pages, 5 figures, 2 table

The Prospect of Responsive Spacecraft Using Aeroassisted, Trans-Atmospheric Maneuvers

Comprised of exo- and trans-atmospheric trajectory segments, atmospheric re-entry represents a complex dynamical event which traditionally signals the mission end-of-life for low-Earth orbit spacecraft. Transcending this paradigm, atmospheric re-entry can be employed as a means of operational maneuver whereby aerodynamic forces can be exploited to create an aeroassisted maneuver. Utilizing a notional trans-atmospheric, lifting re-entry vehicle with L/D =6, the first phase of research demonstrates the terrestrial reachability potential for skip entry aeroassisted maneuvers. By overflying a geographically diverse set of ground targets, comparative analysis indicates a significant savings in delta V expenditure for skip entry compared with exo-atmospheric maneuvers. In the second phase, the Design of Experiments method of orthogonal arrays provides optimal vehicle and skip entry trajectory designs by employing main effects and Pareto front analysis. Depending on re-circularization altitude, the coupled optimal design can achieve an inclination change of 19.91° with 50-85% less delta V than a simple plane change. Finally, the third phase introduces the descent-boost aeroassisted maneuver as an alternative to combined Hohmann and bi-elliptic transfers in order to perform LEO injection. Compared with bi-elliptic transfers, simulations demonstrate that a lifting re-entry vehicle performing a descent-boost maneuver requires 6-12% less for injection into orbits lower than 650 km. In addition, the third phase also introduces the Maneuver Performance Number as a dimensionless means of comparative maneuver effectiveness analysis

Spectral properties of zero temperature dynamics in a model of a compacting granular column

The compacting of a column of grains has been studied using a one-dimensional Ising model with long range directed interactions in which down and up spins represent orientations of the grain having or not having an associated void. When the column is not shaken (zero 'temperature') the motion becomes highly constrained and under most circumstances we find that the generator of the stochastic dynamics assumes an unusual form: many eigenvalues become degenerate, but the associated multi-dimensional invariant spaces have but a single eigenvector. There is no spectral expansion and a Jordan form must be used. Many properties of the dynamics are established here analytically; some are not. General issues associated with the Jordan form are also taken up.Comment: 34 pages, 4 figures, 3 table

Computational studies of blood flow at arterial branches in relation to the localisation of atherosclerosis

Atherosclerotic lesions are non-uniformly distributed at arterial bends and branch sites, suggesting an important role for haemodynamic factors, particularly wall shear stress (WSS), in their development. Using computational flow simulations in idealised and anatomically realistic models of aortic branches, this thesis investigates the role of haemodynamics in the localisation of atherosclerosis. The pattern of atherosclerotic lesions is different between species and ages. Such differences have been most completely documented for the origins of intercostal arteries within the descending thoracic aorta. The first part of the thesis deals with the analysis of wall shear stresses and flow field near the wall in the vicinity of model intercostal branch ostia using high-order spectral/hp element methods. An idealised model of an intercostal artery emerging perpendicularly from the thoracic aorta was developed, initially, to study effects of Reynolds number and flow division under steady flow conditions. Patterns of flow and WSS were strikingly dependent on these haemodynamic parameters. Incorporation of more realistic geometrical features had only minor effects. The WSS distribution in an anatomically correct geometry of a pair of intercostal arteries resembled in character the pattern seen in the idealised geometry. Under unsteady and non-reversing flow conditions, the effect of pulsatility was small. However, significantly different patterns were generated for reversing aortic near-wall flow and reversing side branch flow. The work was extended to study the wall shear stress distribution within the aortic arch and proximal branches of mice, in comparison to that of men. Mice are increasingly used as models to study atherosclerosis and it has been shown that, in knockout mice lacking the low density lipoprotein receptor and apolipoprotein E, lesions develop in vivo at the proximal wall of the entrance to the brachiocephalic artery. Three aortic arch geometries from wild-type mice were reconstructed from MRI images using in-house and commercial software, and the WSS distribution was calculated under steady flow conditions to establish the mouse haemodynamic environment and mouse-to-mouse variability. Approximated human aortic arch geometries were further considered to enable comparison of the flow and WSS fields with that of mice. The haemodynamic environment of the aortic arch varied between the two species. The overall distribution of wall shear stress was more heterogeneous in the human aortic arch than in the mouse arch, although some features were similar. Intraspecies differences in mice were small and influenced primarily by the detailed anatomical geometry and the Reynolds number. A number of simplifications were made in the above flow analyses, and clearly, relaxing these assumptions would increase complexity. Nonetheless, this thesis demonstrates the fundamental features of flow, which underlie the disparate patterns of WSS in different species and/or ages, for simplified cases, and the results are expected to be relevant to more complex ones. Aspects of the observed WSS patterns in the simplified model of intercostal artery correlate with, and may explain, some of the lesion patterns in human, rabbit and mouse aortas. WSS distributions in the aortic arch of wild-type mice associate with lesion locations seen in diseased mice.Open acces

Toward accurate polynomial evaluation in rounded arithmetic

Given a multivariate real (or complex) polynomial $p$ and a domain $\cal D$, we would like to decide whether an algorithm exists to evaluate $p(x)$ accurately for all $x \in {\cal D}$ using rounded real (or complex) arithmetic. Here ``accurately'' means with relative error less than 1, i.e., with some correct leading digits. The answer depends on the model of rounded arithmetic: We assume that for any arithmetic operator $op(a,b)$, for example $a+b$ or $a \cdot b$, its computed value is $op(a,b) \cdot (1 + \delta)$, where $| \delta |$ is bounded by some constant $\epsilon$ where $0 < \epsilon \ll 1$, but $\delta$ is otherwise arbitrary. This model is the traditional one used to analyze the accuracy of floating point algorithms.Our ultimate goal is to establish a decision procedure that, for any $p$ and $\cal D$, either exhibits an accurate algorithm or proves that none exists. In contrast to the case where numbers are stored and manipulated as finite bit strings (e.g., as floating point numbers or rational numbers) we show that some polynomials $p$ are impossible to evaluate accurately. The existence of an accurate algorithm will depend not just on $p$ and $\cal D$, but on which arithmetic operators and which constants are are available and whether branching is permitted. Toward this goal, we present necessary conditions on $p$ for it to be accurately evaluable on open real or complex domains ${\cal D}$. We also give sufficient conditions, and describe progress toward a complete decision procedure. We do present a complete decision procedure for homogeneous polynomials $p$ with integer coefficients, {\cal D} = \C^n, and using only the arithmetic operations $+$, $-$ and $\cdot$.Comment: 54 pages, 6 figures; refereed version; to appear in Foundations of Computational Mathematics: Santander 2005, Cambridge University Press, March 200

Fraction-free algorithm for the computation of diagonal forms matrices over Ore domains using Gr{\"o}bner bases

This paper is a sequel to "Computing diagonal form and Jacobson normal form of a matrix using Groebner bases", J. of Symb. Computation, 46 (5), 2011. We present a new fraction-free algorithm for the computation of a diagonal form of a matrix over a certain non-commutative Euclidean domain over a computable field with the help of Gr\"obner bases. This algorithm is formulated in a general constructive framework of non-commutative Ore localizations of $G$-algebras (OLGAs). We split the computation of a normal form of a matrix into the diagonalization and the normalization processes. Both of them can be made fraction-free. For a matrix $M$ over an OLGA we provide a diagonalization algorithm to compute $U,V$ and $D$ with fraction-free entries such that $UMV=D$ holds and $D$ is diagonal. The fraction-free approach gives us more information on the system of linear functional equations and its solutions, than the classical setup of an operator algebra with rational functions coefficients. In particular, one can handle distributional solutions together with, say, meromorphic ones. We investigate Ore localizations of common operator algebras over $K[x]$ and use them in the unimodularity analysis of transformation matrices $U,V$. In turn, this allows to lift the isomorphism of modules over an OLGA Euclidean domain to a polynomial subring of it. We discuss the relation of this lifting with the solutions of the original system of equations. Moreover, we prove some new results concerning normal forms of matrices over non-simple domains. Our implementation in the computer algebra system {\sc Singular:Plural} follows the fraction-free strategy and shows impressive performance, compared with methods which directly use fractions. Since we experience moderate swell of coefficients and obtain simple transformation matrices, the method we propose is well suited for solving nontrivial practical problems.Comment: 25 pages, to appear in Journal of Symbolic Computatio