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14
Liegroup methods
 ACTA NUMERICA
, 2000
"... Many differential equations of practical interest evolve on Lie groups or on manifolds acted upon by Lie groups. The retention of Liegroup structure under discretization is often vital in the recovery of qualitatively correct geometry and dynamics and in the minimization of numerical error. Having ..."
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Cited by 96 (18 self)
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Many differential equations of practical interest evolve on Lie groups or on manifolds acted upon by Lie groups. The retention of Liegroup structure under discretization is often vital in the recovery of qualitatively correct geometry and dynamics and in the minimization of numerical error. Having introduced requisite elements of differential geometry, this paper surveys the novel theory of numerical integrators that respect Liegroup structure, highlighting theory, algorithmic issues and a number of applications.
Efficient Quadrature of Highly Oscillatory Integrals Using Derivatives
, 2004
"... In this paper we explore quadrature methods for highly oscillatory integrals. Generalizing the method of stationary phase, we expand such integrals into asymptotic series in inverse powers of the frequency. The outcome are two families of methods, one based on a truncation of the asymptotic series a ..."
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Cited by 25 (5 self)
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In this paper we explore quadrature methods for highly oscillatory integrals. Generalizing the method of stationary phase, we expand such integrals into asymptotic series in inverse powers of the frequency. The outcome are two families of methods, one based on a truncation of the asymptotic series and the other extending an approach implicit in the work of Filon. Both kinds of methods approximate the integral as a linear combination of function values and derivatives, with coefficients that may depend on frequency. We determine asymptotic properties of these methods, proving that, perhaps counterintuitively, their performance drastically improves as frequency grows. The paper is accompanied by numerical results that demonstrate the potential of this set of ideas.
On the numerical quadrature of highlyoscillating integrals I: Fourier transforms
 IMA J. Numer. Anal
"... Highlyoscillatory integrals are allegedly difficult to calculate. The main assertion of this paper is that that impression is incorrect. As long as appropriate quadrature methods are used, their accuracy increases when oscillation becomes faster and suitable choice of quadrature points renders this ..."
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Cited by 22 (4 self)
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Highlyoscillatory integrals are allegedly difficult to calculate. The main assertion of this paper is that that impression is incorrect. As long as appropriate quadrature methods are used, their accuracy increases when oscillation becomes faster and suitable choice of quadrature points renders this welcome phenomenon more pronounced. We focus our analysis on Filontype quadrature and analyse its behaviour in a range of frequency regimes for integrals of the form � h 0 f (x)e iωx w(x)dx, where h> 0issmall and ω  large. Our analysis is applied to modified Magnus methods for highlyoscillatory ordinary differential equations. Keywords: quadrature; high oscillation; Liegroup methods. 1.
On the Global Error of Discretization Methods for HighlyOscillatory Ordinary Differential Equations
, 2000
"... Commencing from a globalerror formula, originally due to Henrici, we investigate the accumulation of global error in the numerical solution of linear highlyoscillating systems of the form y 00 + g(t)y = 0, where g(t) t!1 \Gamma! 1. Using WKB analysis we derive an explicit form of the globalerror ..."
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Cited by 21 (5 self)
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Commencing from a globalerror formula, originally due to Henrici, we investigate the accumulation of global error in the numerical solution of linear highlyoscillating systems of the form y 00 + g(t)y = 0, where g(t) t!1 \Gamma! 1. Using WKB analysis we derive an explicit form of the globalerror envelope for RungeKutta and Magnus methods. Our results are closely matched by numerical experiments. Motivated by the superior performance of Liegroup methods, we present a modification of the Magnus expansion which displays even better longterm behaviour in the presence of oscillations.
Avoiding the curse of dimensionality in dynamic stochastic games
, 2008
"... Discretetime stochastic games with a finite number of states have been widely applied to study the strategic interactions among forwardlooking players in dynamic environments. However, these games suffer from a “curse of dimensionality” since the cost of computing players’ expectations over all po ..."
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Cited by 19 (3 self)
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Discretetime stochastic games with a finite number of states have been widely applied to study the strategic interactions among forwardlooking players in dynamic environments. However, these games suffer from a “curse of dimensionality” since the cost of computing players’ expectations over all possible future states increases exponentially in the number of state variables. We explore the alternative of continuoustime stochastic games with a finite number of states and show that continuous time has substantial advantages. Most important, continuous time avoids the curse of dimensionality, thereby speeding up the computations by orders of magnitude in games with more than a few state variables. This much smaller computational burden greatly extends the range and richness of applications of stochastic games.
ON THE NUMERICAL EVALUATION OF FREDHOLM DETERMINANTS
, 804
"... Abstract. Some significant quantities in mathematics and physics are most naturally expressed as the Fredholm determinant of an integral operator, most notably many of the distribution functions in random matrix theory. Though their numerical values are of interest, there is no systematic numerical ..."
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Cited by 10 (5 self)
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Abstract. Some significant quantities in mathematics and physics are most naturally expressed as the Fredholm determinant of an integral operator, most notably many of the distribution functions in random matrix theory. Though their numerical values are of interest, there is no systematic numerical treatment of Fredholm determinants to be found in the literature. Instead, the few numerical evaluations that are available rely on eigenfunction expansions of the operator, if expressible in terms of special functions, or on alternative, numerically more straightforwardly accessible analytic expressions, e.g., in terms of Painlevé transcendents, that have masterfully been derived in some cases. In this paper we close the gap in the literature by studying projection methods and, above all, a simple, easily implementable, general method for the numerical evaluation of Fredholm determinants that is derived from the classical Nyström method for the solution of Fredholm equations of the second kind. Using Gauss–Legendre or Clenshaw– Curtis as the underlying quadrature rule, we prove that the approximation error essentially behaves like the quadrature error for the sections of the kernel. In particular, we get exponential convergence for analytic kernels, which are typical in random matrix theory. The application of the method to the distribution functions of the Gaussian unitary ensemble (GUE), in the bulk and the edge scaling limit, is discussed in detail. After extending the method to systems of integral operators, we evaluate the twopoint correlation functions of the more recently studied Airy and Airy 1 processes. Key words. Fredholm determinant, Nyström’s method, projection method, trace class operators, random
On the Numerical Evaluation of Distributions in Random Matrix Theory: A Review
, 2010
"... Abstract. In this paper we review and compare the numerical evaluation of those probability distributions in random matrix theory that are analytically represented in terms of Painlevé transcendents or Fredholm determinants. Concrete examples for the Gaussian and Laguerre (Wishart) βensembles and t ..."
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Cited by 6 (0 self)
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Abstract. In this paper we review and compare the numerical evaluation of those probability distributions in random matrix theory that are analytically represented in terms of Painlevé transcendents or Fredholm determinants. Concrete examples for the Gaussian and Laguerre (Wishart) βensembles and their various scaling limits are discussed. We argue that the numerical approximation of Fredholm determinants is the conceptually more simple and efficient of the two approaches, easily generalized to the computation of joint probabilities and correlations. Having the means for extensive numerical explorations at hand, we discovered new and surprising determinantal formulae for the kth largest (or smallest) level in the edge scaling limits of the Orthogonal and Symplectic Ensembles; formulae that in turn led to improved numerical evaluations. The paper comes with a toolbox of Matlab functions that facilitates further mathematical experiments by the reader.
Superconvergence of a Chebyshev spectral collocation method
 J. Sci. Comput
, 2008
"... Abstract We reveal the relationship between a Petrov–Galerkin method and a spectral collocation method at the Chebyshev points of the second kind (±1 and zeros of Uk) forthe twopoint boundary value problem. Derivative superconvergence points are identified as the Chebyshev points of the first kind ..."
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Cited by 2 (2 self)
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Abstract We reveal the relationship between a Petrov–Galerkin method and a spectral collocation method at the Chebyshev points of the second kind (±1 and zeros of Uk) forthe twopoint boundary value problem. Derivative superconvergence points are identified as the Chebyshev points of the first kind (Zeros of Tk). Supergeometric convergent rate is established for a special class of solutions.
From high oscillation to rapid approximation III:Multivariate expansions
, 2007
"... In this paper we expand upon the theme of modified Fourier expansions and extend the theory to a multivariate setting and to expansions in eigenfunctions of the Laplace– Neumann operator. We pay detailed attention to expansions in a ddimensional cube and to an effective derivation of expansion coef ..."
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Cited by 1 (0 self)
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In this paper we expand upon the theme of modified Fourier expansions and extend the theory to a multivariate setting and to expansions in eigenfunctions of the Laplace– Neumann operator. We pay detailed attention to expansions in a ddimensional cube and to an effective derivation of expansion coefficients there by means of quadratures of highly oscillatory integrals. Thus, we present asymptotic and Filontype formulæ for an effective derivation of expansion coefficients and discuss their design and relative advantages. Such methods are effective only for large indices, hence we introduce and analyse alternative quadrature schemes that require relatively modest number of additional function evaluations. 1
quadrature
"... The Fejér and Clenshaw–Curtis rules for numerical integration exhibit a curious phenomenon when applied to certain analytic functions. When N (the number of points in the integration rule) increases, the error does not decay to zero evenly but does so in two distinct stages. For N less than a critic ..."
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The Fejér and Clenshaw–Curtis rules for numerical integration exhibit a curious phenomenon when applied to certain analytic functions. When N (the number of points in the integration rule) increases, the error does not decay to zero evenly but does so in two distinct stages. For N less than a critical value, the error behaves like O(ϱ −2N), where ϱ is a constant greater than 1. For these values of N the accuracy of both the Fejér and Clenshaw–Curtis rules is almost indistinguishable from that of the more celebrated Gauss–Legendre quadrature rule. For larger N, however, the error decreases at the rate O(ϱ −N), i.e., only half as fast as before. Convergence curves typically display a kink where the convergence rate cuts in half. In this paper we derive explicit as well as asymptotic error formulas that provide a complete description of this phenomenon.