On Non-Oscillating Integrals for Computing Inhomogeneous Airy Functions

Integral representations are considered of solutions of the inhomogeneous Airy differential equation $w''-z w=\pm1/\pi$. The solutions of these equations are also known as Scorer functions. Certain functional relations for these functions are used to confine the discussion to one function and to a certain sector in the complex plane. By using steepest descent methods from asymptotics, the standard integral representations of the Scorer functions are modified in order to obtain non-oscillating integrals for complex values of $z$. In this way stable representations for numerical evaluations of the functions are obtained. The methods are illustrated with numerical results.Comment: 12 pages, 5 figure

Matrix algorithms for solving (in)homogeneous bound state equations

In the functional approach to quantum chromodynamics, the properties of hadronic bound states are accessible via covariant integral equations, e.g. the Bethe-Salpeter equations for mesons. In particular, one has to deal with linear, homogeneous integral equations which, in sophisticated model setups, use numerical representations of the solutions of other integral equations as part of their input. Analogously, inhomogeneous equations can be constructed to obtain off-shell information in addition to bound-state masses and other properties obtained from the covariant analogue to a wave function of the bound state. These can be solved very efficiently using well-known matrix algorithms for eigenvalues (in the homogeneous case) and the solution of linear systems (in the inhomogeneous case). We demonstrate this by solving the homogeneous and inhomogeneous Bethe-Salpeter equations and find, e.g. that for the calculation of the mass spectrum it is more efficient to use the inhomogeneous equation. This is valuable insight, in particular for the study of baryons in a three-quark setup and more involved systems.Comment: 11 pages, 7 figure

Effect of Polydispersity and Anisotropy in Colloidal and Protein Solutions: an Integral Equation Approach

Application of integral equation theory to complex fluids is reviewed, with particular emphasis to the effects of polydispersity and anisotropy on their structural and thermodynamic properties. Both analytical and numerical solutions of integral equations are discussed within the context of a set of minimal potential models that have been widely used in the literature. While other popular theoretical tools, such as numerical simulations and density functional theory, are superior for quantitative and accurate predictions, we argue that integral equation theory still provides, as in simple fluids, an invaluable technique that is able to capture the main essential features of a complex system, at a much lower computational cost. In addition, it can provide a detailed description of the angular dependence in arbitrary frame, unlike numerical simulations where this information is frequently hampered by insufficient statistics. Applications to colloidal mixtures, globular proteins and patchy colloids are discussed, within a unified framework.Comment: 17 pages, 7 figures, to appear in Interdiscip. Sci. Comput. Life Sci. (2011), special issue dedicated to Prof. Lesser Blu

On non-oscillating integrals for computing inhomogeneous Airy functions

Integral representations are considered of solutions of the inhomogeneous Airy differential equation $w''-z,w=pm1/pi$. The solutions of these equations are also known as Scorer functions. Certain functional relations for these functions are used to confine the discussion to one function and to a certain sector in the complex plane. By using steepest descent methods from asymptotics, the standard integral representations of the Scorer functions are modified in order to obtain non-oscillating integrals for complex values of $z$. In this way stable representations for numerical evaluations of the functions are obtained. The methods are illustrated with numerical results

Some exact results for the three-layer Zamolodchikov model

In this paper we continue the study of the three-layer Zamolodchikov model started in our previous works. We analyse numerically the solutions to the Bethe ansatz equations. We consider two regimes I and II which differ by the signs of the spherical sides (a1,a2,a3)->(-a1,-a2,-a3). We accept the two-line hypothesis for the regime I and the one-line hypothesis for the regime II. In the thermodynamic limit we derive integral equations for distribution densities and solve them exactly. We calculate the partition function for the three-layer Zamolodchikov model and check a compatibility of this result with the functional relations. We also do some numerical checkings of our results.Comment: LaTeX, 27 pages, 9 figure

Generalized Differential-Integral Quadrature and the Solution of the Boundary Layer Equations. G.U. Aero Report 9122

The global methods of generalized differential quadrature (GDQ) and generalized integral quadrature (GIQ) for solutions of partial differential and integral equations are presented in this paper. These methods approximate the derivatives and integrals by a linear combination of all the functional values in the overall domain, where the weighting coefficients can be readily identified. The error estimations of GDQ and GIQ have also been analysed. Application of GDQ and GIQ to solve boundary layer equations demonstrated that accurate numerical results can be obtained using just a few grid points

Hyperelliptic Theta-Functions and Spectral Methods

A code for the numerical evaluation of hyperelliptic theta-functions is presented. Characteristic quantities of the underlying Riemann surface such as its periods are determined with the help of spectral methods. The code is optimized for solutions of the Ernst equation where the branch points of the Riemann surface are parameterized by the physical coordinates. An exploration of the whole parameter space of the solution is thus only possible with an efficient code. The use of spectral approximations allows for an efficient calculation of all quantities in the solution with high precision. The case of almost degenerate Riemann surfaces is addressed. Tests of the numerics using identities for periods on the Riemann surface and integral identities for the Ernst potential and its derivatives are performed. It is shown that an accuracy of the order of machine precision can be achieved. These accurate solutions are used to provide boundary conditions for a code which solves the axisymmetric stationary Einstein equations. The resulting solution agrees with the theta-functional solution to very high precision.Comment: 25 pages, 12 figure

Information-Theoretic Stochastic Optimal Control via Incremental Sampling-based Algorithms

This paper considers optimal control of dynamical systems which are represented by nonlinear stochastic differential equations. It is well-known that the optimal control policy for this problem can be obtained as a function of a value function that satisfies a nonlinear partial differential equation, namely, the Hamilton-Jacobi-Bellman equation. This nonlinear PDE must be solved backwards in time, and this computation is intractable for large scale systems. Under certain assumptions, and after applying a logarithmic transformation, an alternative characterization of the optimal policy can be given in terms of a path integral. Path Integral (PI) based control methods have recently been shown to provide elegant solutions to a broad class of stochastic optimal control problems. One of the implementation challenges with this formalism is the computation of the expectation of a cost functional over the trajectories of the unforced dynamics. Computing such expectation over trajectories that are sampled uniformly may induce numerical instabilities due to the exponentiation of the cost. Therefore, sampling of low-cost trajectories is essential for the practical implementation of PI-based methods. In this paper, we use incremental sampling-based algorithms to sample useful trajectories from the unforced system dynamics, and make a novel connection between Rapidly-exploring Random Trees (RRTs) and information-theoretic stochastic optimal control. We show the results from the numerical implementation of the proposed approach to several examples.Comment: 18 page