Recent advances in higher order quasi-Monte Carlo methods

In this article we review some of recent results on higher order quasi-Monte Carlo (HoQMC) methods. After a seminal work by Dick (2007, 2008) who originally introduced the concept of HoQMC, there have been significant theoretical progresses on HoQMC in terms of discrepancy as well as multivariate numerical integration. Moreover, several successful and promising applications of HoQMC to partial differential equations with random coefficients and Bayesian estimation/inversion problems have been reported recently. In this article we start with standard quasi-Monte Carlo methods based on digital nets and sequences in the sense of Niederreiter, and then move onto their higher order version due to Dick. The Walsh analysis of smooth functions plays a crucial role in developing the theory of HoQMC, and the aim of this article is to provide a unified picture on how the Walsh analysis enables recent developments of HoQMC both for discrepancy and numerical integration

A dynamic modelling of safety nets

The nonlinear dynamic modelling of safety net systems is approached at different scales. For this purpose, the fundamental rope dynamic tests are the reference for two basic tools. One hand an anaytical bidimensional model with explicit geometrical nonlinearity and bilnear material law is proposed for preliminary design. On the other hand, a nonlinear explicit finite element is defined for numerical modelling of net systems. Semi-scale and full scale dynamic tests are performed to validate complete finite element models, suitable for global qualification of safety systems. The direct applications of these tools deal with explicit certification of safety systems for high-speed sport, such as downhill competitions

Matrix-geometric solution of infinite stochastic Petri nets

We characterize a class of stochastic Petri nets that can be solved using matrix geometric techniques. Advantages of such on approach are that very efficient mathematical technique become available for practical usage, as well as that the problem of large state spaces can be circumvented. We first characterize the class of stochastic Petri nets of interest by formally defining a number of constraints that have to be fulfilled. We then discuss the matrix geometric solution technique that can be employed and present some boundary conditions on tool support. We illustrate the practical usage of the class of stochastic Petri nets with two examples: a queueing system with delayed service and a model of connection management in ATM network

Automatic synthesis of analog layout : a survey

A review of recent research in the automatic synthesis of physical geometry for analog integrated circuits is presented. On introduction, an explanation of the difficulties involved in analog layout as opposed to digital layout is covered. Review of the literature then follows. Emphasis is placed on the exposition of general methods for addressing problems specific to analog layout, with the details of specific systems only being given when they surve to illustrate these methods well. The conclusion discusses problems remaining and offers a prediction as to how technology will evolve to solve them. It is argued that although progress has been and will continue to be made in the automation of analog IC layout, due to fundamental differences in the nature of analog IC design as opposed to digital design, it should not be expected that the level of automation of the former will reach that of the latter any time soon

Difference Methods for Boundary Value Problems in Ordinary Differential Equations

A general theory of difference methods for problems of the form Ny ≡ y' - f(t,y) = O, a ≦ t ≦ b, g(y(a),y(b))= 0, is developed. On nonuniform nets, t_0 = a, t_j = t_(j-1) + h_j, 1 ≦ j ≦ J, t_J = b, schemes of the form N_(h)u_j = G_j(u_0,•••,u_J) = 0, 1 ≦ j ≦ J, g(u_0,u_J) = 0 are considered. For linear problems with unique solutions, it is shown that the difference scheme is stable and consistent for the boundary value problem if and only if, upon replacing the boundary conditions by an initial condition, the resulting scheme is stable and consistent for the initial value problem. For isolated solutions of the nonlinear problem, it is shown that the difference scheme has a unique solution converging to the exact solution if (i) the linearized difference equations are stable and consistent for the linearized initial value problem, (ii) the linearized difference operator is Lipschitz continuous, (iii) the nonlinear difference equations are consistent with the nonlinear differential equation. Newton’s method is shown to be valid, with quadratic convergence, for computing the numerical solution

Accurate difference methods for linear ordinary differential systems subject to linear constraints

We consider the general system of n first order linear ordinary differential equations y'(t)=A(t)y(t)+g(t), a<t< b, subject to "boundary" conditions, or rather linear constraints, of the form Σ^(N)_(ν=1) B_(ν)y(τ_ν)=β Here y(t), g(t) and II are n-vectors and A(t), Bx,..., BN are n × n matrices. The N distinct points {τ_ν} lie in [a,b] and we only require N ≧ 1. Thus as special cases initial value problems, N=1, are included as well as the general 2-point boundary value problem, N=2, with τ_1=a, τ_2=b. (More general linear constraints are also studied, see (5.1) and (5.17).

Efficient calculation of the worst-case error and (fast) component-by-component construction of higher order polynomial lattice rules

We show how to obtain a fast component-by-component construction algorithm for higher order polynomial lattice rules. Such rules are useful for multivariate quadrature of high-dimensional smooth functions over the unit cube as they achieve the near optimal order of convergence. The main problem addressed in this paper is to find an efficient way of computing the worst-case error. A general algorithm is presented and explicit expressions for base~2 are given. To obtain an efficient component-by-component construction algorithm we exploit the structure of the underlying cyclic group. We compare our new higher order multivariate quadrature rules to existing quadrature rules based on higher order digital nets by computing their worst-case error. These numerical results show that the higher order polynomial lattice rules improve upon the known constructions of quasi-Monte Carlo rules based on higher order digital nets

A Multivariate Fast Discrete Walsh Transform with an Application to Function Interpolation

For high dimensional problems, such as approximation and integration, one cannot afford to sample on a grid because of the curse of dimensionality. An attractive alternative is to sample on a low discrepancy set, such as an integration lattice or a digital net. This article introduces a multivariate fast discrete Walsh transform for data sampled on a digital net that requires only $O(N \log N)$ operations, where $N$ is the number of data points. This algorithm and its inverse are digital analogs of multivariate fast Fourier transforms. This fast discrete Walsh transform and its inverse may be used to approximate the Walsh coefficients of a function and then construct a spline interpolant of the function. This interpolant may then be used to estimate the function's effective dimension, an important concept in the theory of numerical multivariate integration. Numerical results for various functions are presented