Are quasi-Monte Carlo algorithms efficient for two-stage stochastic programs?

Quasi-Monte Carlo algorithms are studied for designing discrete approximations of two-stage linear stochastic programs with random right-hand side and continuous probability distribution. The latter should allow for a transformation to a distribution with independent marginals. The two-stage integrands are piecewise linear, but neither smooth nor lie in the function spaces considered for QMC error analysis. We show that under some weak geometric condition on the two-stage model all terms of their ANOVA decomposition, except the one of highest order, are continuously differentiable and that first and second order ANOVA terms have mixed first order partial derivatives. Hence, randomly shifted lattice rules (SLR) may achieve the optimal rate of convergence not depending on the dimension if the effective superposition dimension is at most two. We discuss effective dimensions and dimension reduction for two-stage integrands. The geometric condition is shown to be satisfied almost everywhere if the underlying probability distribution is normal and principal component analysis (PCA) is used for transforming the covariance matrix. Numerical experiments for a large scale two-stage stochastic production planning model with normal demand show that indeed convergence rates close to the optimal are achieved when using SLR and randomly scrambled Sobol' point sets accompanied with PCA for dimension reduction

A first look at quasi-Monte Carlo for lattice field theory problems

In this project we initiate an investigation of the applicability of Quasi-Monte Carlo methods to lattice field theories in order to improve the asymptotic error behavior of observables for such theories. In most cases the error of an observable calculated by averaging over random observations generated from an ordinary Monte Carlo simulation behaves like N−1/2, where N is the number of observations. By means of Quasi-Monte Carlo methods it is possible to improve this behavior for certain problems to up to N−1. We adapted and applied this approach to simple systems like the quantum harmonic and anharmonic oscillator and verified an improved error scaling.Peer Reviewe

Lattice meets lattice: Application of lattice cubature to models in lattice gauge theory

High dimensional integrals are abundant in many fields of research including quantum physics. The aim of this paper is to develop efficient recursive strategies to tackle a class of high dimensional integrals having a special product structure with low order couplings, motivated by models in lattice gauge theory from quantum field theory. A novel element of this work is the potential benefit in using lattice cubature rules. The group structure within lattice rules combined with the special structure in the physics integrands may allow efficient computations based on Fast Fourier Transforms. Applications to the quantum mechanical rotor and compact $U(1)$ lattice gauge theory in two and three dimensions are considered

Quasi-Monte Carlo methods for two-stage stochastic mixed-integer programs

We consider randomized QMC methods for approximating the expected recourse in two-stage stochastic optimization problems containing mixed-integer decisions in the second stage. It is known that the second-stage optimal value function is piecewise linear-quadratic with possible kinks and discontinuities at the boundaries of certain convex polyhedral sets. This structure is exploited to provide conditions implying that first and higher order terms of the integrand’s ANOVA decomposition (Math. Comp. 79 (2010), 953–966) have mixed weak first order partial derivatives. This leads to a good smooth approximation of the integrand and, hence, to good convergence rates of randomized QMC methods if the effective (superposition) dimension is low.Peer Reviewe

Successive Coordinate Search and Component-by-Component Construction of Rank-1 Lattice Rules

The (fast) component-by-component (CBC) algorithm is an efficient tool for the construction of generating vectors for quasi-Monte Carlo rank-1 lattice rules in weighted reproducing kernel Hilbert spaces. We consider product weights, which assign a weight to each dimension. These weights encode the effect a certain variable (or a group of variables by the product of the individual weights) has. Smaller weights indicate less importance. Kuo proved that CBC constructions achieve the optimal rate of convergence in the respective function spaces, but this does not imply the algorithm will find the generating vector with the smallest worst-case error. In fact it does not. We investigate a generalization of the component-by-component construction that allows for a general successive coordinate search (SCS), based on an initial generating vector, and with the aim of getting closer to the smallest worst-case error. The proposed method admits the same type of worst-case error bounds as the CBC algorithm, independent of the choice of the initial vector. Under the same summability conditions on the weights as in Kuo's paper the error bound of the algorithm can be made independent of the dimension d and we achieve the same optimal order of convergence for the function spaces from Kuo's paper. Moreover, a fast version of our method, based on the fast CBC algorithm by Nuyens and Cools, is available, reducing the computational cost of the algorithm to O(d nlog(n)) operations, where n denotes the number of function evaluations. Numerical experiments seeded by a Korobov-type generating vector show that the new SCS algorithm will find better choices than the CBC algorithm and the effect is better for slowly decaying weights.status: accepte

Are Quasi-Monte Carlo algorithms efficient for two-stage stochastic programs?

Quasi-Monte Carlo algorithms are studied for designing discrete approximations of two-stage linear stochastic programs with random right-hand side and continuous probability distribution. The latter should allow for a transformation to a distribution with independent marginals. The twostage integrands are piecewise linear, but neither smooth nor lie in the function spaces considered for QMC error analysis. We show that under some weak geometric condition on the two-stage model all terms of their ANOVA decomposition, except the one of highest order, are continuously differentiable and that first and second order ANOVA terms have mixed first order partial derivatives and belong to L2. Hence, randomly shifted lattice rules (SLR) may achieve the optimal rate of convergence O(n-1+delta) with 2 (0; 1 2 ] and a constant not depending on the dimension if the effective superposition dimension is at most two. We discuss effective dimensions and dimension reduction for two-stage integrands. The geometric condition is shown to be satisfied almost everywhere if the underlying probability distribution is normal and principal component analysis (PCA) is used for transforming the covariance matrix. Numerical experiments for a large scale two-stage stochastic production planning model with normal demand show that indeed convergence rates close to the optimal are achieved when using SLR and randomly scrambled Sobol’ point sets accompanied with PCA for dimension reduction

New polynomially exact integration rules on U(N)U(N) and SU(N)SU(N)

In lattice Quantum Field Theory, we are often presented with integrals over polynomials of coefficients of matrices in U(N) or SU(N) with respect to the Haar measure. In some physical situations, e.g., in presence of a chemical potential, these integrals are numerically very difficult since their integrands are highly oscillatory which manifests itself in form of the sign problem. In these cases, Monte Carlo methods often fail to be adequate, rendering such computations practically impossible. We propose a new class of integration rules on U(N) and SU(N) which are derived from polynomially exact rules on spheres. We will examine these quadrature rules and their efficiency at the example of a 0+1 dimensional QCD for a non-zero quark mass and chemical potential. In particular, we will demonstrate the failure of Monte Carlo methods in such applications and that we can obtain polynomially exact, arbitrary precision results using the new integration rules

New polynomially exact integration rules on U(N)U(N) and SU(N)SU(N)

In lattice Quantum Field Theory, we are often presented with integrals over polynomials of coefficients of matrices in U(N) or SU(N) with respect to the Haar measure. In some physical situations, e.g., in presence of a chemical potential, these integrals are numerically very difficult since their integrands are highly oscillatory which manifests itself in form of the sign problem. In these cases, Monte Carlo methods often fail to be adequate, rendering such computations practically impossible. We propose a new class of integration rules on U(N) and SU(N) which are derived from polynomially exact rules on spheres. We will examine these quadrature rules and their efficiency at the example of a 0+1 dimensional QCD for a non-zero quark mass and chemical potential. In particular, we will demonstrate the failure of Monte Carlo methods in such applications and that we can obtain polynomially exact, arbitrary precision results using the new integration rules