Higher order scrambled digital nets are randomized quasi-Monte Carlo rules which have recently been introduced in [J. Dick, Ann. Statist., 39 (2011), 1372-1398 and shown to achieve the optimal rate of convergence of the root mean square error for numerical integration of smooth functions defined on the s-dimensional unit cube. The key ingredient there is a digit interlacing function applied to the components of a randomly scrambled digital net whose number of components is ds, where the integer d is the so-called interlacing factor. In this paper, we replace the randomly scrambled digital nets by randomly scrambled polynomial lattice point sets, which allows us to obtain a better dependence on the dimension while still achieving the optimal rate of convergence. Our results apply to Owen's full scrambling scheme as well as the simplifications studied by Hickernell, Matoušek and Owen. We consider weighted function spaces with general weights, whose elements have square integrable partial mixed derivatives of order up to α ≥ 1, and derive an upper bound on the variance of the estimator for higher order scrambled polynomial lattice rules. Employing our obtained bound as a quality criterion, we prove that the component-by-component construction can be used to obtain explicit constructions of good polynomial lattice point sets. By first constructing classical polynomial lattice point sets in base b and dimension ds, to which we then apply the interlacing scheme of order d, we obtain a construction cost of the algorithm of order O(dsmb m ) operations 1 using O(b m ) memory in case of product weights, where b m is the number of points in the polynomial lattice point set.

In this paper we develop a very efficient approach to the Monte Carlo estimation of the expected value of partial perfect information (EVPPI) that measures the average benefit of knowing the value of a subset of uncertain parameters involved in a decision model. The calculation of EVPPI is inherently a nested expectation problem, with an outer expectation with respect to one random variable X and an inner conditional expectation with respect to the other random variable Y . We tackle this problem by using a Multilevel Monte Carlo (MLMC) method (Giles 2008) in which the number of inner samples for Y increases geometrically with level, so that the accuracy of estimating the inner conditional expectation improves and the cost also increases with level. We construct an antithetic MLMC estimator and provide sufficient assumptions on a decision model under which the antithetic property of the estimator is well exploited, and consequently a root-mean-square accuracy of ε can be achieved at a cost of O(ε −2 ). Numerical results confirm the considerable computational savings compared to the standard, nested Monte Carlo method for some simple testcases and a more realistic medical application.

Quadrature rules using higher order digital nets and sequences are known to exploit the smoothness of a function for numerical integration and to achieve an improved rate of convergence as compared to classical digital nets and sequences for smooth functions. A construction principle of higher order digital nets and sequences based on a digit interlacing function was introduced in [J. Dick, SIAM J. Numer. Anal., 45 (2007) pp. , which interlaces classical digital nets or sequences whose number of components is a multiple of the dimension.In this paper, we study the use of polynomial lattice point sets for interlaced components. We call quadrature rules using such point sets interlaced polynomial lattice rules. We consider weighted Walsh spaces containing smooth functions and derive two upper bounds on the worstcase error for interlaced polynomial lattice rules, both of which can be employed as a quality criterion for the construction of interlaced polynomial lattice rules. We investigate the component-by-component construction and the Korobov construction as a means of explicit constructions of good interlaced polynomial lattice rules that achieve the optimal rate of the worst-case error. Through this approach we are able to obtain a good dependence of the worst-case error bounds on the dimension under certain conditions on the weights, while significantly reducing the construction cost as compared to higher order polynomial lattice rules.

The expected information gain is an important quality criterion of Bayesian experimental designs, which measures how much the information entropy about uncertain quantity of interest θ is reduced on average by collecting relevant data Y . However, estimating the expected information gain has been considered computationally challenging since it is defined as a nested expectation with an outer expectation with respect to Y and an inner expectation with respect to θ. In fact, the standard, nested Monte Carlo method requires a total computational cost of O(ε −3 ) to achieve a root-mean-square accuracy of ε. In this paper we develop an efficient algorithm to estimate the expected information gain by applying a multilevel Monte Carlo (MLMC) method. To be precise, we introduce an antithetic MLMC estimator for the expected information gain and provide a sufficient condition on the data model under which the antithetic property of the MLMC estimator is well exploited such that optimal complexity of O(ε −2 ) is achieved. Furthermore, we discuss how to incorporate importance sampling techniques within the MLMC estimator to avoid arithmetic underflow. Numerical experiments show the considerable computational cost savings compared to the nested Monte Carlo method for a simple test case and a more realistic pharmacokinetic model.

In the geological sequestration of carbon dioxide (CO2), residual gas trapping plays an important role in immobilizing CO2. In this study, we investigate the propagation of gravity currents with residual gas trapping in a two-layered porous medium. We first formulate a model for a constant-flux release of a relatively less dense fluid (CO2) from a point source into a porous medium bounded above by a horizontal less-permeable seal. After a constant-flux release ceases, a fraction of the released fluid remains within the porous spaces at the trailing edge because of the capillary forces. This capillary retention is formulated in a model of gravity currents of a finite-volume release in the two-layered medium. In the latter model, the plume shape at the end of the constant-flux release is used as an initial profile. Using these models sequentially, the propagation of both cross-sectional and axisymmetric currents is quantitatively examined.