A global method for coupling transport with chemistry in heterogeneous porous media

Modeling reactive transport in porous media, using a local chemical equilibrium assumption, leads to a system of advection-diffusion PDE's coupled with algebraic equations. When solving this coupled system, the algebraic equations have to be solved at each grid point for each chemical species and at each time step. This leads to a coupled non-linear system. In this paper a global solution approach that enables to keep the software codes for transport and chemistry distinct is proposed. The method applies the Newton-Krylov framework to the formulation for reactive transport used in operator splitting. The method is formulated in terms of total mobile and total fixed concentrations and uses the chemical solver as a black box, as it only requires that on be able to solve chemical equilibrium problems (and compute derivatives), without having to know the solution method. An additional advantage of the Newton-Krylov method is that the Jacobian is only needed as an operator in a Jacobian matrix times vector product. The proposed method is tested on the MoMaS reactive transport benchmark.Comment: Computational Geosciences (2009) http://www.springerlink.com/content/933p55085742m203/?p=db14bb8c399b49979ba8389a3cae1b0f&pi=1

On two-stage convex chance constrained problems

In this paper we develop approximation algorithms for two-stage convex chance constrainedproblems. Nemirovski and Shapiro [16] formulated this class of problems and proposed anellipsoid-like iterative algorithm for the special case where the impact function f (x, h) is bi-affine.We show that this algorithm extends to bi-convex f (x, h) in a fairly straightforward fashion.The complexity of the solution algorithm as well as the quality of its output are functions of theradius r of the largest Euclidean ball that can be inscribed in the polytope defined by a randomset of linear inequalities generated by the algorithm [16]. Since the polytope determining ris random, computing r is diffiult. Yet, the solution algorithm requires r as an input. Inthis paper we provide some guidance for selecting r. We show that the largest value of r isdetermined by the degree of robust feasibility of the two-stage chance constrained problem –the more robust the problem, the higher one can set the parameter r. Next, we formulate ambiguous two-stage chance constrained problems. In this formulation,the random variables defining the chance constraint are known to have a fixed distribution;however, the decision maker is only able to estimate this distribution to within some error. Weconstruct an algorithm that solves the ambiguous two-stage chance constrained problem whenthe impact function f (x, h) is bi-affine and the extreme points of a certain “dual” polytope areknown explicitly

The power of a collectivity to act in weighted voting games with many small voters

We analyze the propensity to approve a random proposal of a large committee that makes decisions by weighted voting. The approach is a generalized version of James Coleman's "power of a collectivity to act". Throughout the paper it is assumed that the voters are of two kinds: a fixed (possibly empty) set of "major" (big) voters with fixed weights, and an ever-increasing number of "minor" (small) voters, whose total weight is also fixed, but where each individual's weight becomes negligible. As our main result, we obtain that asymptotically many minor voters act like a modification of the quota for the vote among major voters. The paper estimates the rate of convergence which turns out to be very high if the weight distribution among the small voters is not too skewed. The results obtained are illustrated by evaluating the decision rules for the Council of Ministers of the EU for various scenarios of EU enlargement. © 2007 Springer-Verlag