The Grid Dependence of Well Inflow Performance in Reservoir Simulation

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Computing Optimal Designs of multiresponse Experiments reduces to Second-Order Cone Programming

Elfving's Theorem is a major result in the theory of optimal experimental design, which gives a geometrical characterization of $c-$optimality. In this paper, we extend this theorem to the case of multiresponse experiments, and we show that when the number of experiments is finite, $c-,A-,T-$ and $D-$optimal design of multiresponse experiments can be computed by Second-Order Cone Programming (SOCP). Moreover, our SOCP approach can deal with design problems in which the variable is subject to several linear constraints. We give two proofs of this generalization of Elfving's theorem. One is based on Lagrangian dualization techniques and relies on the fact that the semidefinite programming (SDP) formulation of the multiresponse $c-$optimal design always has a solution which is a matrix of rank $1$. Therefore, the complexity of this problem fades. We also investigate a \emph{model robust} generalization of $c-$optimality, for which an Elfving-type theorem was established by Dette (1993). We show with the same Lagrangian approach that these model robust designs can be computed efficiently by minimizing a geometric mean under some norm constraints. Moreover, we show that the optimality conditions of this geometric programming problem yield an extension of Dette's theorem to the case of multiresponse experiments. When the number of unknown parameters is small, or when the number of linear functions of the parameters to be estimated is small, we show by numerical examples that our approach can be between 10 and 1000 times faster than the classic, state-of-the-art algorithms

Enhancement of Sandwich Algorithms for Approximating Higher Dimensional Convex Pareto Sets

In many fields, we come across problems where we want to optimize several conflicting objectives simultaneously. To find a good solution for such multi-objective optimization problems, an approximation of the Pareto set is often generated. In this paper, we con- sider the approximation of Pareto sets for problems with three or more convex objectives and with convex constraints. For these problems, sandwich algorithms can be used to de- termine an inner and outer approximation between which the Pareto set is 'sandwiched'. Using these two approximations, we can calculate an upper bound on the approximation error. This upper bound can be used to determine which parts of the approximations must be improved and to provide a quality guarantee to the decision maker. In this paper, we extend higher dimensional sandwich algorithms in three different ways. Firstly, we introduce the new concept of adding dummy points to the inner approx- imation of a Pareto set. By using these dummy points, we can determine accurate inner and outer approximations more e±ciently, i.e., using less time-consuming optimizations. Secondly, we introduce a new method for the calculation of an error measure which is easy to interpret. The combination of easy calculation and easy interpretation makes this measure very suitable for sandwich algorithms. Thirdly, we show how transforming cer- tain objective functions can improve the results of sandwich algorithms and extend their applicability to certain non-convex problems. The calculation of the introduced error measure when using transformations will also be discussed. To show the effect of these enhancements, we make a numerical comparison using four test cases, including a four-dimensional case from the field of intensity-modulated radiation therapy (IMRT). The results of the different cases show that we can indeed achieve an accurate approximation using significantly fewer optimizations by using the enhancements.Convexity;e-efficiency;e-Pareto optimality;Geometric programming;Higher dimensional;Inner and outer approximation;IMRT;Pareto set;Multi-objective optimiza- tion;Sandwich algorithms;Transformations