A linear domain decomposition method for partially saturated flow in porous media

The Richards equation is a nonlinear parabolic equation that is commonly used for modelling saturated/unsaturated flow in porous media. We assume that the medium occupies a bounded Lipschitz domain partitioned into two disjoint subdomains separated by a fixed interface $\Gamma$. This leads to two problems defined on the subdomains which are coupled through conditions expressing flux and pressure continuity at $\Gamma$. After an Euler implicit discretisation of the resulting nonlinear subproblems a linear iterative ($L$-type) domain decomposition scheme is proposed. The convergence of the scheme is proved rigorously. In the last part we present numerical results that are in line with the theoretical finding, in particular the unconditional convergence of the scheme. We further compare the scheme to other approaches not making use of a domain decomposition. Namely, we compare to a Newton and a Picard scheme. We show that the proposed scheme is more stable than the Newton scheme while remaining comparable in computational time, even if no parallelisation is being adopted. Finally we present a parametric study that can be used to optimize the proposed scheme.Comment: 34 pages, 13 figures, 7 table

On Revenue-Optimal Dynamic Auctions for Bidders with Interdependent Values

In a dynamic market, being able to update one’s value based on information available to other bidders currently in the market can be critical to having profitable transactions. This is nicely captured by the model of interdependent values (IDV): a bidder’s value can explicitly depend on the private information of other bidders. In this paper we present preliminary results about the revenue properties of dynamic auctions for IDV bidders. We adopt a computational approach to design single-item revenue-optimal dynamic auctions with known arrivals and departures but (private) signals that arrive online. In leveraging a characterization of truthful auctions, we present a mixed-integer programming formulation of the design problem. Although a discretization is imposed on bidder signals the solution is a mechanism applicable to continuous signals. The formulation size grows exponentially in the dependence of bidders’ values on other bidders’ signals. We highlight general properties of revenue-optimal dynamic auctions in a simple parametrized example and study the sensitivity of prices and revenue to model parameters.Engineering and Applied Science

Online Auctions for Bidders with Interdependent Values

Interdependent values (IDV) is a valuation model allowing bidders in an auction to express their value for the item(s) to sell as a function of the other bidders' information. We investigate the incentive compatibility (IC) of single-item auctions for IDV bidders in dynamic environments. We provide a necessary and sufficient characterization for IC in this setting. We show that if bidders can misreport departure times and private signals, no reasonable auction can be IC. We present a reasonable IC auction for the case where bidders cannot misreport departures.Engineering and Applied Science

More on the Power of Demand Queries in Combinatorial Auctions: Learning Atomic Languages and Handling Incentives

Query learning models from computational learning theory (CLT) can be adopted to perform elicitation in combinatorial auctions. Indeed, a recent elicitation framework demonstrated that the equivalence queries of CLT can be usefully simulated with price-based demand queries. In this paper, we validate the flexibility of this framework by defining a learning algorithm for atomic bidding languages, a class that includes XOR and OR. We also handle incentives, characterizing the communication requirements of the Vickrey-Clarke-Groves outcome rule. This motivates an extension to the earlier learning framework that brings truthful responses to queries into an equilibrium.Engineering and Applied Science