A numerical scheme for free boundary problems that are gradient flows for the area functional

Europ. J. Appl. Math., 11, issue 2, pp. 61-80
A numerical scheme for free boundary problems that are gradient flows for the area functional
eBay Authors

Many moving boundary problems that are driven in some way by the curvature of the free boundary are gradient flows for the area of the moving interface. Examples are the Mullins-Sekerka flow, the Hele-Shaw flow, flow by mean curvature, and flow by averaged mean curvature. The gradient flow structure suggests an implicit finite differences approach to compute numerical solutions.

The proposed numerical scheme will allow to treat such free boundary problems in both R2 and R3. The advantage of such an approach is the re-usability of much of the setup for all of the different problems. As an example of the method we will compute solutions to the averaged mean curvature flow that exhibit the formation of a singularity.

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Canary in the e-Commerce Coal Mine: Detecting and Predicting Poor Experiences Using Buyer-to-Seller Messages

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Reputation and feedback systems in online marketplaces are often biased, making it difficult to ascertain the quality of sellers. We use post-transaction, buyer-to-seller message traffic to detect signals of unsatisfactory transactions on eBay. We posit that a message sent after the item was paid for serves as a reliable indicator that the buyer may be unhappy with that purchase, particularly when the message included words associated with a negative experience. The fraction of a seller's message traffic that was negative predicts whether a buyer who transacts with this seller will stop purchasing on eBay, implying that platforms can use these messages as an additional signal of seller quality.

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A numerical scheme for axisymmetric solutions of curvature driven free boundary problems, with applications to the Willmore Flow

Uwe Mayer, Gieri Simonett

We present a numerical scheme for radially symmetric solutions to curvature driven moving boundary problems governed by a local law of motion, e.g. the mean curvature flow, the surface diffusion flow, and the Willmore flow. We then present several numerical experiments for the Willmore flow. In particular, we provide numerical evidence that the Willmore flow can develop singularities in finite time.