Abstract. We introduce a three-dimensional (3D) semi-implicit complementary volume numerical scheme for solving the level set formulation of (Riemannian) mean curvature flow problem. We apply the scheme to segmentation of objects (with interrupted edges) in 3D images. The method is unconditionally stable and efficient regarding computational times. The study of its experimental order of convergence on 3D examples shows its second order accuracy for smooth solutions and first order accuracy for highly singular solutions with vanishing gradients as arising in image segmentation. 1. Introduction. In this paper we introduce a fast and stable computational method for a three-dimensional (3D) image segmentation based on a solution of the following Riemannian mean curvature flow equation:

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Highlights A point-to-point response to reviewers’ comments is given in the response letter to them. In the submitted revision all changes are highlighted in blue.

We consider the l1-regularized Markowitz model, where a l1-penalty term is added to the objective function of the classical mean-variance one to stabilize the solution process, promoting sparsity in the solution. The l1-penalty term can also be interpreted in terms of short sales, on which several financial markets have posed restrictions. The choice of the regularization parameter plays a key role to obtain optimal portfolios that meet the financial requirements. We propose an updating rule for the regularization parameter in Bregman iteration to control both the sparsity and the number of short positions. We show that the modified scheme preserves the properties of the original one. Numerical tests are reported, which show the effectiveness of the approach.

In this paper, we present a transform-based algorithm for pricing discretely monitored arithmetic Asian options with remarkable accuracy in a general affine stochastic volatility framework.The accuracy is justified both theoretically and experimentally. In addition, to speed up the valuation process, we employ high-performance computing technologies. More specifically, we develop a parallel option pricing system that can be easily reproduced on parallel computers, also realized as a cluster of personal computers. Numerical results showing the accuracy and efficiency of the procedure are reported in the paper.

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