Piecewise Constant Policy Approximations to Hamilton-Jacobi-Bellman Equations

An advantageous feature of piecewise constant policy timestepping for Hamilton-Jacobi-Bellman (HJB) equations is that different linear approximation schemes, and indeed different meshes, can be used for the resulting linear equations for different control parameters. Standard convergence analysis suggests that monotone (i.e., linear) interpolation must be used to transfer data between meshes. Using the equivalence to a switching system and an adaptation of the usual arguments based on consistency, stability and monotonicity, we show that if limited, potentially higher order interpolation is used for the mesh transfer, convergence is guaranteed. We provide numerical tests for the mean-variance optimal investment problem and the uncertain volatility option pricing model, and compare the results to published test cases

Arbitrary-Lagrangian-Eulerian discontinuous Galerkin schemes with a posteriori subcell finite volume limiting on moving unstructured meshes

We present a new family of high order accurate fully discrete one-step Discontinuous Galerkin (DG) finite element schemes on moving unstructured meshes for the solution of nonlinear hyperbolic PDE in multiple space dimensions, which may also include parabolic terms in order to model dissipative transport processes. High order piecewise polynomials are adopted to represent the discrete solution at each time level and within each spatial control volume of the computational grid, while high order of accuracy in time is achieved by the ADER approach. In our algorithm the spatial mesh configuration can be defined in two different ways: either by an isoparametric approach that generates curved control volumes, or by a piecewise linear decomposition of each spatial control volume into simplex sub-elements. Our numerical method belongs to the category of direct Arbitrary-Lagrangian-Eulerian (ALE) schemes, where a space-time conservation formulation of the governing PDE system is considered and which already takes into account the new grid geometry directly during the computation of the numerical fluxes. Our new Lagrangian-type DG scheme adopts the novel a posteriori sub-cell finite volume limiter method, in which the validity of the candidate solution produced in each cell by an unlimited ADER-DG scheme is verified against a set of physical and numerical detection criteria. Those cells which do not satisfy all of the above criteria are flagged as troubled cells and are recomputed with a second order TVD finite volume scheme. The numerical convergence rates of the new ALE ADER-DG schemes are studied up to fourth order in space and time and several test problems are simulated. Finally, an application inspired by Inertial Confinement Fusion (ICF) type flows is considered by solving the Euler equations and the PDE of viscous and resistive magnetohydrodynamics (VRMHD).Comment: 39 pages, 21 figure

A Partial parametric path algorithm for multiclass classification

The objective functions of Support Vector Machine methods (SVMs) often includeparameters to weigh the relative importance of margins and training accuracies.The values of these parameters have a direct effect both on the optimal accuraciesand the misclassification costs. Usually, a grid search is used to find appropriatevalues for them. This method requires the repeated solution of quadraticprograms for different parameter values, and it may imply a large computationalcost, especially in a setting of multiclass SVMs and large training datasets. Formulti-class classification problems, in the presence of different misclassificationcosts, identifying a desirable set of values for these parameters becomes evenmore relevant. In this paper, we propose a partial parametric path algorithm, basedon the property that the path of optimal solutions of the SVMs with respect tothe preceding parameters is piecewise linear. This partial parametric path algorithmrequires the solution of just one quadratic programming problem, and anumber of linear systems of equations. Thus it can significantly reduce the computationalrequirements of the algorithm. To systematically explore the differentweights to assign to the misclassification costs, we combine the partial parametricpath algorithm with a variable neighborhood search method. Our numerical experimentsshow the efficiency and reliability of the proposed partial parametricpath algorithm