Operator splitting methods for pricing American options under stochastic volatility

We consider the numerical pricing of American options under Heston’s stochastic volatility model. The price is given by a linear complementarity problem with a two-dimensional parabolic partial differential operator. We propose operator splitting methods for performing time stepping after a finite difference space discretization. The idea is to decouple the treatment of the early exercise constraint and the solution of the system of linear equations into separate fractional time steps. With this approach an efficient numerical method can be chosen for solving the system of linear equations in the first fractional step before making a simple update to satisfy the early exercise constraint. Our analysis suggests that the Crank–Nicolson method and the operator splitting method based on it have the same asymptotic order of accuracy. The numerical experiments show that the operator splitting methods have comparable discretization errors. They also demonstrate the efficiency of the operator splitting methods when a multigrid method is used for solving the systems of linear equations.     

A hybrid discontinuous Galerkin method for computing the ground state solution of Bose–Einstein condensates

A numerical method for computing the ground state solution of Bose–Einstein condensates modeled by the Gross–Pitaevskii equation is presented. In this method, the three-dimensional computational domain is divided into hexahedral elements in which the solution is approximated by a sum of basis functions. Both polynomial and plane wave bases are considered for this purpose, and Lagrange multipliers are introduced to weakly enforce the interelement continuity of the solution. The ground state is computed by an iterative procedure for minimizing the energy. The performance results obtained for several numerical experiments demonstrate that the proposed method is more computationally efficient than similar solution approaches based on the standard higher-order finite element method.     

Objective Supervised Machine Learning-Based Classification and Inference of Biological Neuronal Networks

The classification of biological neuron types and networks poses challenges to the full understanding of the human brain’s organisation and functioning. In this paper, we develop a novel objective classification model of biological neuronal morphology and electrical types and their networks, based on the attributes of neuronal communication using supervised machine learning solutions. This presents advantages compared to the existing approaches in neuroinformatics since the data related to mutual information or delay between neurons obtained from spike trains are more abundant than conventional morphological data. We constructed two open-access computational platforms of various neuronal circuits from the Blue Brain Project realistic models, named Neurpy and Neurgen. Then, we investigated how we could perform network tomography with cortical neuronal circuits for the morphological, topological and electrical classification of neurons. We extracted the simulated data of 10,000 network topology combinations with five layers, 25 morphological type (m-type) cells, and 14 electrical type (e-type) cells. We applied the data to several different classifiers (including Support Vector Machine (SVM), Decision Trees, Random Forest, and Artificial Neural Networks). We achieved accuracies of up to 70%, and the inference of biological network structures using network tomography reached up to 65% of accuracy. Objective classification of biological networks can be achieved with cascaded machine learning methods using neuron communication data. SVM methods seem to perform better amongst used techniques. Our research not only contributes to existing classification efforts but sets the road-map for future usage of brain–machine interfaces towards an in vivo objective classification of neurons as a sensing mechanism of the brain’s structure.