Serial and Parallel Krylov Methods for Implicit Finite Difference Schemes Arising in Multivariate Option Pricing

This paper investigates computational and implementation issues for the valuation of options on three underlying assets, focusing on the use of the finite difference methods. We demonstrate that implicit methods, which have good convergence and stability prooperties, can now be implemented efficiently due to the recent development of techniques that allow the efficient solution of large and sparse linear systems. In the trivariate option valuation problem, we use nonstationary iterative methods (also called Krylov methods) for the solution of the large and sparse linear systems arising while using implicit methods. Krylov methods are investigated both in serial and in parallel implementations. Computational results show that the parallel implementation is particularly efficient if a fine grid space is needed.Multivariate option pricing, finite difference methods; Krylov methods; parallel Krylov methods

Space-time discontinuous Galerkin discretizations for linear first-order hyperbolic evolution systems

We introduce a space-time discretization for linear first-order hyperbolic evolution systems using a discontinuous Galerkin approximation in space and a Petrov-Galerkin scheme in time. We show well-posedness and convergence of the discrete system. Then we introduce an adaptive strategy based on goal-oriented dual-weighted error estimation. The full space-time linear system is solved with a parallel multilevel preconditioner. Numerical experiments for the linear transport equation and the Maxwell equation in 2D underline the effciency of the overall adaptive solution process

Space-time discontinuous Galerkin discretizations for linear first-order hyperbolic evolution systems. Revised March 2016

We introduce a space-time discretization for linear first-order hyperbolic evolution systems using a discontinuous Galerkin approximation in space and a Petrov-Galerkin scheme in time. We show well-posedness and convergence of the discrete system. Then we introduce an adaptive strategy based on goal-oriented dual-weighted error estimation. The full space-time linear system is solved with a parallel multilevel preconditioner. Numerical experiments for the linear transport equation and the Maxwell equation in 2D underline the effciency of the overall adaptive solution process

On the Solution of Linear Programming Problems in the Age of Big Data

The Big Data phenomenon has spawned large-scale linear programming problems. In many cases, these problems are non-stationary. In this paper, we describe a new scalable algorithm called NSLP for solving high-dimensional, non-stationary linear programming problems on modern cluster computing systems. The algorithm consists of two phases: Quest and Targeting. The Quest phase calculates a solution of the system of inequalities defining the constraint system of the linear programming problem under the condition of dynamic changes in input data. To this end, the apparatus of Fejer mappings is used. The Targeting phase forms a special system of points having the shape of an n-dimensional axisymmetric cross. The cross moves in the n-dimensional space in such a way that the solution of the linear programming problem is located all the time in an "-vicinity of the central point of the cross.Comment: Parallel Computational Technologies - 11th International Conference, PCT 2017, Kazan, Russia, April 3-7, 2017, Proceedings (to be published in Communications in Computer and Information Science, vol. 753

Sparse Matrix Algorithms on Distributed Memory Multiprocessors

Progress was made in creating algorithms and software for large-scale sparse matrix computations on advanced distributed-memory parallel machines during the past year. This report is divided into: large-scale linear systems; highly parallel triangular solution; spectral nested dissection orderings; parallel multifrontal factorization; structure of orthogonal factors; and sparse bases for the range space and the null space

Simulating a Family of Tissue P Systems Solving SAT on the GPU

In order to provide e cient software tools to deal with large membrane systems, high-throughput simulators are required. Parallel computing platforms are good candidates, since they are capable of partially implementing the inherently parallel nature of the model. In this concern, today GPUs (Graphics Processing Unit) are considered as highly parallel processors, and they are being consolidated as accelerators for scienti c applications. In fact, previous attempts to design P systems simulators on GPUs have shown that a parallel architecture is better suited in performance than traditional single CPUs. In 2010, a GPU-based simulator was introduced for a family of P systems with active membranes solving SAT in linear time. This is the starting point of this paper, which presents a new GPU simulator for another polynomial-time solution to SAT by means of tissue P systems with cell division, trading space for time. The aim of this simulator is to further study which ingredients of di erent P systems models are well suited to be managed by the GPU.Junta de Andalucía P08-TIC04200Ministerio de Economía y Competitividad TIN2012-3743