On multigrid for anisotropic equations and variational inequalities: pricing multi-dimensional European and American options

Partial differential operators in finance often originate in bounded linear stochastic processes. As a consequence, diffusion over these boundaries is zero and the corresponding coefficients vanish. The choice of parameters and stretched grids lead to additional anisotropies in the discrete equations or inequalities. In this study various block smoothers are tested in numerical experiments for equations of Black–Scholes-type (European options) in several dimensions. For linear complementarity problems, as they arise from optimal stopping time problems (American options), the choice of grid transfer is also crucial to preserve complementarity conditions on all grid levels. We adapt the transfer operators at the free boundary in a suitable way and compare with other strategies including cascadic approaches and full approximation schemes

Efficient hierarchical approximation of high-dimensional option pricing problems

A major challenge in computational finance is the pricing of options that depend on a large number of risk factors. Prominent examples are basket or index options where dozens or even hundreds of stocks constitute the underlying asset and determine the dimensionality of the corresponding degenerate parabolic equation. The objective of this article is to show how an efficient discretisation can be achieved by hierarchical approximation as well as asymptotic expansions of the underlying continuous problem. The relation to a number of state-of-the-art methods is highlighted

G-CSC Report 2010

The present report gives a short summary of the research of the Goethe Center for Scientific Computing (G-CSC) of the Goethe University Frankfurt. G-CSC aims at developing and applying methods and tools for modelling and numerical simulation of problems from empirical science and technology. In particular, fast solvers for partial differential equations (i.e. pde) such as robust, parallel, and adaptive multigrid methods and numerical methods for stochastic differential equations are developed. These methods are highly adanvced and allow to solve complex problems.. The G-CSC is organised in departments and interdisciplinary research groups. Departments are localised directly at the G-CSC, while the task of interdisciplinary research groups is to bridge disciplines and to bring scientists form different departments together. Currently, G-CSC consists of the department Simulation and Modelling and the interdisciplinary research group Computational Finance

On multigrid for anisotropic equations and variational inequalities: pricing multi-dimensional European and American options

Partial differential operators in finance often originate in bounded linear stochastic processes. As a consequence, diffusion over these boundaries is zero and the corresponding coefficients vanish. The choice of parameters and stretched grids lead to additional anisotropies in the discrete equations or inequalities. In this study various block smoothers are tested in numerical experiments for equations of Black–Scholes-type (European options) in several dimensions. For linear complementarity problems, as they arise from optimal stopping time problems (American options), the choice of grid transfer is also crucial to preserve complementarity conditions on all grid levels. We adapt the transfer operators at the free boundary in a suitable way and compare with other strategies including cascadic approaches and full approximation schemes

Adaptive Local Multigrid Methods for the Solution of Time Harmonic Eddy Current Problems

The efficient computation of large eddy current problems with finite elements requires adaptive methods and fast optimal iterative solvers like multigrid methods. This paper provides an overview of the most important implementation aspects of an adaptive multigrid scheme for time-harmonic eddy currents. It is shown how the standard multigrid scheme can be modified to yield an O(N) complexity even for general adaptive refinement strategies, where the number of unknowns N can grow slowly from one to the next refinement level. Algorithmic details and numerical examples are given

Mathematical modeling of the Drosophila neuromuscular junction

Poster presentation: An important challenge in neuroscience is understanding how networks of neurons go about processing information. Synapses are thought to play an essential role in cellular information processing however quantitative and mathematical models of the underlying physiologic processes that occur at synaptic active zones are lacking. We are generating mathematical models of synaptic vesicle dynamics at a well-characterized model synapse, the Drosophila larval neuromuscular junction. This synapse's simplicity, accessibility to various electrophysiological recording and imaging techniques, and the genetic malleability intrinsic to Drosophila system make it ideal for computational and mathematical studies. We have employed a reductionist approach and started by modeling single presynaptic boutons. Synaptic vesicles can be divided into different pools; however, a quantitative understanding of their dynamics at the Drosophila neuromuscular junction is lacking [4]. We performed biologically realistic simulations of high and low release probability boutons [3] using partial differential equations (PDE) taking into account not only the evolution in time but also the spatial structure in two dimensions (the extension to three dimensions will be implemented soon). PDEs are solved using UG, a program library for the calculation of multi-dimensional PDEs solved using a finite volume approach and implicit time stepping methods leading to extended linear equation systems be solvedwith multi-grid methods [3,4]. Numerical calculations are done on multi-processor computers for fast calculations using different parameters in order to asses the biological feasibility of different models. In preliminary simulations, we modeled vesicle dynamics as a diffusion process describing exocytosis as Neumann streams at synaptic active zones. The initial results obtained with these models are consistent with experimental data. However, this should be regarded as a work in progress. Further refinements will be implemented, including simulations using morphologically realistic geometries which were generated from confocal scans of the neuromuscular junction using NeuRA (a Neuron Reconstruction Algorithm). Other parameters such as glutamate diffusion and reuptake dynamics, as well as postsynaptic receptor kinetics will be incorporated as well

A new model for computing the evolution of the extracellular, innercellular and membrane potential simultaneously

Poster Presentation from Nineteenth Annual Computational Neuroscience Meeting: CNS*2010 San Antonio, TX, USA. 24-30 July 2010 In order to model extracellular potentials the Line-Source method provides [1] a very powerful and accurate approach. In this method transmembane fluxes are understood as sources for potential distributions which obey the Poission-equation with zero boundary conditions in the infinity. Its solutions reveal that the waveforms are proportional to local transmembrane net currents. The extracellular potentials are comparable small in amplitude and with the aid of their second special derivatives, it is possible to interpret them as additional fluxes to be included into the cable equation having an impact on the membrane potential of surrounding cells [2]. On this basis ephaptic interactions have been studied and have been considered to play a minor role in the network activity. This modeling study provides a new approach based on the first principle of the conservation of charges which leads to a generalized form of the cable equation taking into account the full three-dimensional detail of the cell’s geometry and the presence of the extracellular potential. So instead of coupling the compartment model and the model for extracellular potentials by means of the transmembrane currents, a non-linear system of partial differential equations is solved. Because the abstraction of deviding the cell’s geometry into compartments falls apart, it is possible to examine the contribution of the precise cell geometry to the signal processing while not neglecting the impact which could result from the extracellular potential. Some simulations of propagating action potentials on ramified geometries are going to be shown as well as the resulting distributions of extracellular action potentials

Insights into manufacturing techniques of archaeological pottery: Industrial X-ray computed tomography as a tool in the examination of cultural material

The application of X-radiography in ceramic studies is becoming an increasingly valued method. Using the potential of industrial X-ray computed tomography (CT) for non-destructive testing as an archaeometric or archaeological method in pottery studies, especially regarding aspects such as manufacturing techniques or pottery abrics, requires controlled data-acquisition and post-processing by scientific computing adjusted to archaeological issues. The first results of this evaluation project show that, despite the difficulties inherent in CT technology, considerable information can be extracted for pottery analysis. The application of surface morphology reconstructions and volumetric measurements based on CT data will open a new field in future non-invasive archaeology