Pricing Financial Derivatives using Radial Basis Function generated Finite Differences with Polyharmonic Splines on Smoothly Varying Node Layouts

In this paper, we study the benefits of using polyharmonic splines and node layouts with smoothly varying density for developing robust and efficient radial basis function generated finite difference (RBF-FD) methods for pricing of financial derivatives. We present a significantly improved RBF-FD scheme and successfully apply it to two types of multidimensional partial differential equations in finance: a two-asset European call basket option under the Black--Scholes--Merton model, and a European call option under the Heston model. We also show that the performance of the improved method is equally high when it comes to pricing American options. By studying convergence, computational performance, and conditioning of the discrete systems, we show the superiority of the introduced approaches over previously used versions of the RBF-FD method in financial applications

Multi-patch discontinuous Galerkin isogeometric analysis for wave propagation: explicit time-stepping and efficient mass matrix inversion

We present a class of spline finite element methods for time-domain wave propagation which are particularly amenable to explicit time-stepping. The proposed methods utilize a discontinuous Galerkin discretization to enforce continuity of the solution field across geometric patches in a multi-patch setting, which yields a mass matrix with convenient block diagonal structure. Over each patch, we show how to accurately and efficiently invert mass matrices in the presence of curved geometries by using a weight-adjusted approximation of the mass matrix inverse. This approximation restores a tensor product structure while retaining provable high order accuracy and semi-discrete energy stability. We also estimate the maximum stable timestep for spline-based finite elements and show that the use of spline spaces result in less stringent CFL restrictions than equivalent piecewise continuous or discontinuous finite element spaces. Finally, we explore the use of optimal knot vectors based on L2 n-widths. We show how the use of optimal knot vectors can improve both approximation properties and the maximum stable timestep, and present a simple heuristic method for approximating optimal knot positions. Numerical experiments confirm the accuracy and stability of the proposed methods

Coupling, Conservation, and Performance in Numerical Simulations

This thesis considers three aspects of the numerical simulations, which arecoupling, conservation, and performance. We conduct a project and addressone challenge from each of these aspects.We propose a novel penalty force to enforce contacts with accurate Coulombfriction. The force is compatible with fully-implicit time integration and theuse of optimization-based integration. In addition to processing collisionsbetween deformable objects, the force can be used to couple rigid bodies todeformable objects or the material point method. The force naturally leads tostable stacking without drift over time, even when solvers are not run toconvergence. The force leads to an asymmetrical system, and we provide apractical solution for handling these.Next we present a new technique for transferring momentum and velocity betweenparticles and MAC grids based on the Affine-Particle-In-Cell (APIC) frameworkpreviously developed for co-locatedgrids. We extend the original APIC paper and show thatthe proposed transfers preserve linear and angular momentum and also satisfyall of the original APIC properties.Early indications in the original APIC paper suggested that APIC might besuitable for simulating high Reynolds fluids due to favorable retention ofvortices, but these properties were not studied further. We use twodimensional Fourier analysis to investigate dissipation in the limit \dt=0.We investigate dissipation and vortex retention numerically to quantify theeffectiveness of APIC compared with other transfer algorithms.Finally we present an efficient solver for problems typically seen inmicrofluidic applications.Microfluidic ``lab on a chip'' devices are small devices that operate on smalllength scales on small volumes of fluid. Designs for microfluidic chips aregenerally composed of standardized and often repeated components connected bylong, thin, straight fluid channels. We propose a novel discretizationalgorithm for simulating the Stokes equations on geometry with these features,which produces sparse linear systems with many repeated matrix blocks. Thediscretization is formally third order accurate for velocity and second orderaccurate for pressure in the $L^\infty$ norm. We also propose a novel linearsystem solver based on cyclic reduction, reordered sparse Gaussian elimination,and operation caching that is designed to efficiently solve systems withrepeated matrix blocks