In this paper, we develop a robust numerical method in pricing options, when the underlying asset follows a jump diffusion model. We demonstrate that, with the quadratic spline collocation method, the integral approximation in the pricing PIDE is intuitively simple, and comes down to the evaluation of the probabilistic moments of the jump density. When combined with a Picard iteration scheme, t...

We study the steady-state Navier-Stokes equations in the context of stochastic finite element discretizations. Specifically, we assume that the viscosity is a random field given in the form of a generalized polynomial chaos expansion. For the resulting stochastic problem, we formulate the model and linearization schemes using Picard and Newton iterations in the framework of the stochastic Galer...

Modeling unsaturated flow using numerical techniques such as the finite element method can be especially difficult because of the highly nonlinear nature of the governing equations. This problem is even more challenging when a steady-state solution is needed. This paper describes the implementation of a pseudo-transient technique to drive the solution to steady-state and gives results for a rea...

Numerical approximations to the Fourier transformed solution of partial differential equations are obtained via Monte Carlo simulation of certain random multiplicative cascades. Two particular equations are considered: linear diffusion equation and viscous Burgers equation. The algorithms proposed exploit the structure of the branching random walks in which the multiplicative cascades are defin...

We study the steady-state Navier-Stokes equations in the context of stochastic finite element discretizations. Specifically, we assume that the viscosity is a random field given in the form of a generalized polynomial chaos expansion. For the resulting stochastic problem, we formulate the model and linearization schemes using Picard and Newton iterations in the framework of the stochastic Galer...

In this paper, we establish a fixed point theorem for multi-valued operators in a complete b−metric space using the concept of Berinde and Berinde [9] on multi-valued weak contractions for the Picard iteration in a metric space. Our main result generalizes, extends and improves some of the recent results of Berinde and Berinde [9] as well as those of Daffer and Kaneko [17] and also unifies seve...

Anderson acceleration (AA) is a technique for accelerating the convergence of fixed-point iterations. In this paper, we apply AA to sequence functions and modify norm in its internal optimization problem H−s norm, some positive integer s, bias it towards low-frequency spectral content residual. We analyze by quantifying improvement over Picard iteration. find that based on H−2 well-suited solve...

We propose a novel second order in time numerical scheme for Cahn-Hilliard-NavierStokes phase field model with matched density. The scheme is based on second order convex-splitting for the Cahn-Hilliard equation and pressure-projection for the Navier-Stokes equation. We show that the scheme is mass-conservative, satisfies a modified energy law and is therefore unconditionally stable. Moreover, ...

We study the Green’s function for the linearized Boltzmann equation. For the short-time period, the Green’s function is dominated by the particle-like waves; and for large-time, by the fluid-like waves exhibiting the weak Huygens principle. The fluidlike waves are constructed by the spectral analysis and complex analytic techniques, making uses of the rotational symmetry of the equation in the ...

Since the advent of Taylor Series, polynomial methods have been used to solve di↵erential equations and learn the properties of di↵erential equations. There are many polynomial methods for solving di↵erential equations and understanding dynamical systems. For example, Taylor polynomials, Chebyshev Polynomials and Adomian Polynomials are used to generate approximate solutions. Automatic di↵erent...