scholarly journals A Robust and Accurate Quasi-Monte Carlo Algorithm for Estimating Eigenvalue of Homogeneous Integral Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
F. Mehrdoust ◽  
B. Fathi Vajargah ◽  
E. Radmoghaddam
Keyword(s):  
Monte Carlo ◽  
Integral Equations ◽  
Numerical Algorithm ◽  
Variance Reduction ◽  
Numerical Results ◽  
Monte Carlo Algorithm ◽  
Computational Time ◽  
Reduction Procedure ◽  
Quasi Monte Carlo

We present an efficient numerical algorithm for computing the eigenvalue of the linear homogeneous integral equations. The proposed algorithm is based on antithetic Monte Carlo algorithm and a low-discrepancy sequence, namely, Faure sequence. To reduce the computational time we reduce the variance by using the antithetic variance reduction procedure. Numerical results show that our scheme is robust and accurate.

scholarly journals p-Refined Multilevel Quasi-Monte Carlo for Galerkin Finite Element Methods with Applications in Civil Engineering

Algorithms ◽  
2020 ◽  
Vol 13 (5) ◽  
pp. 110
Author(s):  
Philippe Blondeel ◽  
Pieterjan Robbe ◽  
Cédric Van hoorickx ◽  
Stijn François ◽  
Geert Lombaert ◽  
...  
Keyword(s):  
Finite Element ◽  
Monte Carlo ◽  
Variance Reduction ◽  
Civil Engineering ◽  
Galerkin Finite Element ◽  
Quasi Monte Carlo ◽  
Stability Of Slopes ◽  
Galerkin Finite Element Methods ◽  
Significant Cost Reduction ◽  
Lattice Rule

Civil engineering applications are often characterized by a large uncertainty on the material parameters. Discretization of the underlying equations is typically done by means of the Galerkin Finite Element method. The uncertain material parameter can be expressed as a random field represented by, for example, a Karhunen–Loève expansion. Computation of the stochastic responses, i.e., the expected value and variance of a chosen quantity of interest, remains very costly, even when state-of-the-art Multilevel Monte Carlo (MLMC) is used. A significant cost reduction can be achieved by using a recently developed multilevel method: p-refined Multilevel Quasi-Monte Carlo (p-MLQMC). This method is based on the idea of variance reduction by employing a hierarchical discretization of the problem based on a p-refinement scheme. It is combined with a rank-1 Quasi-Monte Carlo (QMC) lattice rule, which yields faster convergence compared to the use of random Monte Carlo points. In this work, we developed algorithms for the p-MLQMC method for two dimensional problems. The p-MLQMC method is first benchmarked on an academic beam problem. Finally, we use our algorithm for the assessment of the stability of slopes, a problem that arises in geotechnical engineering, and typically suffers from large parameter uncertainty. For both considered problems, we observe a very significant reduction in the amount of computational work with respect to MLMC.

scholarly journals A QUASI-MONTE CARLO ALGORITHM FOR THE NORMAL INVERSE GAUSSIAN DISTRIBUTION AND VALUATION OF FINANCIAL DERIVATIVES

2006 ◽  
Vol 09 (06) ◽  
pp. 843-867 ◽  
Author(s):  
FRED ESPEN BENTH ◽  
MARTIN GROTH ◽  
PAUL C. KETTLER
Keyword(s):  
Monte Carlo ◽  
Value At Risk ◽  
Scale Parameter ◽  
Monte Carlo Algorithm ◽  
Asian Options ◽  
Inverse Gaussian ◽  
Option Prices ◽  
Quasi Monte Carlo ◽  
Normal Inverse Gaussian Distribution ◽  
Normal Inverse Gaussian

We propose a quasi-Monte Carlo (qMC) algorithm to simulate variates from the normal inverse Gaussian (NIG) distribution. The algorithm is based on a Monte Carlo technique found in Rydberg [13], and is based on sampling three independent uniform variables. We apply the algorithm to three problems appearing in finance. First, we consider the valuation of plain vanilla call options and Asian options. The next application considers the problem of deriving implied parameters for the underlying asset dynamics based on observed option prices. We employ our proposed algorithm together with the Newton Method, and show how we can find the scale parameter of the NIG-distribution of the logreturns in case of a call or an Asian option. We also provide an extensive error analysis for this method. Finally we study the calculation of Value-at-Risk for a portfolio of nonlinear products where the returns are modeled by NIG random variables.

A quasi-Monte Carlo implementation of the ziggurat method

2018 ◽  
Vol 24 (2) ◽  
pp. 93-99
Author(s):  
Nguyet Nguyen ◽  
Linlin Xu ◽  
Giray Ökten
Keyword(s):  
Monte Carlo ◽  
Random Variable ◽  
Numerical Results ◽  
Quasi Monte Carlo ◽  
Low Discrepancy Sequences ◽  
Gamma Distributions ◽  
Variable Generation ◽  
Monte Carlo Implementation

Abstract The ziggurat method is a fast random variable generation method introduced by Marsaglia and Tsang in a series of papers. We discuss how the ziggurat method can be implemented for low-discrepancy sequences, and present algorithms and numerical results when the method is used to generate samples from the normal and gamma distributions.

scholarly journals A Multi-Index Quasi--Monte Carlo Algorithm for Lognormal Diffusion Problems

2017 ◽  
Vol 39 (5) ◽  
pp. S851-S872 ◽  
Author(s):  
Pieterjan Robbe ◽  
Dirk Nuyens ◽  
Stefan Vandewalle
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Algorithm ◽  
Multi Index ◽  
Quasi Monte Carlo ◽  
Diffusion Problems
Sign in / Sign up

Export Citation Format

Share Document