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PCE_ODE_HERMITE\ Hermite Polynomial Chaos Expansion for a Scalar ODE {#pce_ode_hermite-hermite-polynomial-chaos-expansion-for-a-scalar-ode align=”center”} ===================================================

PCE_ODE_HERMITE is a C++ library which defines and solves a time-dependent scalar exponential decay ODE with uncertain decay coefficient, using a polynomial chaos expansion, in terms of Hermite polynomials.

The deterministic equation is

        du/dt = - alpha * u,
        u(0) = u0

In the stochastic version, it is assumed that the decay coefficient ALPHA is a Gaussian random variable with mean value ALPHA_MU and variance ALPHA_SIGMA\^2.

The exact expected value of the stochastic equation is known to be

        u(t) = u0 * exp ( t^2/2)

This should be matched by the first component of the polynomial chaos expansion.

Licensing: {#licensing align=”center”}

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

Languages: {#languages align=”center”}

PCE_ODE_HERMITE is available in a C version and a C++ version and a FORTRAN77 version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs: {#related-data-and-programs align=”center”}

BLACK_SCHOLES, a C++ library which implements some simple approaches to the Black-Scholes option valuation theory, by Desmond Higham.

HERMITE_POLYNOMIAL, a C++ library which evaluates the physicist’s Hermite polynomial, the probabilist’s Hermite polynomial, the Hermite function, and related functions.

PCE_BURGERS, a C++ program which defines and solves a version of the time-dependent viscous Burgers equation, with uncertain viscosity, using a polynomial chaos expansion in terms of Hermite polynomials, by Gianluca Iaccarino.

SDE, a C++ library which illustrates the properties of stochastic differential equations (SDE’s), and common algorithms for their analysis, by Desmond Higham;

STOCHASTIC_RK, a C++ library which applies a Runge Kutta (RK) scheme to a stochastic differential equation.

Reference: {#reference align=”center”}

1. Roger Ghanem, Pol Spanos,\ Stochastic Finite Elements: A Spectral Approach,\ Revised Edition,\ Dover, 2003,\ ISBN: 0486428184,\ LC: TA347.F5.G56.
2. Dongbin Xiu,\ Numerical Methods for Stochastic Computations: A Spectral Method Approach,\ Princeton, 2010,\ ISBN13: 978-0-691-14212-8,\ LC: QA274.23.X58.

Source Code: {#source-code align=”center”}

• pce_ode_hermite.cpp, the source code.
• pce_ode_hermite.hpp, the source code.

Examples and Tests: {#examples-and-tests align=”center”}

• pce_ode_hermite_prb.cpp, a sample calling program.
• pce_ode_hermite_prb_output.txt, the output file.

List of Routines: {#list-of-routines align=”center”}

• HE_DOUBLE_PRODUCT_INTEGRAL: integral of He(i,x)*He(j,x)*e\^(-x\^2/2).
• HE_TRIPLE_PRODUCT_INTEGRAL: integral of He(i,x)*He(j,x)*He(k,x)*e\^(-x\^2/2).
• PCE_ODE_HERMITE applies the polynomial chaos expansion to a scalar ODE.
• R8_FACTORIAL computes the factorial of N.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

You can go up one level to the C++ source codes.

Last modified on 18 March 2012.