PCE_ODE_HERMITE
Hermite Polynomial Chaos Expansion for a Scalar ODE

PCE_ODE_HERMITE is a MATLAB 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:

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

Languages:

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:

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

HERMITE_POLYNOMIAL, a MATLAB library which evaluates the physicist's Hermite polynomial, the probabilist's Hermite polynomial, the Hermite function, and related functions.

PCE_BURGERS, a MATLAB 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.

PCE_LEGENDRE, a MATLAB program which assembles the system matrix of a 2D stochastic PDE, using a polynomal chaos expansion in terms of Legendre polynomials;

SDE, a MATLAB library which illustrates the properties of stochastic differential equations, and common algorithms for their analysis, by Desmond Higham;

STOCHASTIC_DIFFUSION, MATLAB functions which implement several versions of a stochastic diffusivity coefficient.

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

Reference:

Roger Ghanem, Pol Spanos,
Stochastic Finite Elements: A Spectral Approach,
Revised Edition,
Dover, 2003,
ISBN: 0486428184,
LC: TA347.F5.G56.
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:

he_double_product_integral.m, returns integral of He(i,x)*He(j,x)*e^(-x^2/2).
he_triple_product_integral.m, returns integral of He(i,x)*He(j,x)*He(k,x)*e^(-x^2/2).
pce_ode_hermite.m, the main program, which sets up and solves the problem.
r8_factorial.m, computes the factorial function.
timestamp.m, prints the current YMDHMS date as a time stamp.

Examples and Tests:

pce_ode_hermite_test.m, calls all the tests.
pce_ode_hermite_test01.m, tests pce_ode_hermite by plotting the computed and exact expected values over the interval [0,2].
pce_ode_hermite_test02.m, tests pce_ode_hermite by computing the solution up to TF = 2.0 with a sequence of increasing PCE degrees, and then plotting the errors between the computed and exact expected value there.
pce_ode_hermite_output.txt, the output file.
expected_value_plot.png, a PNG plot comparing the computed expected value (the first PCE coefficient at each time) versus the known expected value.
convergence_plot.png, a PNG plot comparing, at time T=2, the known expected value versus the expected values computed by a sequence of PCE's of increasing degree.

You can go up one level to the MATLAB source codes.

Last modified on 16 March 2012.