Polynomial Chaos Expansion using Legendre Polynomials

PCE_LEGENDRE is a MATLAB library which sets up the system matrix for a polynomial chaos expansion, using Legendre polynomials with a linear factor, applied to a 2D PDE with a stochastic diffusion coefficient.

We wish to analyze a stochastic PDE of the form:

        -div A(X,Y) grad U(X,Y) = F(X)

We let X be a space of finite element functions generated by piecewise linear functions associated with a particular triangular dissection of the unit square.

We let Y be the space of polynomials over R^N with total degree at most P.

We seek solutions U in XxY using a polynomial chaos expansion approach.


[ nxy, bxy, fxy ] = pce_legendre_linear_assemble ( n, p )
where the input is: and the output is


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


PCE_LEGENDRE is available in a MATLAB version.

Related Data and Programs:

CNOISE, a MATLAB library which generates samples of noise obeying a 1/f^alpha power law, by Miroslav Stoyanov.

LEGENDRE_POLYNOMIAL, a MATLAB library which evaluates the Legendre polynomial and associated functions.

LEGENDRE_RULE, a MATLAB program which computes a Gauss-Legendre quadrature rule.

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, by Gianluca Iaccarino.

PCE_ODE_HERMITE, a MATLAB program which sets up a simple scalar ODE for exponential decay with an uncertain decay rate, using a polynomial chaos expansion in terms of Hermite polynomials.

POLPAK, a MATLAB library which evaluates a variety of mathematical functions.

STOCHASTIC_DIFFUSION, a MATLAB library which implements several versions of a stochastic diffusivity coefficient.

SUBSET, a MATLAB library which enumerates combinations, partitions, subsets, index sets, and other combinatorial objects.


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    Handbook of Mathematical Functions,
    National Bureau of Standards, 1964,
    ISBN: 0-486-61272-4,
    LC: QA47.A34.
  2. Philip Davis, Philip Rabinowitz,
    Methods of Numerical Integration,
    Second Edition,
    Dover, 2007,
    ISBN: 0486453391,
    LC: QA299.3.D28.
  3. Roger Ghanem, Pol Spanos
    Stochastic Finite Elements: A Spectral Approach,
    Revised Edition,
    Dover, 2003,
    ISBN: 0486428184,
    LC: TA347.F5.G56.
  4. Dongbin Xiu,
    Numerical Methods for Stochastic Computations: A Spectral Method Approach,
    Princeton, 2010,
    ISBN13: 978-0-691-14212-8,
    LC: QA274.23.X58.
  5. Daniel Zwillinger, editor,
    CRC Standard Mathematical Tables and Formulae,
    30th Edition,
    CRC Press, 1996,
    ISBN: 0-8493-2479-3,
    LC: QA47.M315.

Source Code:

Examples and Tests:

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

Last modified on 13 March 2012.