stochastic_diffusion
stochastic_diffusion,
a Python code which
implements several versions of a stochastic diffusivity coefficient.
In the 1D stochastic version of the problem, the diffusivity function DC()
includes the influence of stochastic parameters:
 d/dx ( DC(X;OMEGA) d/dx U(X;OMEGA) ) = F(X).
In the 2D stochastic version of the problem, the diffusivity function DC()
includes the influence of stochastic parameters:
 Del ( DC(X,Y;OMEGA) Del U(X,Y;OMEGA) ) = F(X,Y).
Licensing:
The computer code and data files described and made available on this web page
are distributed under
the GNU LGPL license.
Languages:
stochastic_diffusion is available in
a C version and
a C++ version and
a FORTRAN90 version and
a MATLAB version and
a Python version.
Related Data and Programs:
black_scholes,
a Python code which
implements some simple approaches to
the BlackScholes option valuation theory;
Reference:

Ivo Babuska, Fabio Nobile, Raul Tempone,
A Stochastic Collocation Method for Elliptic Partial Differential Equations
with Random Input Data,
SIAM Journal on Numerical Analysis,
Volume 45, Number 3, 2007, pages 10051034.

Howard Elman, Darran Furnaval,
Solving the stochastic steadystate diffusion problem using multigrid,
IMA Journal on Numerical Analysis,
Volume 27, Number 4, 2007, pages 675688.

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

Xiang Ma, Nicholas Zabaras,
An adaptive hierarchical sparse grid collocation algorithm for the solution
of stochastic differential equations,
Journal of Computational Physics,
Volume 228, pages 30843113, 2009.

Fabio Nobile, Raul Tempone, Clayton Webster,
A Sparse Grid Stochastic Collocation Method for Partial Differential
Equations with Random Input Data,
SIAM Journal on Numerical Analysis,
Volume 46, Number 5, 2008, pages 23092345.

Dongbin Xiu, George Karniadakis,
Modeling uncertainty in steady state diffusion problems via
generalized polynomial chaos,
Computer Methods in Applied Mechanics and Engineering,
Volume 191, 2002, pages 49274948.
Source Code:

diffusivity_1d_pwc.png,
a 1D piecewise constant stochastic diffusivity function.

diffusivity_1d_xk.png,
a 1D stochastic diffusivity function from Xiu and Karniadakis.

diffusivity_2d_bnt.png,
a 2D stochastic diffusivity function from Babuska, Nobile, and Tempone.

diffusivity_2d_elman.png,
a 2D stochastic diffusivity function from Elman.

diffusivity_2d_jvb.png,
a 2D stochastic diffusivity function that generalizes the
Babuska, Nobile, and Tempone, by allowing an arbitrary number of
coefficients in the KL expansion.

diffusivity_2d_ntw.png,
a 2D stochastic diffusivity function from Nobile, Tempone, and Webster.

diffusivity_2d_pwc.png,
a 2D piecewise constant stochastic diffusivity function.
Last modified on 24 March 2019.