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 information on this web page is distributed under the MIT license.

Languages:

stochastic_diffusion is available in a C version and a C++ version and a Fortran77 version and a Fortran90 version and a MATLAB version and an Octave version and a Python version.

Related Data and Programs:

black_scholes, a Python code which implements some simple approaches to the Black-Scholes option valuation theory;

Reference:

1. 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 1005-1034.
2. Howard Elman, Darran Furnaval,
   Solving the stochastic steady-state diffusion problem using multigrid,
   IMA Journal on Numerical Analysis,
   Volume 27, Number 4, 2007, pages 675-688.
3. Roger Ghanem, Pol Spanos,
   Stochastic Finite Elements: A Spectral Approach,
   Revised Edition,
   Dover, 2003,
   ISBN: 0486428184,
   LC: TA347.F5.G56.
4. 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 3084-3113, 2009.
5. 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 2309-2345.
6. 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 4927-4948.

Source Code:

• stochastic_diffusion.py, the source code.
• stochastic_diffusion.sh, runs all the tests.
• stochastic_diffusion.txt, the output file.

• 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.