stochastic_diffusion

stochastic_diffusion, an Octave code which implement several versions of a diffusivity coefficient whose value depends on parameters that may be chosen stochastically.

In the 1D stochastic version of the problem, the diffusivity function rho() includes the influence of a stochastic parameter vector OMEGA:

        - d/dx ( rho(X;OMEGA) d/dx U(X;OMEGA) ) = F(X).
      

In the 2D stochastic version of the problem, the diffusivity function rho() includes the influence of a stochastic parameter vector OMEGA:

        - Del ( rho(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 MIT license

Languages:

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

Related Data and Programs:

stochastic_diffusion_test

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. 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.
4. 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:

• diffusivity_1d_xk.m, a 1D stochastic diffusivity function from Xiu and Karniadakis.
• diffusivity_1d_pwc.m, a 1D stochastic diffusivity function defined as an arbitrary piecewise constant function.
• diffusivity_2d_bnt.m, a 2D stochastic diffusivity function from Babuska, Nobile, and Tempone.
• diffusivity_2d_elman.m, a 2D stochastic diffusivity function from Elman.
• diffusivity_2d_jvb.m, 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.m, a 2D stochastic diffusivity function from Nobile, Tempone, and Webster.
• diffusivity_2d_pwc.m, a 2D stochastic diffusivity function defined as a piecwise constant function over a rectangle.