STOCHASTIC_DIFFUSION
Stochastic Diffusivity

STOCHASTIC_DIFFUSION is a MATLAB library which implement several versions of a stochastic diffusivity coefficient.

The 1D diffusion equation has the form

        - d/dx ( A(X) d/dx U(X) ) = F(X).
      

In the 1D stochastic version of the problem, the diffusivity function includes the influence of stochastic parameters:

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

The 2D diffusion equation has the form

        - Del ( A(X,Y) Del U(X,Y) ) = F(X,Y).
      

In the 2D stochastic version of the problem, the diffusivity function includes the influence of stochastic parameters:

        - Del ( A(X,Y;OMEGA) Del U(X,Y;OMEGA) ) = F(X,Y).
      

The present codes are currently just sketches or untested functions. More work and documentation is needed.

Licensing:

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

Related Data and Programs:

BLACK_SCHOLES, a MATLAB library which implements some simple approaches to the Black-Scholes option valuation theory;

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

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

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.

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

STOCHASTIC_GRADIENT_ND_NOISE, a MATLAB program which solves an optimal control problem involving a functional over a system with stochastic noise.

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:

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

The BNT example is used in a rectangle [-1.5,0]x[-0.4,0.8], with zero source term, and somewhat complicated boundary conditions:

• du/dn = 0 for ( -1.5 <= x <= -1.0 ) and y = 0.8.
• u = 0 for -1.0 <= x <= -0.5 and y = 0.8.
• du/dn = 0 for ( -0.5 <= x <= 0.0 ) and y = 0.8.
• du/dn = 0 for x = -1.5 or for x = 0.
• - a * du/dn = 1 for -1.5 <= x <= 0, y = -0.4.

• diffusivity_2d_bnt.m, a 2D stochastic diffusivity function from Babuska, Nobile, and Tempone.
• bnt_contour.m, a function which plots two realizations of the diffusivity, one using a uniform PDF, the second using a Gaussian (normal) PDF.
• bnt_contour_uniform.png, the contour plot for a diffusivity using a uniform PDF.
• bnt_contour_gaussian.png, the contour plot for a diffusivity using a Gaussian PDF.

• diffusivity_2d_elman.m, a 2D stochastic diffusivity function from Elman.

The NTW example is formally 2D but the diffusivity depends only on X. The full problem is the diffusion equation in a square [0,1]x[0,1] with zero boundary conditions and a deterministic right hand side f(x,y)=cos(x)*sin(y).

• diffusivity_2d_ntw.m, a 2D stochastic diffusivity function from Nobile, Tempone, and Webster.
• ntw_contour.m, a function which plots a realization of the diffusivity.
• ntw_contour.png, the contour plot for a diffusivity.

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