PDE Model Reduction

POD_BASIS_FLOW is a FORTRAN90 program which applies the principal orthogonal direction (POD) analysis to a set of solutions of a PDE that models two dimensional time dependent fluid flow.

This procedure, originally devised by Karl Pearson, has arisen repeatedly in a variety of fields, and hence is known under various names, including:

You almost certainly should rather use the simpler program called SVD_BASIS, which does not assume that the data comes from a particular fluid flow problem!

The purpose of the algorithm is to extract the dominant modes of behavior of the system; these modes could then be used, for instance, in a reduced order model of the physical system.

For the computations considered here, a partial differential equation (PDE) has been defined, specifying the time-dependent flow of a fluid through a region. The PDE specification includes a parameter alpha whose value strongly affects the behavior of the flow. The steady state solution X0 is computed for a particular value of alpha. Then the time-dependent problem is solved over a fixed time interval, with alpha varying from time to time. A set of several hundred solutions X(T(I),alpha(I)) are saved.

The need is to try to extract from this solution data the typical modes of behavior of the solution, that is, a small set of orthogonal vectors V such that "most" of the solution vectors X can be well represented by a linear combination of elements of V. Such a set of modes may then be used as a finite element basis that is highly tuned to the physics of the problem, so that a very small set of basis functions can be used to closely approximate the behavior of the solution over a range of values of alpha.

The method of extracting information from the solution data begins by constructing an M by N matrix A, each of whose columns is one of the solution vectors X. Thus,

A = [ X1 | X2 | ... | XN ]

Then the singular value decomposition of A:

A = U * S * V'
is determined using the DGESVD routine from the linear algebra package LAPACK. A subset of the columns of the orthogonal M by M matrix U, associated with the largest singular values S, is chosen to form the POD basis.

Because the data comes from a finite element computation, and the results may be used as a new reduced basis, it may be desirable to carry out mass matrix preconditioning of the data, so that output POD vectors are orthogonal in the L2 inner product (integration of the product of the finite element functions over the domain).

The current version of the program assumes that a steady state solution SS of the PDE is known, and that a multiple of SS is to be subtracted from each solution vector before processing.

FILES: the program assumes the existence of the following files: (the actual names of the files are specified by the user at run time. The names used here are just suggestions.)

INPUT: at run time, the user specifies:

OUTPUT: the program computes basis_num basis vectors. The first vector is written to the file pod_001.txt; again, the output vectors are written with two values per line, since this represents the two components of velocity at a particular node.


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


POD_BASIS_FLOW is available in a FORTRAN90 version.

Related Data and Programs:

BRAIN_SENSOR_POD, a MATLAB program which applies the method of Proper Orthogonal Decomposition to seek underlying patterns in sets of 40 sensor readings of brain activity.

CVT_BASIS_FLOW, a FORTRAN90 program which is similar to POD_BASIS_FLOW but uses CVT techniques to do the analysis of the data.

LAPACK_EXAMPLES, a FORTRAN90 program which demonstrates the use of the LAPACK linear algebra library.

SVD_BASIS, a FORTRAN90 program which is a simpler version of this program; it does not assume that the underlying data represents solutions of a fluid flow problem.


  1. Edward Anderson, Zhaojun Bai, Christian Bischof, Susan Blackford, James Demmel, Jack Dongarra, Jeremy Du Croz, Anne Greenbaum, Sven Hammarling, Alan McKenney, Danny Sorensen,
    LAPACK User's Guide,
    Third Edition,
    SIAM, 1999,
    LC: QA76.73.F25L36
  2. John Burkardt, Max Gunzburger, Hyung-Chun Lee,
    Centroidal Voronoi Tessellation-Based Reduced-Order Modelling of Complex Systems,
    SIAM Journal on Scientific Computing,
    Volume 28, Number 2, 2006, pages 459-484.
  3. Gal Berkooz, Philip Holmes, John Lumley,
    The proper orthogonal decomposition in the analysis of turbulent flows,
    Annual Review of Fluid Mechanics,
    Volume 25, 1993, pages 539-575.
  4. Lawrence Sirovitch,
    Turbulence and the dynamics of coherent structures, Parts I-III,
    Quarterly of Applied Mathematics,
    Volume XLV, Number 3, 1987, pages 561-590.

Source Code:

Examples and Tests:

PDE solution datasets you may copy include:

Once the reduced basis set is computed, it is possible to set up new finite element problems in which the reduced basis set is used as the finite element basis. To see an example of such a calculation, refer to

This program has been run with a number of different datasets, and with various requirements as to normalization and so on. The purpose of most of the runs is to find a generator set of given size. The input and output of each run is stored in a separate subdirectory.

Each run of the code is stored in a separate subdirectory. Available runs include:

List of Routines:

You can go up one level to the FORTRAN90 source codes.

Last revised on 17 July 2004.