(Beta Release) STDMD (i.e. Streaming Total Dynamic Mode Decomposition) is a variant of a data-driven dimensionality reduction method.

STDMD is being used as a mathematical post-processing tool to compute a set of dominant modes out of a given flow (or dataset) each of which is associated with a constant frequency and decay rate, so that dynamic features of a given flow may become interpretable, and tractable. Among other Dynamic Mode Decomposition (DMD) variants, STDMD is presumed to provide the general DMD method capabilities alongside economised and feasible memory and CPU usage.

The code implementation corresponds to Figs. 15-16 of the first citation below, more broadly to Section 2.4.


STDMD and mode-sorting algorithms (tags:K, HRDC, KZ, HWR):
    Kiewat, M. (2019).
    Streaming modal decomposition approaches for vehicle aerodynamics.
    PhD thesis. Munich: Technical University of Munich.

    Hemati, M. S., Rowley, C. W., Deem, E. A., & Cattafesta, L. N. (2017).
    De-biasing the dynamic mode decomposition for applied Koopman
    spectral analysis of noisy datasets.
    Theoretical and Computational Fluid Dynamics, 31(4), 349-368.

    Kou, J., & Zhang, W. (2017).
    An improved criterion to select dominant modes from dynamic mode
    European Journal of Mechanics-B/Fluids, 62, 109-129.

    Hemati, M. S., Williams, M. O., & Rowley, C. W. (2014).
    Dynamic mode decomposition for large and streaming datasets.
    Physics of Fluids, 26(11), 111701.

Parallel classical Gram-Schmidt process (tag:Ka):
    Katagiri, T. (2003).
    Performance evaluation of parallel Gram-Schmidt re-orthogonalization
    In: Palma J. M. L. M., Sousa A. A., Dongarra J., Hernández V. (eds)
    High Performance Computing for Computational Science — VECPAR 2002.
    Lecture Notes in Computer Science, vol 2565, p. 302-314.
    Berlin, Heidelberg: Springer.

Parallel direct tall-skinny QR decomposition (tags:BGD, DGHL):
    Benson, A. R., Gleich, D. F., & Demmel, J. (2013).
    Direct QR factorizations for tall-and-skinny matrices in MapReduce
    2013 IEEE International Conference on Big Data.

    Demmel, J., Grigori, L., Hoemmen, M., & Langou, J. (2012).
    Communication-optimal parallel and sequential QR and LU
    SIAM Journal on Scientific Computing, 34(1), A206-A239.

DMD properties:
    Brunton S. L. (2018).
    Dynamic mode decomposition overview.
    Seattle, Washington: University of Washington. (Retrieved:24-04-20)


Operand Type Location
input {vol,surface}<Type>Field $FOAM_CASE/<time>/<inpField>
output file dat $FOAM_CASE/postProcessing/<FO>/<time>/<file>(s)
output field volScalarField(s) $FOAM_CASE/<time>/<outField>(s)

where <Type>=Scalar/Vector/SphericalTensor/SymmTensor/Tensor.

Output fields:

modeReal<modeIndex><field> | Real part of a mode field
modeImag<modeIndex><field> | Imaginary part of a mode field

Output files:

uSTDMD.dat                 | Unfiltered STDMD output
STDMD.dat                  | Filtered STDMD output

wherein for each mode, the following quantities are output into files:

Amplitude (real part)
Amplitude (imaginary part)
Eigenvalue (real part)
Eigenvalue (imaginary part)


Example of the STDMD function object by using functions sub-dictionary in system/controlDict file:

    // Mandatory entries (unmodifiable)
    type                STDMD;
    libs                (fieldFunctionObjects);
    field               <inpField>;

    // Conditional mandatory entries (unmodifiable)
    // either of the below
    stdmdInterval       5.5;
    executeInterval     10;

    // Optional entries (unmodifiable)
    modeSorter          kiewat;
    nModes              50;
    maxRank             50;
    nGramSchmidt        5;
    fMin                0;
    fMax                1000000000;

    // Optional entries (run-time modifiable, yet not recommended)
    testEigen           false;
    dumpEigen           false;
    minBasis            0.00000001;
    minEVal             0.00000001;
    absTol              0.001;
    relTol              0.0001;
    sortLimiter         500;

    // Optional (inherited) entries
    writePrecision  8;
    writeToFile     true;
    useUserTime     true;
    region          region0;
    enabled         true;
    log             true;
    timeStart       0;
    timeEnd         1000;
    executeControl  timeStep;
    writeControl    timeStep;
    writeInterval   1;

where the entries mean:

Property Description Type Required Default
type Type name: STDMD word yes -
libs Library name: fieldFunctionObjects word yes -
field Name of the operand field word yes -
stdmdInterval STDMD time-step size [s] scalar conditional executeInterval*(current time-step of the simulation)
modeSorter Mode-sorting algorithm variant word no firstSnap
nModes Number of output modes in input freq range label no GREAT
maxRank Max columns in accumulated matrices label no GREAT
nGramSchmidt Number of Gram-Schmidt iterations label no 5
fMin Min (non-negative) output frequency label no 0
fMax Max output frequency label no GREAT
testEigen Flag to verify eigendecompositions bool no false
dumpEigen Flag to log operands of eigendecomps bool no false
minBasis Orthogonal basis expansion threshold scalar no 1e-8
minEVal Min EVal for below EVals are omitted scalar no 1e-8
absTol Min abs tol in eigendecomposition tests scalar no 1e-4
relTol Relative tol in eigendecomposition tests scalar no 1e-6
sortLimiter Maximum allowable magnitude for mag(eigenvalue)*(number of STDMD steps) to be used in modeSorter={kiewat,kouZhang} to avoid overflow errors scalar no 500.0

The inherited entries are elaborated in:

Options for the modeSorter entry:

kiewat    | Modified weighted amplitude scaling method
kouZhang  | Weighted amplitude scaling method
firstSnap | Method of first snapshot amplitude magnitude

Usage by postProcess utility is not available.

Notes on entries

  • Operations on boundary fields, e.g. wallShearStress, are currently not available.
  • To specify the STDMD time-step size (not necessarily equal to the time step of the simulation), entries of either stdmdInterval or executeInterval must be available (highest to lowest precedence). While stdmdInterval allows users to directly specify the STDMD time-step size in seconds, in absence of stdmdInterval, for convenience, executeInterval allows users to compute the STDMD time-step internally by multiplying itself with the current time-step size of the simulation.
  • Limitation: Restart is currently not available since intermediate writing of STDMD matrices are not supported.
  • Limitation: Non-physical input (e.g. full-zero fields) may upset STDMD.
  • Warning: DMD is an active research area at the time of writing; therefore, there would be cases whereat oddities might be encountered.
  • Warning: This STDMD release is the beta release; therefore, small-to-medium changes in input/output interfaces and internal structures should be expected in the next versions.
  • Warning: In modeSorter algorithms of kiewat and kouZhang, there exists a function of power(x,y) where x is the magnitude of an eigenvalue, and y is the number of STDMD snapshots. This function poses a risk of overflow errors since, for example, if x ends up above 1.5 with 500 snapshots or more, this function automatically throws the floating point error meaning overflow. Therefore, the domain-range of this function is heuristically constrained by the optional entry sortLimiter which skips the evaluation of this function for a given mag(eigenvalue)*(no. of STDMD steps), i.e. x*y, whose magnitude is larger than sortLimiter.

Further information


Source code:

  • Foam::functionObjects::STDMD


  • Introduced in version v2006