(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.
URL:mediatum.ub.tum.de/doc/1482652/1482652.pdf
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.
DOI:10.1007/s00162-017-0432-2
Kou, J., & Zhang, W. (2017).
An improved criterion to select dominant modes from dynamic mode
decomposition.
European Journal of Mechanics-B/Fluids, 62, 109-129.
DOI:10.1016/j.euromechflu.2016.11.015
Hemati, M. S., Williams, M. O., & Rowley, C. W. (2014).
Dynamic mode decomposition for large and streaming datasets.
Physics of Fluids, 26(11), 111701.
DOI:10.1063/1.4901016
Parallel classical Gram-Schmidt process (tag:Ka):
Katagiri, T. (2003).
Performance evaluation of parallel Gram-Schmidt re-orthogonalization
methods.
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.
DOI:10.1007/3-540-36569-9_19
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
architectures.
2013 IEEE International Conference on Big Data.
DOI:10.1109/bigdata.2013.6691583
Demmel, J., Grigori, L., Hoemmen, M., & Langou, J. (2012).
Communication-optimal parallel and sequential QR and LU
factorizations.
SIAM Journal on Scientific Computing, 34(1), A206-A239.
DOI:10.1137/080731992
DMD properties:
Brunton S. L. (2018).
Dynamic mode decomposition overview.
Seattle, Washington: University of Washington.
youtu.be/sQvrK8AGCAo (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:
Frequency Magnitude 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:
STDMD1
{
// 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.
wallShearStress, are currently not available.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.STDMD matrices are not supported.STDMD.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.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.Tutorial:
Source code:
History
1.9.5