### Compressive sampling and dynamic mode decomposition

```Compressive sampling and
dynamic mode decomposition
Steven L. Brunton1, Joshua L. Proctor2, J. Nathan Kutz1，
Journal of Computational Dynamics, Submitted in DEC,2013.
Dynamic Mode Decomposition
 Motivation
• Fluid simulation is very important in engineering. in that
most fluid mechanical systems of interest are
described by the same governing equations: the
Navier–Stokes equations.
• Unfortunately, the Navier–Stokes equations are a set of
nonlinear partial differential equations that give rise to
all manner of dynamics, including those characterized
by bifurcations, limit cycles, resonances, and full-blown
turbulence. In some cases, It’s difficult to find analytic
solutions. This forces people to rely on experiments and
high-performance computations when studying such
systems.
Dynamic Mode Decomposition
 Motivation
• As experiments and computations become more
advanced, they generate ever increasing amounts of
data. Manipulating such data to find anything but the
most obvious trends requires a skillset all its own. This
has led to a growing need for data-driven methods
that can take a dataset and characterize it in
meaningful ways with minimal guidance.
Dynamic Mode Decomposition
 Definition
• Many in fluid mechanics have turned to modal
decomposition as the tool of choice for data-driven
analysis. It is a powerful new technique introduced in
the fluid dynamics community to isolate spatially
coherent modes that oscillate at a fixed frequency.
• Generally, a modal decomposition takes a set of data
and from it computes a set of modes, or characteristic
features. The meaning of the modes depends on the
particular type of decomposition used. However, in all
cases, the hope is that the modes identify features of
the data that elucidate the underlying physics.
Dynamic Mode Decomposition
• DMD is a modal decomposition developed specifically
for analyzing the dynamics of nonlinearly evolving fluid
flows. It was first introduced in 2008.
• Proper orthogonal decomposition (POD), perhaps the
most common modal decomposition in the fluids
community. but the POD modes are not necessarily
optimal for modeling those dynamics. For instance,
when analyzing a time-series, the POD modes remain
unchanged if the data are reordered; the modes do
not depend on the time evolution/dynamics encoded
in the data.
Dynamic Mode Decomposition
 features of DMD
• The DMD is a data-driven and equation-free method
that applies equally well to data from experiments or
simulations. An underlying principle is that even if the
data is high-dimensional, it may be described by a low
dimensional attractor subspace defined by a few
coherent structures.
• DMD not only provides modes, but also a linear model
for how the modes evolve n time.
Compressive Sampling and DMD
 Main contribution
• This Paper deviates from the prior studies combining
compressive sampling and DMD, in that we utilize
sparsity of the spatial coherent structures to
reconstruct full-state DMD modes from few
measurements. The resulting DMD eigenvalues are
equal to DMD eigenvalues from the full-state data. It
is then possible to reconstruct full-state DMD
eigenvectors using l1 minimization.
 Compute DMD
Consider the following data snapshot matrices:
 Compute DMD
 Compressive sampling
• The theory of compressive sampling suggests that instead of
measuring the high-dimensional signal x and then compressing,
it is possible to measure a low-dimensional subsample or random
projection of the data and then directly solve for the few nonzero coefficients in the transform basis.
C is the measurement matrix. If x is sparse, then we would like
to solve the underdetermined system of equations
 Compressive DMD
• This paper establish basic connections between the DMD on fullstate and projected data. These connections facilitate the two
main applied results of this work:
 It is possible to compute DMD on projected data and
reconstruct full-state DMD modes through compressive
sampling
 If full-state measurements are available, it is advantageous
to compress the data, compute the projected DMD, and
then compute full-state DMD modes by linearly combining
snapshots according to the projected DMD transforms.
 Compressive DMD
 Compressive DMD
Invariance of DMD to unitary transformations
 Compressive DMD
Various approaches and algorithms
Compressed DMD
Compressive sampling DMD
Experiments
Double gyre flow
Experiments
Double gyre flow
Experiments
Double gyre flow
Experiments
Flow around a cylinder
Experiments
Flow around a cylinder
```