22,785 research outputs found
Penalized Orthogonal Iteration for Sparse Estimation of Generalized Eigenvalue Problem
We propose a new algorithm for sparse estimation of eigenvectors in
generalized eigenvalue problems (GEP). The GEP arises in a number of modern
data-analytic situations and statistical methods, including principal component
analysis (PCA), multiclass linear discriminant analysis (LDA), canonical
correlation analysis (CCA), sufficient dimension reduction (SDR) and invariant
co-ordinate selection. We propose to modify the standard generalized orthogonal
iteration with a sparsity-inducing penalty for the eigenvectors. To achieve
this goal, we generalize the equation-solving step of orthogonal iteration to a
penalized convex optimization problem. The resulting algorithm, called
penalized orthogonal iteration, provides accurate estimation of the true
eigenspace, when it is sparse. Also proposed is a computationally more
efficient alternative, which works well for PCA and LDA problems. Numerical
studies reveal that the proposed algorithms are competitive, and that our
tuning procedure works well. We demonstrate applications of the proposed
algorithm to obtain sparse estimates for PCA, multiclass LDA, CCA and SDR.
Supplementary materials are available online
Kernel methods for detecting coherent structures in dynamical data
We illustrate relationships between classical kernel-based dimensionality
reduction techniques and eigendecompositions of empirical estimates of
reproducing kernel Hilbert space (RKHS) operators associated with dynamical
systems. In particular, we show that kernel canonical correlation analysis
(CCA) can be interpreted in terms of kernel transfer operators and that it can
be obtained by optimizing the variational approach for Markov processes (VAMP)
score. As a result, we show that coherent sets of particle trajectories can be
computed by kernel CCA. We demonstrate the efficiency of this approach with
several examples, namely the well-known Bickley jet, ocean drifter data, and a
molecular dynamics problem with a time-dependent potential. Finally, we propose
a straightforward generalization of dynamic mode decomposition (DMD) called
coherent mode decomposition (CMD). Our results provide a generic machine
learning approach to the computation of coherent sets with an objective score
that can be used for cross-validation and the comparison of different methods
Optimal wavy surface to suppress vortex shedding using second-order sensitivity to shape changes
A method to find optimal 2nd-order perturbations is presented, and applied to
find the optimal spanwise-wavy surface for suppression of cylinder wake
instability. Second-order perturbations are required to capture the stabilizing
effect of spanwise waviness, which is ignored by standard adjoint-based
sensitivity analyses. Here, previous methods are extended so that (i) 2nd-order
sensitivity is formulated for base flow changes satisfying linearised
Navier-Stokes, and (ii) the resulting method is applicable to a 2D global
instability problem. This makes it possible to formulate 2nd-order sensitivity
to shape modifications. Using this formulation, we find the optimal shape to
suppress the a cylinder wake instability. The optimal shape is then perturbed
by random distributions in full 3D stability analysis to confirm that it is a
local optimal at the given amplitude and wavelength. Furthermore, it is shown
that none of the 10 random wavy shapes alone stabilize the wake flow at Re=50,
while the optimal shape does. At Re=100, surface waviness of maximum height 1%
of the cylinder diameter is sufficient to stabilize the flow. The optimal
surface creates streaks by passively extracting energy from the base flow
derivatives and effectively altering the tangential velocity component at the
wall, as opposed to spanwise-wavy suction which inputs energy to the normal
velocity component at the wall. This paper presents a fully two-dimensional and
computationally affordable method to find optimal 2nd-order perturbations of
generic flow instability problems and any boundary control (such as boundary
forcing, shape modulation or suction).Comment: 19 pages, 6 figure
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