12,430 research outputs found
Spectral Condition Numbers of Orthogonal Projections and Full Rank Linear Least Squares Residuals
A simple formula is proved to be a tight estimate for the condition number of
the full rank linear least squares residual with respect to the matrix of least
squares coefficients and scaled 2-norms. The tight estimate reveals that the
condition number depends on three quantities, two of which can cause
ill-conditioning. The numerical linear algebra literature presents several
estimates of various instances of these condition numbers. All the prior values
exceed the formula introduced here, sometimes by large factors.Comment: 15 pages, 1 figure, 2 table
Modeling of Transitional Channel Flow Using Balanced Proper Orthogonal Decomposition
We study reduced-order models of three-dimensional perturbations in
linearized channel flow using balanced proper orthogonal decomposition (BPOD).
The models are obtained from three-dimensional simulations in physical space as
opposed to the traditional single-wavenumber approach, and are therefore better
able to capture the effects of localized disturbances or localized actuators.
In order to assess the performance of the models, we consider the impulse
response and frequency response, and variation of the Reynolds number as a
model parameter. We show that the BPOD procedure yields models that capture the
transient growth well at a low order, whereas standard POD does not capture the
growth unless a considerably larger number of modes is included, and even then
can be inaccurate. In the case of a localized actuator, we show that POD modes
which are not energetically significant can be very important for capturing the
energy growth. In addition, a comparison of the subspaces resulting from the
two methods suggests that the use of a non-orthogonal projection with adjoint
modes is most likely the main reason for the superior performance of BPOD. We
also demonstrate that for single-wavenumber perturbations, low-order BPOD
models reproduce the dominant eigenvalues of the full system better than POD
models of the same order. These features indicate that the simple, yet accurate
BPOD models are a good candidate for developing model-based controllers for
channel flow.Comment: 35 pages, 20 figure
Angles Between Infinite Dimensional Subspaces with Applications to the Rayleigh-Ritz and Alternating Projectors Methods
We define angles from-to and between infinite dimensional subspaces of a
Hilbert space, inspired by the work of E. J. Hannan, 1961/1962 for general
canonical correlations of stochastic processes. The spectral theory of
selfadjoint operators is used to investigate the properties of the angles,
e.g., to establish connections between the angles corresponding to orthogonal
complements. The classical gaps and angles of Dixmier and Friedrichs are
characterized in terms of the angles. We introduce principal invariant
subspaces and prove that they are connected by an isometry that appears in the
polar decomposition of the product of corresponding orthogonal projectors.
Point angles are defined by analogy with the point operator spectrum. We bound
the Hausdorff distance between the sets of the squared cosines of the angles
corresponding to the original subspaces and their perturbations. We show that
the squared cosines of the angles from one subspace to another can be
interpreted as Ritz values in the Rayleigh-Ritz method, where the former
subspace serves as a trial subspace and the orthogonal projector of the latter
subspace serves as an operator in the Rayleigh-Ritz method. The Hausdorff
distance between the Ritz values, corresponding to different trial subspaces,
is shown to be bounded by a constant times the gap between the trial subspaces.
We prove a similar eigenvalue perturbation bound that involves the gap squared.
Finally, we consider the classical alternating projectors method and propose
its ultimate acceleration, using the conjugate gradient approach. The
corresponding convergence rate estimate is obtained in terms of the angles. We
illustrate a possible acceleration for the domain decomposition method with a
small overlap for the 1D diffusion equation.Comment: 22 pages. Accepted to Journal of Functional Analysi
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