Variational Characterization Of Singular Values

[Melanie Wendelberger]

Thisarticle first discusses the finite-dimensional case and its applications before considering compact operators on infinite-dimensional Hilbert spaces. We will see that for compact operators, the proof of the main theorem uses essentially the same idea from the finite-dimensional argument.

In the case that the operator is non-Hermitian, the theorem provides an equivalent characterization of the associated singular values. The min-max theorem can be extended to self-adjoint operators that are bounded below.

Let A be a compact, Hermitian operator on a Hilbert space H. Recall that the spectrum of such an operator (the set of eigenvalues) is a set of real numbers whose only possible cluster point is zero. It is thus convenient to list the positive eigenvalues of A as

The min-max theorem also applies to (possibly unbounded) self-adjoint operators.[1][2] Recall the essential spectrum is the spectrum without isolated eigenvalues of finite multiplicity. Sometimes we have some eigenvalues below the essential spectrum, and we would like to approximate the eigenvalues and eigenfunctions.

Recall that the singular value decomposition (SVD) of a matrix is a factorization , where and are unitary and , with , where where . We sometimes write to specify the matrix to which the singular value belongs.

A standard technique for obtaining singular value inequalities for is to apply eigenvalue inequalities to the Hermitian positive semidefinite matrices or , whose eigenvalues are the squares of the singular values of , or to the Hermitian matrix

Therefore Theorem 3 shows that removing a column from does not increase any singular value and that when no singular value decreases below . However, when the smallest singular value of may be less than the smallest singular value of .

The reference for Theorem 3 is Stanley C. Eisenstat and Ilse C. F. Ipsen,Relative Perturbation Techniques for Singular Value Problems, If you want to cite this blog post you can use this BibTeX entry:

The singular values of $F$ are the (square roots of ) eigenvalues of $F F^t,$ and the regularity of the latter have been studied half-to-death. See either T. Kato (perturbation theory of linear operators, ch. 1) or Golub-van Loan (Matrix Computations -- they almost certainly talk about singular values directly, without going through eigenvalues, but at worst talk about eigenvalues).

(top) The second RSV v2 of the observability matrix, premultiplied by the square root of the background error correlation matrix ?1/2. (bottom) The result of integrating these fields by the Eady model over a 12-h interval. The (right) lower buoyancy and (middle) upper buoyancy are shown. (left) The corresponding streamfunction field, where the numbers at the top left indicate the maximum magnitude of the streamfunction field.

The evolution of the KE for the cases where the true state is given by the most rapidly (a) growing and (b) decaying mode, and the background state has a phase error. The details are the same as in Fig. 4.

The singular values of the observability matrix that correspond to the first (solid) and second (dashed) pairs of RSVs that contribute to the analysis increment, plotted against the time of the initial observations. In all cases, the final set of observations are given at T + 12.

The singular values of the observability matrix that correspond to the first (solid) and second (dashed) pairs of RSVs that contribute to the analysis increment, plotted against the height of the horizontal line of observations.

The evolution of the KE growth rates for the cases where the background state values are all zero and the true state is given by (a) the most rapidly growing mode and (b) a PV-dipole perturbation. The true state (solid) and the analyses using the different specifications for the variance ratios (see figure labels) are shown.

4DVAR analyses of (top) nondimensional QGPV and (bottom) buoyancy for the cases where the true state is given by (a) an interior QGPV dipole perturbation that exhibits rapid finite-time nonmodal growth. The details are the same as in Fig. 12.

The extent to which the four-dimensional variational data assimilation (4DVAR) is able to use information about the time evolution of the atmosphere to infer the vertical spatial structure of baroclinic weather systems is investigated. The singular value decomposition (SVD) of the 4DVAR observability matrix is introduced as a novel technique to examine the spatial structure of analysis increments. Specific results are illustrated using 4DVAR analyses and SVD within an idealized 2D Eady model setting. Three different aspects are investigated. The first aspect considers correcting errors that result in normal-mode growth or decay. The results show that 4DVAR performs well at correcting growing errors but not decaying errors. Although it is possible for 4DVAR to correct decaying errors, the assimilation of observations can be detrimental to a forecast because 4DVAR is likely to add growing errors instead of correcting decaying errors. The second aspect shows that the singular values of the observability matrix are a useful tool to identify the optimal spatial and temporal locations for the observations. The results show that the ability to extract the time-evolution information can be maximized by placing the observations far apart in time. The third aspect considers correcting errors that result in nonmodal rapid growth. 4DVAR is able to use the model dynamics to infer some of the vertical structure. However, the specification of the case-dependent background error variances plays a crucial role.

The dynamical instability of the atmosphere means that small perturbations that are introduced into the flow may grow rapidly. For example, the flow at midlatitudes is baroclinically unstable due to the vertical shear associated with the meridional temperature gradient. This wave instability provides the dominant mechanism for disturbances to develop into midlatitude weather systems. In such development, the vertical spatial structure of the disturbance plays a fundamental role in governing the development. For example, the normal-mode analysis of simple linear models (Charney 1947; Eady 1949) showed that the fastest-growing structure exhibits a westward tilt with height in the pressure field. This vertical tilt leads to a process known as self-development where the upper- and lower-level waves act to intensify each other, leading to exponential modal growth. In contrast, the fastest-decaying structure exhibits an eastward tilt with height so that the circulations associated with the upper- and lower-level waves act to weaken each other.

More recent studies (Farrell 1982, 1984) showed that it is possible for the growth rate of a disturbance to exceed the exponential growth of the fastest-growing normal mode over a limited period of time. Such disturbances may be found by computing the singular vectors of the linear model (Farrell 1989; Buizza and Palmer 1995). The spatial structures of these disturbances are characterized by localized tilted interior potential vorticity (PV) structures that separate and become vertically aligned under the action of the shear, leading to amplification (Badger and Hoskins 2001). Again, the vertical structure of the disturbance is fundamental for this rapid nonmodal growth. In the presence of model and previous analysis error, it is likely that errors in the background state contain such perturbations that result in rapid growth. Thus, it is vital that data assimilation algorithms are able to analyze the rich vertical structure of such anomalies. The new generation of vertical sounders on satellites are able to provide data with a high vertical resolution, which should be useful for analyzing anomalies with small vertical scales. However, the maximum initial amplitudes of singular vectors typically occur in regions of strong baroclinicity (Palmer 1996), and hence where the data are contaminated by cloud. Thus, the analysis of such structures still remains a significant challenge.

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