Irad Yavneh

Baroclinic Instability and Loss of Balance

Abstract Under the influences of stable density stratification and the earth’s rotation, large-scale flows in the ocean and atmosphere have a mainly balanced dynamics—sometimes called the slow manifold—in the sense that there are diagnostic hydrostatic and gradient-wind momentum balances that constrain the fluid acceleration. The nonlinear balance equations are a widely successful, approximate model for this regime, and mathematically explicit limits of their time integrability have been identified. It is hypothesized that these limits are indicative, at least approximately, of the transition from the larger-scale regime of inverse energy cascades by anisotropic flows to the smaller-scale regime of forward energy cascade to dissipation by more nearly isotropic flows and intermittently breaking inertia–gravity waves. This paper analyzes the particular example of an unbalanced instability of a balanced, horizontally uniform, vertically sheared current, as it occurs within the Boussinesq equations. This ageo...

A Plurality of Sparse Representations Is Better Than the Sparsest One Alone

Cleaning of noise from signals is a classical and long-studied problem in signal processing. Algorithms for this task necessarily rely on an a priori knowledge about the signal characteristics, along with information about the noise properties. For signals that admit sparse representations over a known dictionary, a commonly used denoising technique is to seek the sparsest representation that synthesizes a signal close enough to the corrupted one. As this problem is too complex in general, approximation methods, such as greedy pursuit algorithms, are often employed. In this line of reasoning, we are led to believe that detection of the sparsest representation is key in the success of the denoising goal. Does this mean that other competitive and slightly inferior sparse representations are meaningless? Suppose we are served with a group of competing sparse representations, each claiming to explain the signal differently. Can those be fused somehow to lead to a better result? Surprisingly, the answer to this question is positive; merging these representations can form a more accurate (in the mean-squared-error (MSE) sense), yet dense, estimate of the original signal even when the latter is known to be sparse. In this paper, we demonstrate this behavior, propose a practical way to generate such a collection of representations by randomizing the Orthogonal Matching Pursuit (OMP) algorithm, and produce a clear analytical justification for the superiority of the associated Randomized OMP (RandOMP) algorithm. We show that while the maximum a posteriori probability (MAP) estimator aims to find and use the sparsest representation, the minimum mean- squared-error (MMSE) estimator leads to a fusion of representations to form its result. Thus, working with an appropriate mixture of candidate representations, we are surpassing the MAP and tending towards the MMSE estimate, and thereby getting a far more accurate estimation in terms of the expected lscr2 -norm error.

Anisotropy and coherent vortex structures in planetary turbulence

High-resolution numerical simulations were made of unforced, planetary-scale fluid dynamics. In particular, the simulation was based on the quasi-geostrophic equations for a Boussinesq fluid in a uniformly rotating and stably stratified environment, which is an idealization for large regions of either the atmosphere or ocean. The solutions show significant discrepancies from the long-standing theoretical prediction of isotropy. The discrepancies are associated with the self-organization of the flow into a large population of coherent vortices. Their chaotic interactions govern the subsequent evolution of the flow toward a final configuration that is nonturbulent.

Multigrid multidimensional scaling

Multidimensional scaling (MDS) is a generic name for a family of algorithms that construct a configuration of points in a target metric space from information about inter-point distances measured in some other metric space. Large-scale MDS problems often occur in data analysis, representation and visualization. Solving such problems efficiently is of key importance in many applications. In this paper we present a multigrid framework for MDS problems. We demonstrate the performance of our algorithm on dimensionality reduction and isometric embedding problems, two classical problems requiring efficient large-scale MDS. Simulation results show that the proposed approach significantly outperforms conventional MDS algorithms. Copyright

Accelerated multigrid convergence and high-Reynolds recirculating flows

Techniques are developed for accelerating multigrid convergence in general, and for advection-diffusion and incompressible flow problems with small viscosity in particular. It is shown by analysis that the slowing down of convergence is due mainly to poor coarse-grid correction to certain error components, and means for dealing with this problem are suggested, analyzed, and tested by numerical experiments, showing very significant improvement in convergence rates at little cost.

On multigrid solution of high-reynolds incompressible entering flows☆

An approach is presented for effectively separating the solution process of the elliptic component of high-Reynolds incompressible steady entering flow, for which classical multigrid techniques are well-suited, from that of the non-elliptic part, for which other methods are more effective. It is shown by analysis and numerical calculations that such an approach is very effective in terms of asymptotic convergence as well as reduction of errors well below discretization level in a 1 FMG algorithm.

On red-black SOR smoothing in multigrid

Optimal relaxation parameters are obtained for red-black Gauss-Seidel relaxation in multigrid solvers of a family of elliptic equations. The resulting relaxation schemes are found to retain very high efficiency over an appreciable range of coefficients of the elliptic differential operator, yielding simple, inexpensive, and fully parallelizable smoothers in many situations where less cost-effective block- and alternating-direction schemes are commonly used.

Non-axisymmetric instability of centrifugally stable stratified Taylor-Couette flow

The stability is investigated of the swirling flow between two concentric cylinders in the presence of stable axial linear density stratication, for flows not satisfying the well-known Rayleigh criterion for inviscid centrifugal instability, d(Vr) 2 =d r< 0. We show by a linear stability analysis that a sucient condition for non-axisymmetric instability is, in fact, d(V=r) 2 =d r< 0, which implies a far wider range of instability than previously identied. The most unstable modes are radially smooth and occur for a narrow range of vertical wavenumbers. The growth rate is nearly independent of the stratication when the latter is strong, but it is proportional to it when it is weak, implying stability for an unstratied flow. The instability depends strongly on a nondimensional parameter,S, which represents the ratio between the strain rate and twice the angular velocity of the flow. The instabilities occur for anti-cyclonic flow ( S< 0). The optimal growth rate of the fastest-growing mode, which is non-oscillatory in time, decays exponentially fast as S (which can also be considered a Rossby number) tends to 0. The mechanism of the instability is an arrest and phase-locking of Kelvin waves along the boundaries by the mean shear flow. Additionally, we identify a family of (probably innitely many) unstable modes with more oscillatory radial structure and slower growth rates than the primary instability. We determine numerically that the instabilities persist for nite viscosity, and the unstable modes remain similar to the inviscid modes outside boundary layers along the cylinder walls. Furthermore, the nonlinear dynamics of the anti-cyclonic flow are dominated by the linear instability for a substantial range of Reynolds numbers.

Closed-Form MMSE Estimation for Signal Denoising Under Sparse Representation Modeling Over a Unitary Dictionary

This paper deals with the Bayesian signal denoising problem, assuming a prior based on a sparse representation modeling over a unitary dictionary. It is well known that the maximum a posteriori probability (MAP) estimator in such a case has a closed-form solution based on a simple shrinkage. The focus in this paper is on the better performing and less familiar minimum-mean-squared-error (MMSE) estimator. We show that this estimator also leads to a simple formula, in the form of a plain recursive expression for evaluating the contribution of every atom in the solution. An extension of the model to real-world signals is also offered, considering heteroscedastic nonzero entries in the representation, and allowing varying probabilities for the chosen atoms and the overall cardinality of the sparse representation. The MAP and MMSE estimators are redeveloped for this extended model, again resulting in closed-form simple algorithms. Finally, the superiority of the MMSE estimator is demonstrated both on synthetically generated signals and on real-world signals (image patches).

Practical spherical embedding of manifold triangle meshes

Gotsman et al. (SIGGRAPH 2003) presented the first method to generate a provably bijective parameterization of a closed genus-0 manifold mesh to the unit sphere. This involves the solution of a large system of non-linear equations. However, they did not show how to solve these equations efficiently, so, while theoretically sound, the method has remained impractical till now. We show why simple iterative methods to solve the equations are bound to fail, and provide an efficient numerical scheme that succeeds. Our method uses a number of optimization methods combined with an algebraic multigrid technique. With these, we are able to spherically parameterize meshes containing up to a hundred thousand vertices in a matter of minutes.