The BMO-discrepancy suffers from the curse of dimensionality

We show that the minimal discrepancy of a point set in the d-dimensional unit cube with respect to the BMO seminorm suffers from the curse of dimensionality.     

The curse of dimensionality for numerical integration on general domains

We prove the curse of dimensionality in the worst case setting for multivariate numerical integration for various classes of smooth functions. We prove the results when the domains are isotropic convex bodies with small diameter satisfying a universal ${\psi }_2$-estimate. In particular, we obtain the result for the important class of volume-normalized ${\ell }_p^d$-balls in the complete regime $2\le p\le \infty$. This extends a result in a work of Hinrichs et al. (2014) to the whole range $2\le p\le \infty$, and additionally provides a unified approach. The key ingredient in the proof is a deep result from the theory of Asymptotic Geometric Analysis, the thin-shell volume concentration estimate due to O. Guédon and E. Milman. The connection of Asymptotic Geometric Analysis and Information-based Complexity revealed in this work seems promising and is of independent interest.     

Counting polycubes without the dimensionality curse

d-dimensional polycubes are the generalization of planar polyominoes to higher dimensions. That is, a d-D polycube of size n is a connected set of n cells of a d-dimensional hypercubic lattice, where connectivity is through (d−1)-dimensional faces of the cells. Computing A_d(n), the number of distinct d-dimensional polycubes of size n, is a long-standing elusive problem in discrete geometry. In a previous work we described the generalization from two to higher dimensions of a polyomino-counting algorithm of Redelmeier [D.H. Redelmeier, Counting polyominoes: Yet another attack, Discrete Math. 36 (1981) 191-203]. The main deficiency of the algorithm is that it keeps the entire set of cells that appear in any possible polycube in memory at all times. Thus, the amount of required memory grows exponentially with the dimension. In this paper we present an improved version of the same method, whose order of memory consumption is a (very low) polynomial in both n and d. We also describe how we parallelized the algorithm and ran it through the Internet on dozens of computers simultaneously.     

Challenging the curse of dimensionality in multivariate local linear regression

Local polynomial fitting for univariate data has been widely studied and discussed, but up until now the multivariate equivalent has often been deemed impractical, due to the so-called curse of dimensionality. Here, rather than discounting it completely, we use density as a threshold to determine where over a data range reliable multivariate smoothing is possible, whilst accepting that in large areas it is not. The merits of a density threshold derived from the asymptotic influence function are shown using both real and simulated data sets. Further, the challenging issue of multivariate bandwidth selection, which is known to be affected detrimentally by sparse data which inevitably arise in higher dimensions, is considered. In an effort to alleviate this problem, two adaptations to generalized cross-validation are implemented, and a simulation study is presented to support the proposed method. It is also discussed how the density threshold and the adapted generalized cross-validation technique introduced herein work neatly together.     

The curse of dimensionality for the class of monotone functions and for the class of convex functions

We study the integration and approximation problems for monotone or convex bounded functions that depend on $d$ variables, where $d$ can be arbitrarily large. We consider the worst case error for algorithms that use finitely many function values. We prove that these problems suffer from the curse of dimensionality. That is, one needs exponentially many (in $d$) function values to achieve an error $\epsilon$.     

The instability of non-semi-Fredholm operators under compact perturbations

Methods are presented for calculating the perturbation series for the roots of polynomials which contain parameters in addition to the perturbation parameter. In particular the complicating effects of multiplicities of the unperturbed roots are examined.     

The instability time distribution behavior of slotted ALOHA

This article deals with a quantification of the instability behavior of the slotted multiuser communication network system ALOHA. In (2) U. Schmid and the author could quantify the expected value E (Y) of the instability time Y = max (n: X_n = 0), where X_n denotes the backlog after n slots. Here it is shown that Y is approximately exponentially distributed. Moreover, uniform asymptotic expansions for y_m = P [Y = m] and for the moments M_m = E (Y^m) are given. © 1994 John Wiley & Sons, Inc.     

The role of thermal instability in star formation

The observations tell us that the density in the giant molecularclouds in which stars are formed is inhomogeneous on a varietyof scales, but it seems unlikely that this is due to the actionof gravitational instability. This paper describes numericalcalculations using an adaptive mesh refinement magnetohydrodynamicscode that show that thermal instability may have an importantrole to play in the formation of this structure     

The instability of some gradient methods for ill-posed problems

Summary For the solution of linear ill-posed problems some gradient methods like conjugate gradients and steepest descent have been examined previously in the literature. It is shown that even though these methods converge in the case of exact data their instability makes it impossible to base a-priori parameter choice regularization methods upon them.     

The geometric structures and instability of entropic dynamical models

In this paper, we characterize two entropic dynamical (ED) models from the viewpoint of information geometry and give the geometric structures of the associated statistical manifolds of the models. The scalar curvatures and the geodesics are obtained. Also the instability of entropic dynamical models is studied from the behavior of the geodesics lengths, statistical volume elements and Jacobi vector fields.     

The global instability of uniform flows in non-one-dimensional regions

The conditions for the instability of flows or states, which are independent of time and coordinates, in extended non-one-dimensional regions are considered in a linear approximation. An extension of the idea of global instability, previously introduced for the one-dimensional case, is given. A method is proposed for weakly unstable flows, which enables one to investigate under what conditions perturbations, which grow without limit with time, and which do not depend on the specific form of boundary conditions (provided they are not degenerate), exist. The case of a two-dimensional rectangular region is considered in detail.     

Hyperbolic wavelet thresholding methods and the curse of dimensionality through the maxiset approach

In this paper we compute the maxisets of some denoising methods (estimators) for multidimensional signals based on thresholding coefficients in hyperbolic wavelet bases. That is, we determine the largest functional space over which the risk of these estimators converges at a chosen rate. In the unidimensional setting, refining the choice of the coefficients that are subject to thresholding by pooling information from geometric structures in the coefficient domain (e.g., vertical blocks) is known to provide ‘large maxisets’. In the multidimensional setting, the situation is less straightforward. In a sense these estimators are much more exposed to the curse of dimensionality. However we identify cases where information pooling has a clear benefit. In particular, we identify some general structural constraints that can be related to compound functional models and to a minimal level of anisotropy.     

Dominant strategies, superior information, and winner's curse in second-price auctions

In a general model of common-value second-price auctions with differential information, we show equivalence between the following characteristics of a bidder: (i) having a dominant strategy; (ii) possessing superior information; (iii) being immune from winner's curse. When a dominant strategy exists, it is given by the conditional expectation of the common value with respect to bidder's information field; if the dominant strategy is used, other bidders cannot make a profit. Final version November 2001     

The centrifugal instability of a Stokes layer: asymmetric disturbances

The centrifugal instability of a Stokes layer on a circular cylinder to non-axisymmetric disturbances is studied. The governing equations used previously for this problem are corrected, and numerical results are presented.

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Zusammenfassung Es wird die zentrifugale Instabilität einer Stokes'schen Grenzschicht an einem Kreiszylinder gegenüber nicht-axisymmetrischen Störungen untersucht. Die früher benützten Gleichungen für dieses Problem sind korrigiert und numerisch gelöst worden.

     

The centrifugal instability of a stokes layer: Asymmetric disturbances

The influence of the asymmetry of the perturbations on the instability of viscous incompressible flow originated by a circular cylinder performing harmonic oscillations around its axis is investigated in the linear regime. The critical values of the Taylor number (based on the amplitude of the cylinder tangential velocity and on the thickness of the basic flow) and of the axial wavenumber are found to increase monotonically as the asymmetry parameter (azimuthal wavenumber) increases. This result agrees with previous experimental observations of the author.

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Sommario Si esamina l'influenza che l'asimmetria dei disturbi esercita sulla stabilità lineare del moto indotto da un cilindro, oscillante in modo armonico intorno al suo asse, in un fluido viscoso incomprimibile posto nella regione esterna al cilindro. I valori critici del numero di Taylor (costruitocon l'ampiezza della velocità tangenziale del cilindro e lo spessore dello strato interessato dal moto base) e del numero d'onda assiale si trovano essere monotonicamente crescenti al crescere del parametro che caratterizza l'asimmetria (numero d'onda circonferenziale). Ciò risulta in accordo con osservazioni sperimentali precedenti dell'autore.

     

The instability of the flow in a suddenly blocked pipe

This paper considers the decay of Poiseuille flow within a suddenlyblocked pipe. For small to moderate times the flow is shownto consist of an inviscid core flow coupled with a boundarylayer at the pipe wall. A small-time asymptotic solution isdeveloped and it is shown that this solution is valid for timesup to the point at which the boundary layer fills the wholepipe. A small-time composite solution is used to initiate anumerical marching procedure which overcomes the small-timesingularity that arises in the flow and so allows us to describethe ultimate decay of the flow within a blocked pipe. The stabilityof this flow is then considered using both a quasi-steady approximationand a transient-growth analysis based upon marching solutionsof the linearized Navier–Stokes equations. Our transientstability analysis predicts a critical Reynolds number, fortransition to turbulence, in the range 970 < Re < 1370.