Sharp adaptive estimation of quadratic functionals

Estimation of a quadratic functional of a function observed in the Gaussian white noise model is considered. A data-dependent method for choosing the amount of smoothing is given. The method is based on comparing certain quadratic estimators with each other. It is shown that the method is asymptotically sharp or nearly sharp adaptive simultaneously for the “regular” and “irregular” region. We consider l_p bodies and construct bounds for the risk of the estimator which show that for p=4 the estimator is exactly optimal and for example when p ∈[3,100], then the upper bound is at most 1.055 times larger than the lower bound. We show the connection of the estimator to the theory of optimal recovery. The estimator is a calibration of an estimator which is nearly minimax optimal among quadratic estimators. Writing of this article was financed by Deutsche Forschungsgemeinschaft under project MA1026/6-2, CIES, France, and Jenny and AnttiWihuri Foundation.     

On a quadratic functional occurring in a bivariate scattered data interpolation problem

Sunto L’applicazione di noti metodi che utilizzano funzioni di tipo blending per la costruzione di funzioni bivariate C¹ per l’interpolazione di dati, richiede la conoscenza delle derivate parziali del primo ordine ai vertici di una triangolazione sottostante. In questo lavoro consideriamo il metodo proposto da Nielson, che consiste nel calcolare stime delle derivate parziali del primo ordine minimizzando un opportuno funzionale quadratico, caratterizzato da parametri di tensione non negativi. Scopo del lavoro è l’analisi di alcune proprietà particolari di questo funzionale per la costruzione di algoritmi efficienti e robusti per la determinazione delle stime suddette delle derivate quando si ha a che fare con insiemi di dati di grandi dimensioni. Abstract The application of widely known blending methods for constructingC ¹ bivariate functions interpolating scattered data requires the knowledge of the partial derivatives of first order at the vertices of an underlying triangulation. In this paper we consider the method proposed by Nielson that consists in computing estimates of the first order partial derivatives by minimizing an appropriate quadratic functional, characterized by nonnegative tension parameters. The aim of the paper is to analyse some peculiar properties of this functional in order to construct robust and efficient algorithms for determining the above estimates of the derivatives when we are concerned with extremely large data sets.      

On a particular quadratic network problem

Let c(x), d(x) and h(x) be linear functions defined over a convex polyhedron X = {x | Ax = r and 0 ⩽ x ⩽ u }, where A is the node-edge incidence matrix of a given network. Let f(x) = c(x)d(x) + h(x) be a (particular) quadratic function which we intend to minimize over X. In this paper we use the theory of multiobjective programming do develop algorithms for the semidefinite positive case (f(x) = ϱ[d(x)]² + h(x) and ϱ is a strictly postive real constant) and the semidefinite negative case (f(x) = ϱ[d(x)]² + h (x) and ϱ is a strictly negative real constant). A special indefinite case (h(x) is constant over X) is also studied.In the proposed algorithms, basic solutions are used in all the interactions with possible exception for the last one. Moreover only the concepts of the simplex method are used; consequently these algorithms can be used even when A is not the node-edge incidence matrix of a network. However they are particularly attractive for the network case. In fact, network basic solutions are efficiently manipulated when appropriated data structures are used in the computational implementation of an algorithm.     

On a quadratic functional in a nonlocal boundary value problem for the heat equation

Translated from Matematicheskie Zametki, Vol. 49, No. 1, pp. 135–143, January, 1991.     

On the stability problem for a mixed type of quartic and quadratic functional equation

Let E₁ and E₂ be real linear spaces. In this paper, we determine the general solution for a mixed type functional equation of a quartic and a quadratic mapping f:E₁→E₂,

     

On a functional inequality connected with quadratic functionals

In the present paper we investigate some functional inequalities which are closely connected with quadratic functionals. In particular, we are interested in inequalities of the type

     

On the continuous quadratic knapsack problem

We introduce a new algorithm for the continuous bounded quadratic knapsack problem. This algorithm is motivated by the geometry of the problem, is based on the iterative solution of a series of simple projection problems, and is easy to understand and implement. In practice, the method compares favorably to other well-known algorithms (some of which have superior worst-case complexity) on problem sizes up ton = 4000.     

On the estimation of wavelet coefficients

In wavelet representations, the magnitude of the wavelet coefficients depends on both the smoothness of the represented function f and on the wavelet. We investigate the extreme values of wavelet coefficients for the standard function spaces A_k=f| ∥f^k)∥₂ ≤ 1}, k∈N. In particular, we compare two important families of wavelets in this respect, the orthonormal Daubechies wavelets and the semiorthogonal spline wavelets. Deriving the precise asymptotic values in both cases, we show that the spline constants are considerably smaller. This revised version was published online in June 2006 with corrections to the Cover Date.     

An adaptive wavelet shrinkage approach to the Spektor-Lord-Willis problem

The stereological problem of unfolding the sphere size distribution from linear sections is considered. A minimax estimator of the intensity function of a Poisson process that describes the problem is introduced and an adaptive estimator is constructed that achieves the optimal rate of convergence over Besov balls to within logarithmic factors. The construction of these estimators uses Wavelet-Vaguelette Decomposition (WVD) of the operator that defines our inverse problem.     

On the problem of estimation of the number of cycles in two-dimensional quadratic systems from the viewpoint of nonlinear mechanics

Two-dimensional quadratic systems are considered as a Liénard equation with certain special nonlinearities. Theorems on the existence or absence of cycles are given. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 1, pp. 48–57, January, 1998.     

On the stability of a quadratic Jensen type functional equation

In this paper we obtain the general solution of the quadratic Jensen type functional equation

and prove the stability of this equation in the spirit of Hyers, Ulam, Rassias, and G vruta.     

On restricted functional inequalities associated with quadratic functional equations

Aequationes mathematicae - In this paper it is proved that, for a function $$f:mathcal {X}rightarrow E$$ mapping from a normed linear space $$mathcal {X}$$ into an inner product space E, the...     

On linearizations of the quadratic two-parameter eigenvalue problem

We present several transformations that can be used to solve the quadratic two-parameter eigenvalue problem (QMEP), by formulating an associated linear multiparameter eigenvalue problem. Two of these transformations are generalizations of the well-known linearization of the quadratic eigenvalue problem and linearize the QMEP as a singular two-parameter eigenvalue problem. The third replaces all nonlinear terms by new variables and adds new equations for their relations. The QMEP is thus transformed into a nonsingular five-parameter eigenvalue problem. The advantage of these transformations is that they enable one to solve the QMEP using existing numerical methods for multiparameter eigenvalue problems. We also consider several special cases of the QMEP, where some matrix coefficients are zero     

On a linearly quadratic problem of optimal control of a Stochastic Volterra equation

Translated from Matematicheskie Zametki, Vol. 48, No. 6, pp. 112–117, December, 1990.