We derive minimax rates for estimation in anisotropic smoothness classes. This rate is attained by a coordinatewise thresholded wavelet estimator based on a tensor product basis with separate scale parameter for every dimension. It is shown that this basis is superior to its one-scale multiresolution analog, if different degrees of smoothness in different directions are present.; As an important application we introduce a new adaptive wavelet estimator of the time-dependent spectrum of a locally stationary time series. Using this model which was resently developed by Dahlhaus, we show that the resulting estimator attains nearly the rate, which is optimal in Gaussian white noise, simultaneously over a wide range of smoothness classes. Moreover, by our new approach we overcome the difficulty of how to choose the right amount of smoothing, i.e. how to adapt to the appropriate resolution, for reconstructing the local structure of the evolutionary spectrum in the time-frequency plane.
Many problems arising in (geo)physics and technology can be formulated as compact operator equations of the first kind \(A F = G\). Due to the ill-posedness of the equation a variety of regularization methods are in discussion for an approximate solution, where particular emphasize must be put on balancing the data and the approximation error. In doing so one is interested in optimal parameter choice strategies. In this paper our interest lies in an efficient algorithmic realization of a special class of regularization methods. More precisely, we implement regularization methods based on filtered singular value decomposition as a wavelet analysis. This enables us to perform, e.g., Tikhonov-Philips regularization as multiresolution. In other words, we are able to pass over from one regularized solution to another one by adding or subtracting so-called detail information in terms of wavelets. It is shown that regularization wavelets as proposed here are efficiently applicable to a future problem in satellite geodesy, viz. satellite gravity gradiometry.
An asymptotic-induced scheme for nonstationary transport equations with thediffusion scaling is developed. The scheme works uniformly for all ranges ofmean free paths. It is based on the asymptotic analysis of the diffusion limit ofthe transport equation. A theoretical investigation of the behaviour of thescheme in the diffusion limit is given and an approximation property is proven.Moreover, numerical results for different physical situations are shown and atheuniform convergence of the scheme is established numerically.
Metaharmonic wavelets are introduced for constructing the solution of theHelmholtz equation (reduced wave equation) corresponding to Dirichlet's orNeumann's boundary values on a closed surface approach leading to exactreconstruction formulas is considered in more detail. A scale discrete version ofmultiresolution is described for potential functions metaharmonic outside theclosed surface and satisfying the radiation condition at infinity. Moreover, wediscuss fully discrete wavelet representations of band-limited metaharmonicpotentials. Finally, a decomposition and reconstruction (pyramid) scheme foreconomical numerical implementation is presented for Runge-Walsh waveletapproximation.
In the modeling of biological phenomena, in living organisms whether the measurements are of blood pressure, enzyme levels, biomechanical movements or heartbeats, etc., one of the important aspects is time variation in the data. Thus, the recovery of a "smooth" regression or trend function from noisy time-varying sampled data becomes a problem of particular interest. Here we use non-linear wavelet thresholding to estimate a regression or a trend function in the presence of additive noise which, in contrast to most existing models, does not need to be stationary. (Here, nonstationarity means that the spectral behaviour of the noise is allowed to change slowly over time.). We develop a procedure to adapt existing threshold rules to such situations, e.g., that of a time-varying variance in the errors. Moreover, in the model of curve estimation for functions belonging to a Besov class with locally stationary errors, we derive a near-optimal rate for the L2-risk between the unknown function and our soft or hard threshold estimator, which holds in the general case of an error distribution with bounded cumulants. In the case of Gaussian errors, a lower bound on the asymptotic minimax rate in the wavelet coefficient domain is also obtained. Also it is argued that a stronger adaptivity result is possible by the use of a particular location and level dependent threshold obtained by minimizing Stein's unbiased estimate of the risk. In this respect, our work generalizes previous results, which cover the situation of correlated, but stationary errors. A natural application of our approach is the estimation of the trend function of nonstationary time series under the model of local stationarity. The method is illustrated on both an interesting simulated example and a biostatistical data-set, measurements of sheep luteinizing hormone, which exhibits a clear nonstationarity in its variance.
Starting from the mollified version of the Enskog equation for a hard-sphere fluid, a grid-free algorithm to obtain the solution is proposed. The algorithm is based on the finite pointset method. For illustration, it is applied to a Riemann problem. The shock-wave solution is compared to the results of Frezzotti and Sgarra where a good agreement is found.
Here the self-organization property of one-dimensional Kohonen's algorithm in its 2k-neighbour setting with a general type of stimuli distribution and non-increasing learning rate is considered. We prove that the probability of self-organization for all initial values of neurons is uniformly positive. For the special case of a constant learning rate, it implies that the algorithm self-organizes with probability one.
We develop a test for stationarity of a time series against the alternative of a time-changing covariance structure. Using localized versions of the periodogram, we obtain empirical versions of a reasonable notion of a time-varying spectral density. Coefficients w.r.t. a Haar wavelet series expansion of such a time-varying periodogram are a possible indicator whether there is some deviation from covariance stationarity. We propose a test based on the limit distribution of these empirical coefficients.
In modern approximation methods linear combinations in terms of (space localizing) radial basis functions play an essential role. Areas of application are numerical integration formulas on the uni sphere omega corresponding to prescribed nodes, spherical spline interpolation, and spherical wavelet approximation. the evaluation of such a linear combination is a time consuming task, since a certain number of summations, multiplications and the calculation of scalar products are required. This paper presents a generalization of the panel clustering method in a spherical setup. The economy and efficiency of panel clustering is demonstrated for three fields of interest, namely upward continuation of the earth's gravitational potential, geoid computation by spherical splines and wavelet reconstruction of the gravitational potential.