RBF approximation of large datasets by partition of unity and local stabilization

We present an algorithm to approximate large dataset by Radial Basis Function (RBF) techniques. The method couples a fast domain decomposition procedure with a localized stabilization method. The resulting algorithm can efficiently deal with large problems and it is robust with respect to the typical instability of kernel methods

Partition of unity interpolation using stable kernel-based techniques

In this paper we propose a new stable and accurate approximation technique which is extremely effective for interpolating large scattered data sets. The Partition of Unity (PU) method is performed considering Radial Basis Functions (RBFs) as local approximants and using locally supported weights. In particular, the approach consists in computing, for each PU subdomain, a stable basis. Such technique, taking advantage of the local scheme, leads to a significant benefit in terms of stability, especially for flat kernels. Furthermore, an optimized searching procedure is applied to build the local stable bases, thus rendering the method more efficient

Fontes de nitrogênio e técnicas de propagação de mudas atuam na produtividade de erva-mate.

Avaliou-se a influência de fontes de nitrogênio e técnicas de propagação na produtividade de erva-mate. Em plantio realizado em 2005 em São Mateus do Sul-PR (SMS), no espaçamento 1,2 x 3,0 m com mudas propagadas: por semente (procedência SMS) e por miniestaquia (procedências Bituruna, Cruz Machado e SMS). Em 2010, após a segunda colheita aplicou-se 130 kg ha-1 de nitrogênio na forma de nitrato de amônio, sulfato de amônio e ureia. Na colheita de 2012, com intervalo de 18 meses, quantificou-se a produtividade de erva-mate comercial, galho fino e galho grosso. A produtividade de todos os componentes avaliados foi influenciada pela interação entre fontes de N e procedências. Conclui-se que a preferência da erva-mate pela fonte de nitrogênio é dependente do local de origem da cultura; a miniestaquia é uma técnica eficiente na propagação de erva-mate, recomendada para melhorar a produtividade da cultura

5-th Dolomites Workshop on Constructive Approximation and Applications – Special Issue dedicated to Robert Schaback on the occasion of his 75th birthday

The guest editors discuss the highlights of the 5-th Dolomites Workshop on Constructive Approximation and Applications, and briefly introduce the papers included in this special issue

Stable interpolation with exponential-polynomial splines and node selection via greedy algorithms

In this work we extend some ideas about greedy algorithms, which are well-established tools for, e.g., kernel bases, and exponential-polynomial splines whose main drawback consists in possible overfitting and consequent oscillations of the approximant. To partially overcome this issue, we develop some results on theoretically optimal interpolation points. Moreover, we introduce two algorithms which perform an adaptive selection of the spline interpolation points based on the minimization either of the sample residuals (f-greedy), or of an upper bound for the approximation error based on the spline Lebesgue function (λ-greedy). Both methods allow us to obtain an adaptive selection of the sampling points, i.e., the spline nodes. While the f-greedy selection is tailored to one specific target function, the λ-greedy algorithm enables us to define target-data-independent interpolation nodes

Active surveillance for favorable-risk prostate cancer: Is there a greater psychological impact than previously thought? A systematic, mixed studies literature review.

OBJECTIVE: Active surveillance (AS) allows men with favorable-risk prostate cancer to avoid or postpone active treatment and hence spares potential adverse effects for a significant proportion of these patients. Active surveillance may create an additional emotional burden for these patients. The aim of the review was to determine the psychological impact of AS to inform future study in this area and to provide recommendations for clinical practice. METHODS: Studies were identified through database searching from inception to September 2015. Quantitative or qualitative noninterventional studies published in English that assessed the psychological impact of AS were included. The Mixed Methods Appraisal Tool was used to assess methodological quality. RESULTS: Twenty-three papers were included (20 quantitative and 3 qualitative). Quantitatively, the majority of patients do not report psychological difficulties; however, when appropriateness of study design is considered, the conclusion that AS has minimal impact on well-being may not be accurate. This is due to small sample sizes, inappropriately timed baseline, and inappropriate/lack of comparison groups. In addition, a mismatch in outcome was noted between the outcome of quantitative and qualitative studies in uncertainty, with qualitative studies indicating a greater psychological impact. CONCLUSIONS: Because of methodological concerns, many quantitative studies may not provide a true account of the burden of AS. Further mixed-methods studies are necessary to address the limitations highlighted and to provide clarity on the impact of AS. Practitioners should be aware that despite findings of previous reviews, patients may require additional emotional support

Investigations of polygonal patterned ground in continuous Antarctic permafrost by means of ground penetrating radar and electrical resistivity tomography: Some unexpected correlations

The results of a combined geophysical and geomorphological investigation of thermal-contraction-crack polygons near Gondwana station (Germany) in northern Victoria Land (Antarctica) are reported. An area of about 20,000 m2 characterized by random orthogonal polygons was investigated using integrated ground penetrating radar, electrical resistivity tomography, geomorphological surveys, and two trench excavations. The polygons are well developed only at elevations higher than 6–7 m above current sea level on Holocene-age raised beaches. It is concluded that the polygons are composite in nature because the shallow linear depressions that outline the polygons are underlain by fissures that can contain both sandy gravel and foliated ice (i.e., ice wedges) even in the same polygon network and at distances of just a few meters. Unexpectedly, most of the polygons follow the border of the raised beaches and develop in correspondence with stratigraphic layers dipping toward the sea, imaged by ground penetrating radar (GPR) profiles and interpreted as prograding layers toward the present-day shoreline

Interpolation with the polynomial kernels

The polynomial kernels are widely used in machine learning and they are one of the default choices to develop kernel-based classification and regression models. However, they are rarely used and considered in numerical analysis due to their lack of strict positive definiteness. In particular they do not enjoy the usual property of unisolvency for arbitrary point sets, which is one of the key properties used to build kernel-based interpolation methods. This paper is devoted to establish some initial results for the study of these kernels, and their related interpolation algorithms, in the context of approximation theory. We will first prove necessary and sufficient conditions on point sets which guarantee the existence and uniqueness of an interpolant. We will then study the Reproducing Kernel Hilbert Spaces (or native spaces) of these kernels and their norms, and provide inclusion relations between spaces corresponding to different kernel parameters. With these spaces at hand, it will be further possible to derive generic error estimates which apply to sufficiently smooth functions, thus escaping the native space. Finally, we will show how to employ an efficient stable algorithm to these kernels to obtain accurate interpolants, and we will test them in some numerical experiment. After this analysis several computational and theoretical aspects remain open, and we will outline possible further research directions in a concluding section. This work builds some bridges between kernel and polynomial interpolation, two topics to which the authors, to different extents, have been introduced under the supervision or through the work of Stefano De Marchi. For this reason, they wish to dedicate this work to him in the occasion of his 60th birthday