On tractability of path integration

Do we really need to use randomized algorithms for path integrals? Perhaps we can find a deterministic algorithm that is more effective even in the worst case setting. To answer this question we study the worst case complexity of path integration which roughly speaking is defined as the minimal number of the integrand evaluations needed to compute an approximation with error at most e. We consider path integration with respect to a Gaussian measure and for various classes of integrands

Average Case Optimal Algorithms in Hilbert Spaces

We study optimal algorithms and optimal information for an average case model. This is done for linear problems in a separable Hilbert space equipped with a probability measure. We show, in particular, that for any measure, an affine spline algorithm is optimal among affine algorithms. The affine spline algorithm is defined in terms of the correlation operator and the mean element of the measure. We provide a condition on the measure which guarantees that the affine spline algorithm is optimal among all algorithms. The problem of optimal information is also solved

Average Case Optimality for Linear Problems

We introduce an average case model and define general notions of optimal algorithm and optimal information. We prove that the same algorithm and information are optimal in the worst and average cases and that adaptive information is not more powerful than nonadaptive information

Can Adaption Help on the Average?

We study adaptive information for approximation of linear problems in a separable Hilbert space equipped with a probability measure μ. It is known that adaption does not help in the worst case for linear problems. We prove that adaption also does not help on the average. That is, there exists nonadaptive information which is as powerful as adaptive information. This result holds for "orthogonally invariant" measures. We provide necessary and sufficient conditions for a measure to be orthogonally invariant. Examples of orthogonally invariant measures include Gaussian measures and, in the finite dimensional case, weighted Lebesgue measures

When Is Nonadaptive Information as Powerful as Adaptive Information?

Information based complexity is a unified treatment of problems where only partial or approximate information is available. In this approach one states how well a problem should be solved and indicates the type of information available. The theory then tells one optimal information and optimal algorithm and yields bounds on the problem complexity. In this paper we survey some recent results addressing one of the problems studied in information based complexity. The problem deals with nonadaptive and adaptive information both for the worst case and average case settings

African swine fever in wild boar

The European Commission requested EFSA to compare the reliability of wild boar density estimates across the EU and to provide guidance to improve data collection methods. Currently, the only EU-wide available data are hunting data. Their collection methods should be harmonised to be comparable and to improve predictive models for wild boar density. These models could be validated by more precise density data, collected at local level e.g. by camera trapping. Based on practical and theoretical considerations, it is currently not possible to establish wild boar density thresholds that do not allow sustaining African swine fever (ASF). There are many drivers determining if ASF can be sustained or not, including heterogeneous population structures and human-mediated spread and there are still unknowns on the importance of different transmission modes in the epidemiology. Based on extensive literature reviews and observations from affected Member States, the efficacy of different wild boar population reduction and separation methods is evaluated. Different wild boar management strategies at different stages of the epidemic are suggested. Preventive measures to reduce and stabilise wild boar density, before ASF introduction, will be beneficial both in reducing the probability of exposure of the population to ASF and the efforts needed for potential emergency actions (i.e. less carcass removal) if an ASF incursion were to occur. Passive surveillance is the most effective and efficient method of surveillance for early detection of ASF in free areas. Following focal ASF introduction, the wild boar populations should be kept undisturbed for a short period (e.g. hunting ban on all species, leave crops unharvested to provide food and shelter within the affected area) and drastic reduction of the wild boar population may be performed only ahead of the ASF advance front, in the free populations. Following the decline in the epidemic, as demonstrated through passive surveillance, active population management should be reconsidered.info:eu-repo/semantics/publishedVersio

African swine fever in wild boar

The European Commission requested EFSA to compare the reliability of wild boar density estimates across the EU and to provide guidance to improve data collection methods. Currently, the only EU-wide available data are hunting data. Their collection methods should be harmonised to be comparable and to improve predictive models for wild boar density. These models could be validated by more precise density data, collected at local level e.g. by camera trapping. Based on practical and theoretical considerations, it is currently not possible to establish wild boar density thresholds that do not allow sustaining African swine fever (ASF). There are many drivers determining if ASF can be sustained or not, including heterogeneous population structures and human-mediated spread and there are still unknowns on the importance of different transmission modes in the epidemiology. Based on extensive literature reviews and observations from affected Member States, the efficacy of different wild boar population reduction and separation methods is evaluated. Different wild boar management strategies at different stages of the epidemic are suggested. Preventive measures to reduce and stabilise wild boar density, before ASF introduction, will be beneficial both in reducing the probability of exposure of the population to ASF and the efforts needed for potential emergency actions (i.e. less carcass removal) if an ASF incursion were to occur. Passive surveillance is the most effective and efficient method of surveillance for early detection of ASF in free areas. Following focal ASF introduction, the wild boar populations should be kept undisturbed for a short period (e.g. hunting ban on all species, leave crops unharvested to provide food and shelter within the affected area) and drastic reduction of the wild boar population may be performed only ahead of the ASF advance front, in the free populations. Following the decline in the epidemic, as demonstrated through passive surveillance, active population management should be reconsidered.info:eu-repo/semantics/publishedVersio

The Exponent of Discrepancy is At Most 1.4778...

We study discrepancy with arbitrary weights in the L 2 norm over the d dimensional unit cube. The exponent p of discrepancy is defined as the smallest p for which there exists a positive number K such that for all d and all " 1 there exist K " \Gammap points with discrepancy at most ". It is well known that p 2 (1; 2]. We improve the upper bound by showing that p 1:478841:::: : This is done by using relations between discrepancy and integration in the average case setting with the Wiener sheet measure. Our proof is not constructive. The known constructive bound on the exponent p is 2:454. 1 Introduction We study discrepancy with arbitrary weights in the L 2 norm over the d dimensional unit cube [0; 1] d . This problem is defined as finding n points from [0; 1] d which approximate the volumes of rectangles (starting from zero) with minimal error, see [8, 9] for the precise definition, history and basic properties. Discrepancy has been extensively studied in number..