Efficient cosmological parameter sampling using sparse grids

We present a novel method to significantly speed up cosmological parameter sampling. The method relies on constructing an interpolation of the CMB-log-likelihood based on sparse grids, which is used as a shortcut for the likelihood-evaluation. We obtain excellent results over a large region in parameter space, comprising about 25 log-likelihoods around the peak, and we reproduce the one-dimensional projections of the likelihood almost perfectly. In speed and accuracy, our technique is competitive to existing approaches to accelerate parameter estimation based on polynomial interpolation or neural networks, while having some advantages over them. In our method, there is no danger of creating unphysical wiggles as it can be the case for polynomial fits of a high degree. Furthermore, we do not require a long training time as for neural networks, but the construction of the interpolation is determined by the time it takes to evaluate the likelihood at the sampling points, which can be parallelised to an arbitrary degree. Our approach is completely general, and it can adaptively exploit the properties of the underlying function. We can thus apply it to any problem where an accurate interpolation of a function is needed.Comment: Submitted to MNRAS, 13 pages, 13 figure

Differential Evolution Methods for the Fuzzy Extension of Functions

The paper illustrates a differential evolution (DE) algorithm to calculate the level-cuts of the fuzzy extension of a multidimensional real valued function to fuzzy numbers. The method decomposes the fuzzy extension engine into a set of "nested" min and max box-constrained op- timization problems and uses a form of the DE algorithm, based on multi populations which cooperate during the search phase and specialize, a part of the populations to find the the global min (corresponding to lower branch of the fuzzy extension) and a part of the populations to find the global max (corresponding to the upper branch), both gaining efficiency from the work done for a level-cut to the subsequent ones. A special ver- sion of the algorithm is designed to the case of differentiable functions, for which a representation of the fuzzy numbers is used to improve ef- ficiency and quality of calculations. The included computational results indicate that the DE method is a promising tool as its computational complexity grows on average superlinearly (of degree less than 1.5) in the number of variables of the function to be extended.Fuzzy Sets, Differential Evolution Method, Fuzzy Extension of Functions

Low rank surrogates for polymorphic fields with application to fuzzy-stochastic partial differential equations

We consider a general form of fuzzy-stochastic PDEs depending on the interaction of probabilistic and non-probabilistic ("possibilistic") influences. Such a combined modelling of aleatoric and epistemic uncertainties for instance can be applied beneficially in an engineering context for real-world applications, where probabilistic modelling and expert knowledge has to be accounted for. We examine existence and well-definedness of polymorphic PDEs in appropriate function spaces. The fuzzy-stochastic dependence is described in a high-dimensional parameter space, thus easily leading to an exponential complexity in practical computations. To aleviate this severe obstacle in practise, a compressed low-rank approximation of the problem formulation and the solution is derived. This is based on the Hierarchical Tucker format which is constructed with solution samples by a non-intrusive tensor reconstruction algorithm. The performance of the proposed model order reduction approach is demonstrated with two examples. One of these is the ubiquitous groundwater flow model with Karhunen-Loeve coefficient field which is generalized by a fuzzy correlation length

B-splines for sparse grids : algorithms and application to higher-dimensional optimization

In simulation technology, computationally expensive objective functions are often replaced by cheap surrogates, which can be obtained by interpolation. Full grid interpolation methods suffer from the so-called curse of dimensionality, rendering them infeasible if the parameter domain of the function is higher-dimensional (four or more parameters). Sparse grids constitute a discretization method that drastically eases the curse, while the approximation quality deteriorates only insignificantly. However, conventional basis functions such as piecewise linear functions are not smooth (continuously differentiable). Hence, these basis functions are unsuitable for applications in which gradients are required. One example for such an application is gradient-based optimization, in which the availability of gradients greatly improves the speed of convergence and the accuracy of the results. This thesis demonstrates that hierarchical B-splines on sparse grids are well-suited for obtaining smooth interpolants for higher dimensionalities. The thesis is organized in two main parts: In the first part, we derive new B-spline bases on sparse grids and study their implications on theory and algorithms. In the second part, we consider three real-world applications in optimization: topology optimization, biomechanical continuum-mechanics, and dynamic portfolio choice models in finance. The results reveal that the optimization problems of these applications can be solved accurately and efficiently with hierarchical B-splines on sparse grids.In der Simulationstechnik werden zeitaufwendige Zielfunktionen oft durch einfache Surrogate ersetzt, die durch Interpolation gewonnen werden können. Vollgitter-Interpolationsmethoden leiden unter dem sogenannten Fluch der Dimensionalität, der sie unbrauchbar macht, falls der Parameterbereich der Funktion höherdimensional ist (vier oder mehr Parameter). Dünne Gitter sind eine Diskretisierungsmethode, die den Fluch drastisch lindert und die Approximationsqualität nur leicht verschlechtert. Leider sind konventionelle Basisfunktionen wie die stückweise linearen Funktionen nicht glatt (stetig differenzierbar). Daher sind sie für Anwendungen ungeeignet, in denen Gradienten benötigt werden. Ein Beispiel für eine solche Anwendung ist gradientenbasierte Optimierung, in der die Verfügbarkeit von Gradienten die Konvergenzgeschwindigkeit und die Ergebnisgenauigkeit deutlich verbessert. Diese Dissertation demonstriert, dass hierarchische B-Splines auf dünnen Gittern hervorragend geeignet sind, um glatte Interpolierende für höhere Dimensionalitäten zu erhalten. Die Dissertation ist in zwei Hauptteile gegliedert: Der erste Teil leitet neue B-Spline-Basen auf dünnen Gittern her und untersucht ihre Implikationen bezüglich Theorie und Algorithmen. Der zweite Teil behandelt drei Realwelt-Anwendungen aus der Optimierung: Topologieoptimierung, biomechanische Kontinuumsmechanik und Modelle der dynamischen Portfolio-Wahl in der Finanzmathematik. Die Ergebnisse zeigen, dass die Optimierungsprobleme dieser Anwendungen durch hierarchische B-Splines auf dünnen Gittern genau und effizient gelöst werden können

Low rank surrogates for polymorphic fields with application to fuzzy-stochastic partial differential equations

We consider a general form of fuzzy-stochastic PDEs depending on the interaction of probabilistic and non-probabilistic ("possibilistic") influences. Such a combined modelling of aleatoric and epistemic uncertainties for instance can be applied beneficially in an engineering context for real-world applications, where probabilistic modelling and expert knowledge has to be accounted for. We examine existence and well-definedness of polymorphic PDEs in appropriate function spaces. The fuzzy-stochastic dependence is described in a high-dimensional parameter space, thus easily leading to an exponential complexity in practical computations. To aleviate this severe obstacle in practise, a compressed low-rank approximation of the problem formulation and the solution is derived. This is based on the Hierarchical Tucker format which is constructed with solution samples by a non-intrusive tensor reconstruction algorithm. The performance of the proposed model order reduction approach is demonstrated with two examples. One of these is the ubiquitous groundwater flow model with Karhunen-Loeve coefficient field which is generalized by a fuzzy correlation length

Review of Low Voltage Load Forecasting: Methods, Applications, and Recommendations

The increased digitalisation and monitoring of the energy system opens up numerous opportunities to decarbonise the energy system. Applications on low voltage, local networks, such as community energy markets and smart storage will facilitate decarbonisation, but they will require advanced control and management. Reliable forecasting will be a necessary component of many of these systems to anticipate key features and uncertainties. Despite this urgent need, there has not yet been an extensive investigation into the current state-of-the-art of low voltage level forecasts, other than at the smart meter level. This paper aims to provide a comprehensive overview of the landscape, current approaches, core applications, challenges and recommendations. Another aim of this paper is to facilitate the continued improvement and advancement in this area. To this end, the paper also surveys some of the most relevant and promising trends. It establishes an open, community-driven list of the known low voltage level open datasets to encourage further research and development.Comment: 37 pages, 6 figures, 2 tables, review pape