Prohibición de Reglamentos por decisión de la Ley

In this thesis, the problem of uncertainty propagation with a computationally expensive model is considered. Two variants are discussed: one where the distribution is known explicitly (for instance described by a probability density function or by samples) and one where the distribution is calibrated and therefore depends on the model.

If the distribution is known explicitly, there are various existing approaches to con-struct nodal sets for interpolation or integration. However, many of these approaches require stringent assumptions on the input distribution. Moreover these assumption are not applicable in wind turbine load calculations. In this thesis alternatives are developed, see Chapters3and4. These are applicable to a wide range of problems. Mathematically, the accuracy of the approaches is assessed using test functions. To illustrate the

applica-bility to computationally complex problems, the example of computing the flow over an airfoil incorporating uncertainty (which is a relevant problem for wind turbine blade design) is considered various times in this thesis. Moreover, the approaches are directly applicable to load calculations, which is demonstrated in Chapter5.

If the distribution is not known explicitly, the problem is more subtle. The usual approach, i.e. Markov chain Monte Carlo, is often not applicable if the model under consideration is computationally complex. Recently alternatives to alleviate this have been developed, which are further extended and generalized in this thesis, see Chapter6 and7. Again, the example of calculating the flow over an airfoil is considered.

A geometrical interpretation of interpolatory quadrature rules

Quadrature rules are collocation methods that are used oftentimes in this thesis to determine weighted integrals such as (2.1). As discussed in Chapter2, a quadrature rules converges if it has positive weights and integrates a large number of polynomials, or in other words, has high degree. Moreover, from a computational perspective we are mainly interested in nested quadrature rules, which allow for straightforward refinement of quadrature rule estimations. Many quadrature rules exist that have two of these three properties (i.e. nested, high degree, and positive weights) and it is non-trivial to directly construct rules that have all three properties. The focus of this chapter is not directly on constructing quadrature rules, but on gaining insight and deriving procedures that modify existing quadrature rules, which aid the construction of quadrature rules later in this thesis.

The approach taken is to derive a mathematical framework describing modifications of univariate interpolatory quadrature rules. More specifically, three elementary opera-tions are proposed: the addition of nodes, the removal of nodes, and the replacement of nodes. All operations are designed to preserve positive weights, keep the quadrature rule interpolatory, and by construction yield nested quadrature rules. They form the key ingredients of the proposed methodologies in subsequent chapters.

3.1. Introduction

As discussed briefly in Section2.1.3, the best-known interpolatory quadrature rule is possibly the Gaussian quadrature rule [74], which exists for virtually any probability

The majority of this chapter is based on the following article: L. M. M. van den Bos and B. Sanderse. A geometric approach for the addition of nodes to an interpolatory quadrature rule with positive weights. Under review, 2019. arXiv:1902.07477 [math.NA].

Section3.3, that describes the removal of nodes, is based on the following article: L. M. M. van den Bos, B. Koren, and R. P. Dwight. Non-intrusive uncertainty quantification using reduced cubature rules. Journal of Computational Physics, 332:418–445, 2017. DOI:10.1016/j.jcp.2016.12.011. arXiv:1905.06177 [math.NA].

distribution with finite moments. It has positive weights and maximal polynomial degree. However, the nodes are not nested. The Gauss–Kronrod quadrature rule is an extension of a Gaussian quadrature rule, such that two nested rules with positive weights are obtained [103,123,185]. The Gauss–Kronrod–Patterson quadrature rule [124, 142] (or simply Gauss–Patterson quadrature rule) further extends this idea by repeatedly applying the same algorithm, such that a sequence of nested rules is obtained. However, it does not exist for any distribution [93,94]. Even though many other extensions have been proposed over the years [67,102,115], in general it is difficult to obtain a series of nested quadrature rules with positive weights based on Gaussian rules [125]. Moreover often the smallest possible granularity between two consecutive nested quadrature rules can only be found by exhaustive search [22].

Another large group of well-known quadrature rules is formed by the Clenshaw–

Curtis quadrature rules [35], or simply those quadrature rules that are based on Cheby-shev approximations. The Clenshaw–Curtis rule is formed by the ChebyCheby-shev extrema and symbolic expressions of its nodes are known, i.e. see (2.16). Besides having excellent interpolation properties [84], it is well known that these quadrature rules have posi-tive weights if the distribution under consideration is uniform (explicit expressions are known [180]). Moreover for non-uniform distributions, the condition number of the quadrature rule converges to unity [24]. However, the vanilla Clenshaw–Curtis nodes are only nested for exponentially growing numbers of nodes [79].

Both the Gaussian and Clenshaw–Curtis quadrature rules have explicitly predefined nodes based on the roots of orthogonal polynomials. This results in accurate quadrature rules, but the construction of an accurate nested quadrature rule with fine granularity based on these rules remains notoriously difficult.

The goal of this chapter is to propose a geometrical framework that mathematically describes the three elementary operations mentioned in the beginning of this chapter, i.e. the addition, removal, and replacement of nodes. All operations are based on the geometrical interpretation of the linear system describing the nodes and the weights [15, 41,144], which yields necessary and sufficient conditions for a quadrature rule to have positive weights. The removal, addition, or replacement of a single node can be deter-mined analytically, whereas numerical methods are required to determine all sequences of multiple nodes that can be added or replaced in a quadrature rule. The focus of this chapter is mainly on the mathematical aspects and not on the numerical construction of quadrature rules.

This chapter is structured as follows. In Section3.2the nomenclature and prob-lem setting considered in this chapter is discussed, which is based on the notation introduced in Section2.1.3. Then the three operations discussed above are considered.

First, the removal of nodes is introduced in Section3.3. It is always possible to remove a node from a quadrature rule such that the obtained rule has positive weights. By inverting the operation of removing a node, the addition of one or multiple nodes can be described. The problem of adding a single node can be solved analytically, which is done in Section3.4. Contrary to the removal of nodes, it is not always possible to add a node to a quadrature rule such that the obtained rule has positive weights. Therefore the theory is extended to adding multiple nodes in Section3.5, where the results developed for adding a single node will be used extensively. It is always possible to add multiple

nodes to a quadrature rule, provided that any number of nodes may be added to the rule. The replacement of one or multiple nodes follows from combining both operations, i.e. firstly adding a node and secondly removing a node. Any node in the quadrature rule can be replaced by a new node and all possible values of these new nodes can be described analytically. Section3.6contains numerical examples of quadrature rules to demonstrate simple applications of the proposed framework. The chapter is concluded in Section3.7.