New Generalized Definitions of Rough Membership Relations and Functions from Topological Point of View

In this paper, we shall integrate some ideas in terms of concepts in topology. In fact, we introduce two different views to define generalized membership relations and functions as mathematical tools to classify the sets and help for measuring exactness and roughness of sets. Moreover, we define several types of fuzzy sets. Comparisons between the induced operations were discussed. Finally, many results, examples and counter examples to indicate connections are investigated

Inductive machine learning of optimal modular structures: Estimating solutions using support vector machines

Structural optimization is usually handled by iterative methods requiring repeated samples of a physics-based model, but this process can be computationally demanding. Given a set of previously optimized structures of the same topology, this paper uses inductive learning to replace this optimization process entirely by deriving a function that directly maps any given load to an optimal geometry. A support vector machine is trained to determine the optimal geometry of individual modules of a space frame structure given a specified load condition. Structures produced by learning are compared against those found by a standard gradient descent optimization, both as individual modules and then as a composite structure. The primary motivation for this is speed, and results show the process is highly efficient for cases in which similar optimizations must be performed repeatedly. The function learned by the algorithm can approximate the result of optimization very closely after sufficient training, and has also been found effective at generalizing the underlying optima to produce structures that perform better than those found by standard iterative methods

Isolating Vector Boson Scattering at the LHC: gauge cancellations and the Equivalent Vector Boson Approximation vs complete calculations

We have studied the possibility of extracting the $W^+W^-\to W^+W^-$ signal using the process $us\to cd W^+W^-$ as a test case. We have investigated numerically the strong gauge cancellations between signal and irreducible background, analysing critically the reliability of the Equivalent Vector Boson Approximation which is commonly used to define the signal. Complete matrix elements are necessary to study Electro--Weak Symmetry Breaking effects at high $WW$ invariant mass.Comment: Final version published in PR

Data-efficient Neuroevolution with Kernel-Based Surrogate Models

Surrogate-assistance approaches have long been used in computationally expensive domains to improve the data-efficiency of optimization algorithms. Neuroevolution, however, has so far resisted the application of these techniques because it requires the surrogate model to make fitness predictions based on variable topologies, instead of a vector of parameters. Our main insight is that we can sidestep this problem by using kernel-based surrogate models, which require only the definition of a distance measure between individuals. Our second insight is that the well-established Neuroevolution of Augmenting Topologies (NEAT) algorithm provides a computationally efficient distance measure between dissimilar networks in the form of "compatibility distance", initially designed to maintain topological diversity. Combining these two ideas, we introduce a surrogate-assisted neuroevolution algorithm that combines NEAT and a surrogate model built using a compatibility distance kernel. We demonstrate the data-efficiency of this new algorithm on the low dimensional cart-pole swing-up problem, as well as the higher dimensional half-cheetah running task. In both tasks the surrogate-assisted variant achieves the same or better results with several times fewer function evaluations as the original NEAT.Comment: In GECCO 201

On spatially irregular ordinary differential equations and a pathwise volatility modelling framework

This thesis develops a new framework for modelling price processes in finance, such as an equity price or foreign exchange rate. This can be related to the conventional Ito calculus-based framework through the time integral of a price's squared volatility. In the new framework, corresponding processes are strictly increasing, solve random ODEs, and are composed with geometric Brownian motion to obtain price processes. The new framework has no dependence on stochastic calculus, so processes can be studied on a pathwise basis using probability-free ODE techniques and functional analysis. The ODEs considered depend on continuous driving functions which are `spatially irregular', meaning they need not have any spatial regularity properties such as Holder continuity. They are however strictly increasing in time, thus temporally asymmetric. When sensible initial values are chosen, IVP solutions are also strictly increasing, and the IVPs' solution set is shown to contain all differentiable bijections on the non-negative reals. This enables the modelling of any non-negative volatility path which is not zero over intervals, via the time derivative of solutions. Despite this generality, new well-posedness results establish the uniqueness of solutions going forwards in time, and continuity of the IVPs' solution map. Motivation to explore this framework comes from its connection with the Heston volatility model. The framework explains how Heston price processes can converge to an interval-valued generalisation of the NIG Levy process, and reveals a deeper relationship between integrated CIR processes and the IG Levy process. Within this framework, a `Riemann-Liouville-Heston' martingale model is defined which generalises these relationships to fractional counterparts. Implied volatilities from this model are simulated, and exhibit features characteristic of leading `rough' volatility models.Comment: The author's PhD thesis. Major extension of v2. 211 pages, 22 figure

Evolutionary Approaches to Optimization Problems in Chimera Topologies

Chimera graphs define the topology of one of the first commercially available quantum computers. A variety of optimization problems have been mapped to this topology to evaluate the behavior of quantum enhanced optimization heuristics in relation to other optimizers, being able to efficiently solve problems classically to use them as benchmarks for quantum machines. In this paper we investigate for the first time the use of Evolutionary Algorithms (EAs) on Ising spin glass instances defined on the Chimera topology. Three genetic algorithms (GAs) and three estimation of distribution algorithms (EDAs) are evaluated over $1000$ hard instances of the Ising spin glass constructed from Sidon sets. We focus on determining whether the information about the topology of the graph can be used to improve the results of EAs and on identifying the characteristics of the Ising instances that influence the success rate of GAs and EDAs.Comment: 8 pages, 5 figures, 3 table