6,661 research outputs found
Tensor product approach to modelling epidemics on networks
To improve mathematical models of epidemics it is essential to move beyond the traditional assumption of homogeneous well--mixed population and involve more precise information on the network of contacts and transport links by which a stochastic process of the epidemics spreads. In general, the number of states of the network grows exponentially with its size, and a master equation description suffers from the curse of dimensionality. Almost all methods widely used in practice are versions of the stochastic simulation algorithm (SSA), which is notoriously known for its slow convergence. In this paper we numerically solve the chemical master equation for an SIR model on a general network using recently proposed tensor product algorithms. In numerical experiments we show that tensor product algorithms converge much faster than SSA and deliver more accurate results, which becomes particularly important for uncovering the probabilities of rare events, e.g. for number of infected people to exceed a (high) threshold
Self-supervised learning for transferable representations
Machine learning has undeniably achieved remarkable advances thanks to large labelled datasets and supervised learning. However, this progress is constrained by the labour-intensive annotation process. It is not feasible to generate extensive labelled datasets for every problem we aim to address. Consequently, there has been a notable shift in recent times toward approaches that solely leverage raw data. Among these, self-supervised learning has emerged as a particularly powerful approach, offering scalability to massive datasets and showcasing considerable potential for effective knowledge transfer. This thesis investigates self-supervised representation learning with a strong focus on computer vision applications. We provide a comprehensive survey of self-supervised methods across various modalities, introducing a taxonomy that categorises them into four distinct families while also highlighting practical considerations for real-world implementation. Our focus thenceforth is on the computer vision modality, where we perform a comprehensive benchmark evaluation of state-of-the-art self supervised models against many diverse downstream transfer tasks. Our findings reveal that self-supervised models often outperform supervised learning across a spectrum of tasks, albeit with correlations weakening as tasks transition beyond classification, particularly for datasets with distribution shifts. Digging deeper, we investigate the influence of data augmentation on the transferability of contrastive learners, uncovering a trade-off between spatial and appearance-based invariances that generalise to real-world transformations. This begins to explain the differing empirical performances achieved by self-supervised learners on different downstream tasks, and it showcases the advantages of specialised representations produced with tailored augmentation. Finally, we introduce a novel self-supervised pre-training algorithm for object detection, aligning pre-training with downstream architecture and objectives, leading to reduced localisation errors and improved label efficiency. In conclusion, this thesis contributes a comprehensive understanding of self-supervised representation learning and its role in enabling effective transfer across computer vision tasks
Classical and quantum algorithms for scaling problems
This thesis is concerned with scaling problems, which have a plethora of connections to different areas of mathematics, physics and computer science. Although many structural aspects of these problems are understood by now, we only know how to solve them efficiently in special cases.We give new algorithms for non-commutative scaling problems with complexity guarantees that match the prior state of the art. To this end, we extend the well-known (self-concordance based) interior-point method (IPM) framework to Riemannian manifolds, motivated by its success in the commutative setting. Moreover, the IPM framework does not obviously suffer from the same obstructions to efficiency as previous methods. It also yields the first high-precision algorithms for other natural geometric problems in non-positive curvature.For the (commutative) problems of matrix scaling and balancing, we show that quantum algorithms can outperform the (already very efficient) state-of-the-art classical algorithms. Their time complexity can be sublinear in the input size; in certain parameter regimes they are also optimal, whereas in others we show no quantum speedup over the classical methods is possible. Along the way, we provide improvements over the long-standing state of the art for searching for all marked elements in a list, and computing the sum of a list of numbers.We identify a new application in the context of tensor networks for quantum many-body physics. We define a computable canonical form for uniform projected entangled pair states (as the solution to a scaling problem), circumventing previously known undecidability results. We also show, by characterizing the invariant polynomials, that the canonical form is determined by evaluating the tensor network contractions on networks of bounded size
Algorithms and complexity for approximately counting hypergraph colourings and related problems
The past decade has witnessed advancements in designing efficient algorithms for approximating the number of solutions to constraint satisfaction problems (CSPs), especially in the local lemma regime. However, the phase transition for the computational tractability is not known. This thesis is dedicated to the prototypical problem of this kind of CSPs, the hypergraph colouring. Parameterised by the number of colours q, the arity of each hyperedge k, and the vertex maximum degree Î, this problem falls into the regime of LovĂĄsz local lemma when ΠⲠqáľ. In prior, however, fast approximate counting algorithms exist when ΠⲠqáľ/Âł, and there is no known inapproximability result. In pursuit of this, our contribution is two-folded, stated as follows.
⢠When q, k ⼠4 are evens and Π⼠5¡qáľ/², approximating the number of hypergraph colourings is NP-hard.
⢠When the input hypergraph is linear and ΠⲠqáľ/², a fast approximate counting algorithm does exist
Optimising heating and cooling of smart buildings
This thesis is concerned with optimization techniques to improve the efficiency of heating and
cooling of both existing and new buildings. We focus on the thermal demand-side and we make
novel contributions to the optimality of both design and operational questions. We demonstrate
that our four novel contributions can reduce operations cost and consumption, optimize retrofit
and estimate relevant parameters of the built environment. The ultimate objective of this work is
to provide affordable and cost-effective solutions that take advantage of local existing resources.
This work addresses four gaps in the state-of-the-art. First, we contribute to current building
practice that is mostly based on human experience and simulations, which often leads to oversized
heating systems and low efficiency. The results in this thesis show the advantages of using
optimization approaches for thermal aspects in buildings. We propose models that seek optimal
decisions for one specific design day, as well as an approach that optimizes multiple day-scenarios
to more accurately represent a whole year.
Second, we study the full potential of buildingsâ thermal mass and design. This has not been
fully explored due to two factors: the complexity of the mathematics involved, and the fast developing
and variety of emerging technologies and approaches. We tackle the mathematical challenge by
solving non-linear non-convex models with integer decisions and by estimating buildingâs thermal
mass. We support rapid architectural development by studying flexible models able to adapt to
the latest building technologies such as passive house design, smart façades, and dynamic shadings.
Third, we consider flexibility provision to significantly reduce total energy costs. Flexibility
studies often only focus on flexible building loads but do not consider heating, which is often the
largest load of a building and is less flexible. Because of that, we study and model a buildingâs
heating demand and we propose optimization techniques to support greater flexibility of heating
loads, allowing buildings to participate more efficiently in providing demand response.
Fourth, we consider a building as an integrated system, unlike many other modelling approaches that focus on single aspects. We model a building as a complex system comprising the buildingâs structure, weather conditions and usersâ requirements. Furthermore, we account for design decisions and for new and emerging technologies, such as heat pumps and thermal storage. Optimal decisions come from the joint analysis of all these interconnected factors.
The thesis is structured in three parts: the introduction, the main body and the conclusions. The main body is made by five chapters, each of which focuses on one research project and has the
following structure: overview, introduction, literature review, mathematical framework description,
application and results section, conclusion and future works. The first two chapters discuss the
optimization of operational aspects. The first focuses on a single thermal zone and the second in
two connected ones. The third chapter is a continuation of the first two, and presents an approach
to optimize both operations and design of buildings in a heat community. This approach integrates
the use of an energy software already in the market. The fourth chapter discusses an approach to
find the optimal refurbishment of an existing building at minimum cost. The fifth chapter shows
an inferring model to represent a house of a building stock. We study the common case where the
houseâs data is lacking or inaccurate, and we present a model that is able to estimate the required
thermal parameters for modelling the house using only heating demand
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
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