13 research outputs found
Caustic Skeleton & Cosmic Web
We present a general formalism for identifying the caustic structure of an
evolving mass distribution in an arbitrary dimensional space. For the class of
Hamiltonian fluids the identification corresponds to the classification of
singularities in Lagrangian catastrophe theory. Based on this we develop a
theoretical framework for the formation of the cosmic web, and specifically
those aspects that characterize its unique nature: its complex topological
connectivity and multiscale spinal structure of sheetlike membranes, elongated
filaments and compact cluster nodes. The present work represents an extension
of the work by Arnol'd et al., who classified the caustics for the 1- and
2-dimensional Zel'dovich approximation. His seminal work established the role
of emerging singularities in the formation of nonlinear structures in the
universe. At the transition from the linear to nonlinear structure evolution,
the first complex features emerge at locations where different fluid elements
cross to establish multistream regions. The classification and characterization
of these mass element foldings can be encapsulated in caustic conditions on the
eigenvalue and eigenvector fields of the deformation tensor field. We introduce
an alternative and transparent proof for Lagrangian catastrophe theory, and
derive the caustic conditions for general Lagrangian fluids, with arbitrary
dynamics, including dissipative terms and vorticity. The new proof allows us to
describe the full 3-dimensional complexity of the gravitationally evolving
cosmic matter field. One of our key findings is the significance of the
eigenvector field of the deformation field for outlining the spatial structure
of the caustic skeleton. We consider the caustic conditions for the
3-dimensional Zel'dovich approximation, extending earlier work on those for 1-
and 2-dimensional fluids towards the full spatial richness of the cosmic web
Integrable Matrix Models in Discrete Space-Time
We introduce a class of integrable dynamical systems of interacting classical
matrix-valued fields propagating on a discrete space-time lattice, realized as
many-body circuits built from elementary symplectic two-body maps. The models
provide an efficient integrable Trotterization of non-relativistic
-models with complex Grassmannian manifolds as target spaces,
including, as special cases, the higher-rank analogues of the Landau-Lifshitz
field theory on complex projective spaces. As an application, we study
transport of Noether charges in canonical local equilibrium states. We find a
clear signature of superdiffusive behavior in the Kardar-Parisi-Zhang
universality class, irrespectively of the chosen underlying global unitary
symmetry group and the quotient structure of the compact phase space, providing
a strong indication of superuniversal physics.Comment: v2, 60 pages, 10 figures, 1 tabl
Tissue-scale, patient-specific modeling and simulation of prostate cancer growth
Programa Oficial de Doutoramento en Enxeñaría Civil . 5011V01[Abstract]
Prostate cancer is a major health problem among aging men worldwide. This pathology
is easier to cure in its early stages, when it is still organ-confined. However, it hardly
ever produces any symptom until it becomes excessively large or has invaded other
tissues. Hence, the current approach to combat prostate cancer is a combination of
prevention and regular screening for early detection. Indeed, most cases of prostate
cancer are diagnosed and treated when it is localized within the organ. Despite the wealth
of accumulated knowledge on the biological basis and clinical management of the disease,
we lack a comprehensive theoretical model into which we can organize and understand
the abundance of data on prostate cancer. Additionally, the standard clinical practice
in oncology is largely based on statistical patterns, which is not sufficiently accurate to
individualize the diagnosis, prediction of prognosis, treatment, and follow-up.
Recently, mathematical modeling and simulation of cancer and their treatments have
enabled the prediction of clinical outcomes and the design of optimal therapies on a
patient-specific basis. This new trend in medical research has been termed mathematical
oncology. Prostate cancer is an ideal candidate to benefit from this technology for several
reasons. First, patient-specific clinical approaches may contribute to reduce the rates of
overtreatment and undertreatment of prostate cancer. Multiparametric magnetic resonance
is increasingly used to monitor and diagnose this disease. This imaging technology can
provide abundant information to build a patient-specific mathematical model of prostate
cancer growth. Moreover, the prostate is a sufficiently small organ to pursue tissue-scale
predictive simulations. Prostate cancer growth can also be estimated using the serum
concentration of a biomarker known as the prostate specific antigen. Additionally, some
prostate cancer patients do not receive any treatment but are clinically monitored and
periodically imaged, which opens the door to in vivo model validation. The advent of
versatile and powerful technologies in computational mechanics permits to address the
challenges posed by the prostate anatomy and the resolution of the mathematical models.
Finally, mathematical oncology technologies can guide the future research on prostate
cancer, e.g., proposing new treatment strategies or unveiling mechanisms involved in
tumor growth. Therefore, the aim of this thesis is to provide a computational framework for the tissuescale,
patient-specific modeling and simulation of organ-confined PCa growth within
the context of mathematical oncology. We present a model for localized prostate cancer
growth that reproduces the growth patterns of the disease observed in experimental
and clinical studies. To capture the coupled dynamics of healthy and tumoral tissue,
we use the phase-field method together with reaction-diffusion equations for nutrient
consumption and prostate specific antigen production. We leverage this model to run the
first tissue-scale, patient-specific simulations of prostate cancer growth over the organ
anatomy extracted from medical images. Our results show similar tumor progression as
observed in clinical practice.
We leverage isogeometric analysis to handle the nonlinearity of our set of equations,
as well as the complex anatomy of the prostate and the intricate tumoral morphologies.
We further advocate dynamical mesh adaptivity to speed up calculations, rationalize
computational resources, and facilitate simulation in a clinically relevant time. We
present a set of efficient algorithms to accommodate local h-refinement and h-coarsening
of hierarchical splines in isogeometric analysis. Our methods are based on Bézier
projection, which we extend to hierarchical spline spaces. We also introduce a balance
parameter to control the overlapping of basis functions across the levels of the hierarchy,
leading to improved numerical conditioning. Our simulations of cancer growth show
remarkable accuracy with very few degrees of freedom in comparison to the uniform
mesh that the same simulation would require.
Finally, we study the interaction between prostate cancer and benign prostatic hyperplasia,
another common prostate pathology that causes the organ to gradually enlarge. In
particular, we investigate why tumors originating in larger prostates present favorable
pathological features. We perform a qualitative simulation study by extending our
mathematical model of prostate cancer growth to include the equations of mechanical
equilibrium and the coupling terms between them and tumor dynamics. We assume that
the deformation of the prostate is a quasistatic phenomenon and we model prostatic tissue
as a linear elastic, heterogeneous, isotropic material. This model is calibrated by studying
the deformation caused by either disease independently. Our simulations show that a
history of benign prostatic hyperplasia creates mechanical stress fields in the prostate that
hamper prostatic tumor growth and limit its invasiveness.[Resumen]
El cáncer de próstata es un gran problema de salud en hombres de edad avanzada en
todo el mundo. Esta patología es más fácil de curar en sus estadios iniciales, cuando
aún es órgano-confinada. Sin embargo, casi nunca produce ningún síntoma hasta que es
demasiado grande o ha invadido otros tejidos. Por tanto, el enfoque actual para combatir
el cáncer de próstata es una combinación de prevención y exámenes rutinarios para una
detección precoz. De hecho, la mayoría de casos de cáncer de próstata son diagnosticados
y tratados cuando aún está localizado dentro del órgano. A pesar de la riqueza del
conocimiento acumulado sobre las bases biológicas y la gestión clínica de la enfermedad,
carecemos de un modelo teórico completo en el que podamos organizar y comprender
la enorme cantidad de datos existentes sobre el cáncer de próstata. Además, la práctica
clínica estándar en oncología está basada en gran medida en patrones estadísticos, lo
cual no es suficientemente preciso para individualizar el diagnóstico, la predicción de la
prognosis, el tratamiento y el seguimiento.
Recientemente, la modelización y la simulación matemáticas del cáncer y sus tratamientos
han permitido predecir resultados clínicos y el diseño de terapias óptimas de
forma personalizada. Esta nueva corriente de investigación médica se ha denominado
oncología matemática. El cáncer de próstata es un candidato ideal para beneficiarse de
esta tecnología por varios motivos. En primer lugar, un enfoque clínico personalizado
podría contribuir a reducir las tasas de tratamiento excesivo o insuficiente de cáncer de
próstata. La resonancia magnética multiparamétrica se usa cada vez más para monitorizar
y diagnosticar esta enfermedad. Esta tecnología de imagen puede proporcionar abundante
información para construir un modelo matemático de crecimiento de cáncer de próstata
personalizado. Además, la próstata es un órgano suficientemente pequeño para perseguir
la realización de simulaciones predictivas a escala tisular. El crecimiento del cáncer de
próstata también se puede estimar usando la concentración en sangre de un biomarcador
conocido como el antígeno prostático específico. Adicionalmente, algunos pacientes de
cáncer de próstata no reciben tratamiento pero son monitorizados clínicamente y se les
toman imágenes médicas periódicamente, lo que abre la puerta a la validación in vivo de
modelos. El desarrollo de tecnologías versátiles y potentes en mecánica computacional permite hacer frente a los retos derivados de la anatomía prostática y la resolución de los
modelos matemáticos. Finalmente, las tecnologías de oncología matemática pueden guiar
las investigaciones futuras sobre cáncer de próstata, por ejemplo, proponiendo nuevas
estrategias de tratamiento o descubriendo mecanismos involucrados en el crecimiento
tumoral.
Por tanto, el objeto de esta tesis es proporcionar un marco computacional para la modelización
y simulación del crecimiento del cáncer de próstata órgano-confinado de
forma personalizada y a escala tisular dentro del contexto de la oncología matemática.
Presentamos un modelo de crecimiento de cáncer de próstata localizado que reproduce
los patrones de crecimiento de la enfermedad observados en estudios experimentales y
clínicos. Para capturar las dinámicas acopladas de los tejidos sano y tumoral, usamos el
método de campo de fase junto con ecuaciones de reacción-difusión para el consumo
de nutriente y la producción de antígeno prostático específico. Empleamos este modelo
para realizar las primeras simulaciones personalizadas a escala tisular del crecimiento de
cáncer de próstata sobre la anatomía del órgano extraída de imágenes médicas. Nuestros
resultados muestran una progresión tumoral similar a la observada en la práctica clínica.
Utilizamos el análisis isogeométrico para resolver la no-linealidad de nuestro sistema de
ecuaciones, así como la compleja anatomía de la próstata y las intricadas morfologías
tumorales. Adicionalmente, proponemos el uso de adaptatividad dinámica de malla para
acelerar los cálculos, racionalizar los recursos computacionales y facilitar la simulación en
un tiempo clínicamente relevante. Presentamos un conjunto de algoritmos eficientes para
introducir el refinamiento y el engrosado locales tipo h en análisis isogeométrico. Nuestros
métodos están basados en la proyección de Bézier, que extendemos a los espacios de
splines jerárquicas. También introducimos un parámetro de balance para controlar la
superposición de funciones de base a través de los niveles de la jerarquía, lo cual conduce
a un condicionamiento numérico mejorado. Nuestras simulaciones de crecimiento de
cáncer muestran una notable precisión con muy pocos grados de libertad en comparación
con la malla uniforme que la misma simulación requeriría.
Finalmente, estudiamos la interacción entre el cáncer de próstata y la hiperplasia benigna
de próstata, otra patología prostática común que hace crecer al órgano gradualmente. En
particular, investigamos por qué los tumores que se originan en próstatas más grandes
presentan características patológicas favorables. Realizamos un estudio de simulación
cualitativo extendiendo nuestro modelo matemático de crecimiento de cáncer de próstata
para incluir las ecuaciones de equilibrio mecánico y los términos de acoplamiento entre
estas y la dinámica tumoral. Asumimos que la deformación de la próstata es un fenómeno
cuasiestático y modelamos el tejido prostático como un material elástico lineal, heterogéneo
e isotrópico. Este modelo es calibrado estudiando la deformación causada por
cada enfermedad independientemente. Nuestras simulaciones muestran que un historial
de hiperplasia benigna de próstata crea campos de tensión mecánica en la próstata que
obstaculizan el crecimiento del cáncer de próstata y limitan su invasividad.[Resumo]
O cancro de próstata é un gran problema de saúde en homes de idade avanzada en todo
o mundo. Esta patoloxía é máis fácil de curar nos seus estadios iniciais, cando aínda
é órgano-confinada. Porén, case nunca produce ningún síntoma ata que é demasiado
grande ou ten invadido outros tecidos. Polo tanto, o enfoque actual para combater o
cancro de próstata é unha combinación de prevención e exames rutinarios para unha
detección precoz. De feito, a maioría de casos de cancro de próstata son diagnosticados e
tratados cando aínda está localizado dentro do órgano. Malia a riqueza do coñecemento
acumulado sobre as bases biolóxicas e a xestión clínica da doenza, carecemos dun modelo
teórico completo no que podamos organizar e comprender a enorme cantidade de datos
existentes sobre o cancro de próstata. Ademais, a práctica clínica estándar en oncoloxía
está baseada en gran medida en patróns estatísticos, o cal non é suficientemente preciso
para individualizar a diagnose, a predición da prognose, o tratamento e o seguimento.
Recentemente, a modelización e a simulación matemáticas do cancro e os seus tratamentos
permitiron predicir resultados clínicos e o deseño de terapias óptimas de forma
personalizada. Esta nova corrente de investigación médica denomínase oncoloxía matemática.
O cancro de próstata é un candidato ideal para beneficiarse desta tecnoloxía por
varios motivos. En primeiro lugar, un enfoque clínico personalizado podería contribuír a
reducir as taxas de tratamento excesivo ou insuficiente de cancro de próstata. A resonancia
magnética multiparamétrica úsase cada vez máis para monitorizar e diagnosticar esta
enfermidade. Esta tecnoloxía de imaxe pode proporcionar abundante información para
construír un modelo matemático de crecemento de cancro de próstata personalizado.
Ademais, a próstata é un órgano suficientemente pequeno para perseguir a realización
de simulacións preditivas a escala tisular. O crecemento do cancro de próstata tamén
se pode estimar usando a concentración en sangue dun biomarcador coñecido como o
antíxeno prostático específico. Adicionalmente, algúns pacientes de cancro de próstata
non reciben tratamento pero son monitorizados clinicamente e se lles toman imaxes
médicas periodicamente, o que abre a porta á validación in vivo de modelos. O desenvolvemento
de tecnoloxías versátiles e potentes en mecánica computacional permite facer
fronte aos retos derivados da anatomía prostática e a resolución dos modelos matemáticos. Finalmente, as tecnoloxías de oncoloxía matemática poden guiar as investigacións futuras
sobre cancro de próstata, por exemplo, propoñendo novas estratexias de tratamento ou
descubrindo mecanismos involucrados no crecemento tumoral.
Polo tanto, o obxecto desta tese é proporcionar un marco computacional para a modelización
e simulación do crecemento do cancro de próstata órgano-confinado de forma
personalizada e a escala tisular dentro do contexto da oncoloxía matemática. Presentamos
un modelo de crecemento de cancro de próstata localizado que reproduce os patróns de
crecemento da enfermidade observados en estudos experimentais e clínicos. Para capturar
as dinámicas acopladas dos tecidos san e tumoral, usamos o método de campo de fase
xunto con ecuacións de reacción-difusión para o consumo de nutriente e a produción
de antíxeno prostático específico. Empregamos este modelo para realizar as primeiras
simulacións personalizadas a escala tisular do crecemento de cancro de próstata sobre
a anatomía do órgano extraída de imaxes médicas. Os nosos resultados amosan unha
progresión tumoral similar á observada na práctica clínica.
Utilizamos a análise isoxeométrica para resolver a non-linealidade do noso sistema
de ecuacións, así como a complexa anatomía da próstata e as intricadas morfoloxías
tumorais. Adicionalmente, propoñemos o uso de adaptatividade dinámica de malla para
acelerar os cálculos, racionalizar os recursos computacionais e facilitar a simulación nun
tempo clinicamente relevante. Presentamos un conxunto de algoritmos eficientes para
introducir o refinamento e o engrosado locais tipo h en análise isoxeométrica. Os nosos
métodos están baseados na proxección de Bézier, que estendemos aos espazos de splines
xerárquicas. Tamén introducimos un parámetro de balance para controlar a superposición
de funcións de base a través dos niveis da xerarquía, o cal conduce a un condicionamento
numérico mellorado. As nosas simulacións de crecemento de cancro amosan unha notable
precisión con moi poucos graos de liberdade en comparación coa malla uniforme que a
mesma simulación requiriría.
Finalmente, estudamos a interacción entre o cancro de próstata e a hiperplasia benigna
de próstata, outra patoloxía prostática común que fai crecer ao órgano gradualmente. En
particular, investigamos por que os tumores que se orixinan en próstatas máis grandes
presentan características patolóxicas favorables. Realizamos un estudo de simulación
cualitativo estendendo o noso modelo matemático de crecemento de cancro de próstata
para incluír as ecuacións de equilibrio mecánico e os termos de acoplamento entre estas e
a dinámica tumoral. Asumimos que a deformación da próstata é un fenómeno cuasiestático
e modelamos o tecido prostático como un material elástico lineal, heteroxéneo e
isotrópico. Este modelo é calibrado estudando a deformación causada por cada enfermidade
independientemente. As nosas simulacións amosan que un historial de hiperplasia
benigna de próstata crea campos de tensión mecánica na próstata que obstaculizan o
crecemento do cancro de próstata e limitan a súa invasividade
Path Integrals in the Sky: Classical and Quantum Problems with Minimal Assumptions
Cosmology has, after the formulation of general relativity, been transformed from a branch of philosophy into an active field in physics. Notwithstanding the significant improvements in our understanding of our Universe, there are still many open questions on both its early and late time evolution. In this thesis, we investigate a range of problems in classical and quantum cosmology, using advanced mathematical tools, and making only minimal assumptions. In particular, we apply Picard-Lefschetz theory, catastrophe theory, infinite dimensional measure theory, and weak-value theory.
To study the beginning of the Universe in quantum cosmology, we apply Picard-Lefschetz theory to the Lorentzian path integral for gravity. We analyze both the Hartle-Hawking no-boundary proposal and Vilenkin's tunneling proposal, and demonstrate that the Lorentzian path integral corresponding to the mini-superspace formulation of the two proposals is well-defined. However, when including fluctuations, we show that the path integral predicts the existence of large fluctuations. This indicates that the Universe cannot have had a smooth beginning in Euclidean de Sitter space. In response to these conclusions, the scientific community has made a series of adapted formulations of the no-boundary and tunneling proposals. We show that these new proposals suffer from similar issues.
Second, we generalize the weak-value interpretation of quantum mechanics to relativistic systems. We apply this formalism to a relativistic quantum particle in a constant electric field. We analyze the evolution of the relativistic particle in both the classical and the quantum regime and evaluate the back-reaction of the Schwinger effect on the electric field in -dimensional spacetime, using analytical methods. In addition, we develop a numerical method to evaluate both the wavefunction and the corresponding weak-values in more general electric and magnetic fields.
We conclude the quantum part of this thesis with a chapter on Lorentzian path integrals. We propose a new definition of the real-time path integral in terms of Brownian motion on the Lefschetz thimble of the theory. We prove the existence of a -measure for the path integral of the non-relativistic free particle, the (inverted) harmonic oscillator, and the relativistic particle in a range of potentials. We also describe how this proposal extends to more general path integrals.
In the classical part of this thesis, we analyze two problems in late-time cosmology. Multi-dimensional oscillatory integrals are prevalent in physics, but notoriously difficult to evaluate. We develop a new numerical method, based on multi-dimensional Picard-Lefschetz theory, for the evaluation of these integrals. The virtue of this method is that its efficiency increases when integrals become more oscillatory. The method is applied to interference patterns of lensed images near caustics described by catastrophe theory. This analysis can help us understand the lensing of astrophysical sources by plasma lenses, which is especially relevant in light of the proposed lensing mechanism for fast radio bursts.
Finally, we analyze large-scale structure formation in terms of catastrophe theory. We show that the geometric structure of the three-dimensional cosmic-web is determined by both the eigenvalue and the eigenvector fields of the deformation tensor. We formulate caustic conditions, classifying caustics using properties of these fields. When applied to the Zel'dovich approximation of structure formation, the caustic conditions enable us to construct a caustic skeleton of the three-dimensional cosmic-web in terms of the initial conditions
Chiral Random Matrix Theory: Generalizations and Applications
Kieburg M. Chiral Random Matrix Theory: Generalizations and Applications. Bielefeld: Fakultät für Physik; 2015
Categorical Quantum Dynamics
We use strong complementarity to introduce dynamics and symmetries within the
framework of CQM, which we also extend to infinite-dimensional separable
Hilbert spaces: these were long-missing features, which open the way to a
wealth of new applications. The coherent treatment presented in this work also
provides a variety of novel insights into the dynamics and symmetries of
quantum systems: examples include the extremely simple characterisation of
symmetry-observable duality, the connection of strong complementarity with the
Weyl Canonical Commutation Relations, the generalisations of Feynman's clock
construction, the existence of time observables and the emergence of quantum
clocks.
Furthermore, we show that strong complementarity is a key resource for
quantum algorithms and protocols. We provide the first fully diagrammatic,
theory-independent proof of correctness for the quantum algorithm solving the
Hidden Subgroup Problem, and show that strong complementarity is the feature
providing the quantum advantage. In quantum foundations, we use strong
complementarity to derive the exact conditions relating non-locality to the
structure of phase groups, within the context of Mermin-type non-locality
arguments. Our non-locality results find further application to quantum
cryptography, where we use them to define a quantum-classical secret sharing
scheme with provable device-independent security guarantees.
All in all, we argue that strong complementarity is a truly powerful and
versatile building block for quantum theory and its applications, and one that
should draw a lot more attention in the future.Comment: Thesis submitted for the degree of Doctor of Philosophy, Oxford
University, Michaelmas Term 2016 (273 pages
Exploiting Novel Deep Learning Architecture in Character Animation Pipelines
This doctoral dissertation aims to show a body of work proposed for improving different blocks in the character animation pipelines resulting in less manual work and more realistic character animation. To that purpose, we describe a variety of cutting-edge deep learning approaches that have been applied to the field of human motion modelling and character animation.
The recent advances in motion capture systems and processing hardware have shifted from physics-based approaches to data-driven approaches that are heavily used in the current game production frameworks. However, despite these
significant successes, there are still shortcomings to address. For example, the existing production pipelines contain processing steps such as marker
labelling in the motion capture pipeline or annotating motion primitives, which should be done manually. In addition, most of the current approaches for character animation used in game production are limited by the amount of stored animation data resulting in many duplicates and repeated patterns.
We present our work in four main chapters. We first present a large dataset of human motion called MoVi. Secondly, we show how machine learning approaches can be used to automate proprocessing data blocks of optical motion capture pipelines. Thirdly, we show how generative models can be used to generate batches of synthetic motion sequences given only weak control signals. Finally, we show how novel generative models can be applied to real-time character control in the game production
Exploiting Novel Deep Learning Architecture in Character Animation Pipelines
This doctoral dissertation aims to show a body of work proposed for improving different blocks in the character animation pipelines resulting in less manual work and more realistic character animation. To that purpose, we describe a variety of cutting-edge deep learning approaches that have been applied to the field of human motion modelling and character animation.
The recent advances in motion capture systems and processing hardware have shifted from physics-based approaches to data-driven approaches that are heavily used in the current game production frameworks. However, despite these
significant successes, there are still shortcomings to address. For example, the existing production pipelines contain processing steps such as marker
labelling in the motion capture pipeline or annotating motion primitives, which should be done manually. In addition, most of the current approaches for character animation used in game production are limited by the amount of stored animation data resulting in many duplicates and repeated patterns.
We present our work in four main chapters. We first present a large dataset of human motion called MoVi. Secondly, we show how machine learning approaches can be used to automate proprocessing data blocks of optical motion capture pipelines. Thirdly, we show how generative models can be used to generate batches of synthetic motion sequences given only weak control signals. Finally, we show how novel generative models can be applied to real-time character control in the game production
A Unified Framework for Gradient-based Hyperparameter Optimization and Meta-learning
Machine learning algorithms and systems are progressively becoming part of our societies, leading to a growing need of building a vast multitude of accurate, reliable and interpretable models which should possibly exploit similarities among tasks. Automating segments of machine learning itself seems to be a natural step to undertake to deliver increasingly capable systems able to perform well in both the big-data and the few-shot learning regimes. Hyperparameter optimization (HPO) and meta-learning (MTL) constitute two building blocks of this growing effort. We explore these two topics under a unifying perspective, presenting a mathematical framework linked to bilevel programming that captures existing similarities and translates into procedures of practical interest rooted in algorithmic differentiation. We discuss the derivation, applicability and computational complexity of these methods and establish several approximation properties for a class of objective functions of the underlying bilevel programs. In HPO, these algorithms generalize and extend previous work on gradient-based methods. In MTL, the resulting framework subsumes classic and emerging strategies and provides a starting basis from which to build and analyze novel techniques. A series of examples and numerical simulations offer insight and highlight some limitations of these approaches. Experiments on larger-scale problems show the potential gains of the proposed methods in real-world applications. Finally, we develop two extensions of the basic algorithms apt to optimize a class of discrete hyperparameters (graph edges) in an application to relational learning and to tune online learning rate schedules for training neural network models, an old but crucially important issue in machine learning
Precision theoretical methods for large-scale structure of the Universe
We develop new analytic methods to accurately describe the formation of cosmic large-scale structure. These methods are based on the path integral formalism and allows one to efficiently address a number of long-standing problems in the field.
We describe the non-linear evolution of the baryon acoustic oscillations (BAO) in the distribution of matter. We argue for the need for resummation of large infrared (IR) enhanced contributions from bulk flows. We show how this can be done via a systematic resummation of Feynman diagrams guided by well-defined power counting rules. We formulate IR resummation both in real and redshift spaces. For the latter we develop a new method that maps cosmological correlation functions from real to redshift space and retains their IR finiteness. Our results agree well with the N-body simulation data at the BAO scales. This establishes IR resummation within our approach as a robust and complete procedure and provides a consistent theoretical model for the BAO feature in the statistics of matter and biased tracers in real and redshift spaces.
Eventually, we perform a non-perturbative calculation of the 1-point probability distribution function (PDF) for the spherically-averaged matter density field. We evaluate the PDF in the saddle-point approximation and show how it factorizes into an exponent given by a spherically symmetric saddle-point solution and a prefactor produced by fluctuations. The prefactor splits into a monopole contribution which is evaluated exactly, and a factor corresponding to aspherical fluctuations. The latter is crucial for the consistency of the calculation: neglecting it would make the PDF incompatible with translational invariance. We compute the aspherical prefactor using a combination of analytic and numerical techniques, identify the sensitivity to the short-scale physics and argue that it must be properly renormalized. Finally, we compare our result with N-body simulation data and find an excellent agreement