6,373 research outputs found
Numerical investigation of a 3D hybrid high-order method for the indefinite time-harmonic Maxwell problem
Hybrid High-Order (HHO) methods are a recently developed class of methods belonging to
the broader family of Discontinuous Sketetal methods. Other well known members of the
same family are the well-established Hybridizable Discontinuous Galerkin (HDG) method,
the nonconforming Virtual Element Method (ncVEM) and the Weak Galerkin (WG) method.
HHO provides various valuable assets such as simple construction, support for fully-polyhedral
meshes and arbitrary polynomial order, great computational efficiency, physical accuracy and
straightforward support for hp-refinement. In this work we propose an HHO method for the
indefinite time-harmonic Maxwell problem and we evaluate its numerical performance. In
addition, we present the validation of the method in two different settings: a resonant cavity
with Dirichlet conditions and a parallel plate waveguide problem with a total/scattered field
decomposition and a plane-wave boundary condition. Finally, as a realistic application, we
demonstrate HHO used on the study of the return loss in a waveguide mode converter
Exploiting Structural Properties in the Analysis of High-dimensional Dynamical Systems
The physical and cyber domains with which we interact are filled with high-dimensional dynamical systems. In machine learning, for instance, the evolution of overparametrized neural networks can be seen as a dynamical system. In networked systems, numerous agents or nodes dynamically interact with each other. A deep understanding of these systems can enable us to predict their behavior, identify potential pitfalls, and devise effective solutions for optimal outcomes. In this dissertation, we will discuss two classes of high-dimensional dynamical systems with specific structural properties that aid in understanding their dynamic behavior.
In the first scenario, we consider the training dynamics of multi-layer neural networks. The high dimensionality comes from overparametrization: a typical network has a large depth and hidden layer width. We are interested in the following question regarding convergence: Do network weights converge to an equilibrium point corresponding to a global minimum of our training loss, and how fast is the convergence rate? The key to those questions is the symmetry of the weights, a critical property induced by the multi-layer architecture. Such symmetry leads to a set of time-invariant quantities, called weight imbalance, that restrict the training trajectory to a low-dimensional manifold defined by the weight initialization. A tailored convergence analysis is developed over this low-dimensional manifold, showing improved rate bounds for several multi-layer network models studied in the literature, leading to novel characterizations of the effect of weight imbalance on the convergence rate.
In the second scenario, we consider large-scale networked systems with multiple weakly-connected groups. Such a multi-cluster structure leads to a time-scale separation between the fast intra-group interaction due to high intra-group connectivity, and the slow inter-group oscillation, due to the weak inter-group connection. We develop a novel frequency-domain network coherence analysis that captures both the coherent behavior within each group, and the dynamical interaction between groups, leading to a structure-preserving model-reduction methodology for large-scale dynamic networks with multiple clusters under general node dynamics assumptions
Symmetries of Riemann surfaces and magnetic monopoles
This thesis studies, broadly, the role of symmetry in elucidating structure. In particular, I investigate the role that automorphisms of algebraic curves play in three specific contexts; determining the orbits of theta characteristics, influencing the geometry of the highly-symmetric Bring’s curve, and in constructing magnetic monopole solutions. On theta characteristics, I show how to turn questions on the existence of invariant characteristics into questions of group cohomology, compute comprehensive tables of orbit decompositions for curves of genus 9 or less, and prove results on the existence of infinite families of curves with invariant characteristics. On Bring’s curve, I identify key points with geometric significance on the curve, completely determine the structure of the quotients by subgroups of automorphisms, finding new elliptic curves in the process, and identify the unique invariant theta characteristic on the curve. With respect to monopoles, I elucidate the role that the Hitchin conditions play in determining monopole spectral curves, the relation between these conditions and the automorphism group of the curve, and I develop the theory of computing Nahm data of symmetric monopoles. As such I classify all 3-monopoles whose Nahm data may be solved for in terms of elliptic functions
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
Quantum-Classical hybrid systems and their quasifree transformations
The focus of this work is the description of a framework for quantum-classical hybrid systems.
The main emphasis lies on continuous variable systems described by canonical commutation relations and, more precisely, the quasifree case.
Here, we are going to solve two main tasks:
The first is to rigorously define spaces of states and observables, which are naturally connected within the general structure.
Secondly, we want to describe quasifree channels for which both the Schrödinger picture and the Heisenberg picture are well defined.
We start with a general introduction to operator algebras and algebraic quantum theory.
Thereby, we highlight some of the mathematical details that are often taken for granted while working with purely quantum systems.
Consequently, we discuss several possibilities and their advantages respectively disadvantages in describing classical systems analogously to the quantum formalism.
The key takeaway is that there is no candidate for a classical state space or observable algebra that can be put easily alongside a quantum system to form a hybrid and simultaneously fulfills all of our requirements for such a partially quantum and partially classical system.
Although these straightforward hybrid systems are not sufficient enough to represent a general approach, we use one of the candidates to prove an intermediate result, which showcases the advantages of a consequent hybrid ansatz:
We provide a hybrid generalization of classical diffusion generators where the exchange of information between the classical and the quantum side is controlled by the induced noise on the quantum system.
Then, we present solutions for our initial tasks.
We start with a CCR-algebra where some variables may commute with all others and hence generate a classical subsystem.
After clarifying the necessary representations, our hybrid states are given by continuous characteristic functions, and the according state space is equal to the state space of a non-unital C*-algebra.
While this C*-algebra is not a suitable candidate for an observable algebra itself, we describe several possible subsets in its bidual which can serve this purpose.
They can be more easily characterized and will also allow for a straightforward definition of a proper Heisenberg picture.
The subsets are given by operator-valued functions on the classical phase space with varying degrees of regularity, such as universal measurability or strong*-continuity.
We describe quasifree channels and their properties, including a state-channel correspondence, a factorization theorem, and some basic physical operations.
All this works solely on the assumption of a quasifree system, but we also show that the more famous subclass of Gaussian systems fits well within this formulation and behaves as expected
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
Exploring the Limits of Controlled Markovian Quantum Dynamics with Thermal Resources
Our aim is twofold: First, we rigorously analyse the generators of
quantum-dynamical semigroups of thermodynamic processes. We characterise a wide
class of GKSL-generators for quantum maps within thermal operations and argue
that every infinitesimal generator of (a one-parameter semigroup of) Markovian
thermal operations belongs to this class. We completely classify and visualise
them and their non-Markovian counterparts for the case of a single qubit.
Second, we use this description in the framework of bilinear control systems
to characterise reachable sets of coherently controllable quantum systems with
switchable coupling to a thermal bath. The core problem reduces to studying a
hybrid control system ("toy model") on the standard simplex allowing for two
types of evolution: (i) instantaneous permutations and (ii) a one-parameter
semigroup of -stochastic maps. We generalise upper bounds of the reachable
set of this toy model invoking new results on thermomajorisation. Using tools
of control theory we fully characterise these reachable sets as well as the set
of stabilisable states as exemplified by exact results in qutrit systems.Comment: 46 pages mai
Pointless Global Bundle Adjustment With Relative Motions Hessians
Bundle adjustment (BA) is the standard way to optimise camera poses and to
produce sparse representations of a scene. However, as the number of camera
poses and features grows, refinement through bundle adjustment becomes
inefficient. Inspired by global motion averaging methods, we propose a new
bundle adjustment objective which does not rely on image features' reprojection
errors yet maintains precision on par with classical BA. Our method averages
over relative motions while implicitly incorporating the contribution of the
structure in the adjustment. To that end, we weight the objective function by
local hessian matrices - a by-product of local bundle adjustments performed on
relative motions (e.g., pairs or triplets) during the pose initialisation step.
Such hessians are extremely rich as they encapsulate both the features' random
errors and the geometric configuration between the cameras. These pieces of
information propagated to the global frame help to guide the final optimisation
in a more rigorous way. We argue that this approach is an upgraded version of
the motion averaging approach and demonstrate its effectiveness on both
photogrammetric datasets and computer vision benchmarks
Sampling with Barriers: Faster Mixing via Lewis Weights
We analyze Riemannian Hamiltonian Monte Carlo (RHMC) for sampling a polytope
defined by inequalities in endowed with the metric defined by the
Hessian of a convex barrier function. The advantage of RHMC over Euclidean
methods such as the ball walk, hit-and-run and the Dikin walk is in its ability
to take longer steps. However, in all previous work, the mixing rate has a
linear dependence on the number of inequalities. We introduce a hybrid of the
Lewis weights barrier and the standard logarithmic barrier and prove that the
mixing rate for the corresponding RHMC is bounded by , improving on the previous best bound of (based on the log barrier). This continues the general parallels
between optimization and sampling, with the latter typically leading to new
tools and more refined analysis. To prove our main results, we have to
overcomes several challenges relating to the smoothness of Hamiltonian curves
and the self-concordance properties of the barrier. In the process, we give a
general framework for the analysis of Markov chains on Riemannian manifolds,
derive new smoothness bounds on Hamiltonian curves, a central topic of
comparison geometry, and extend self-concordance to the infinity norm, which
gives sharper bounds; these properties appear to be of independent interest
Spectral Sparsification for Communication-Efficient Collaborative Rotation and Translation Estimation
We propose fast and communication-efficient optimization algorithms for
multi-robot rotation averaging and translation estimation problems that arise
from collaborative simultaneous localization and mapping (SLAM),
structure-from-motion (SfM), and camera network localization applications. Our
methods are based on theoretical relations between the Hessians of the
underlying Riemannian optimization problems and the Laplacians of suitably
weighted graphs. We leverage these results to design a collaborative solver in
which robots coordinate with a central server to perform approximate
second-order optimization, by solving a Laplacian system at each iteration.
Crucially, our algorithms permit robots to employ spectral sparsification to
sparsify intermediate dense matrices before communication, and hence provide a
mechanism to trade off accuracy with communication efficiency with provable
guarantees. We perform rigorous theoretical analysis of our methods and prove
that they enjoy (local) linear rate of convergence. Furthermore, we show that
our methods can be combined with graduated non-convexity to achieve
outlier-robust estimation. Extensive experiments on real-world SLAM and SfM
scenarios demonstrate the superior convergence rate and communication
efficiency of our methods.Comment: Revised extended technical report (37 pages, 15 figures, 6 tables
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