9,709 research outputs found
Online Network Source Optimization with Graph-Kernel MAB
We propose Grab-UCB, a graph-kernel multi-arms bandit algorithm to learn
online the optimal source placement in large scale networks, such that the
reward obtained from a priori unknown network processes is maximized. The
uncertainty calls for online learning, which suffers however from the curse of
dimensionality. To achieve sample efficiency, we describe the network processes
with an adaptive graph dictionary model, which typically leads to sparse
spectral representations. This enables a data-efficient learning framework,
whose learning rate scales with the dimension of the spectral representation
model instead of the one of the network. We then propose Grab-UCB, an online
sequential decision strategy that learns the parameters of the spectral
representation while optimizing the action strategy. We derive the performance
guarantees that depend on network parameters, which further influence the
learning curve of the sequential decision strategy We introduce a
computationally simplified solving method, Grab-arm-Light, an algorithm that
walks along the edges of the polytope representing the objective function.
Simulations results show that the proposed online learning algorithm
outperforms baseline offline methods that typically separate the learning phase
from the testing one. The results confirm the theoretical findings, and further
highlight the gain of the proposed online learning strategy in terms of
cumulative regret, sample efficiency and computational complexity
Accelerated primal-dual methods with enlarged step sizes and operator learning for nonsmooth optimal control problems
We consider a general class of nonsmooth optimal control problems with
partial differential equation (PDE) constraints, which are very challenging due
to its nonsmooth objective functionals and the resulting high-dimensional and
ill-conditioned systems after discretization. We focus on the application of a
primal-dual method, with which different types of variables can be treated
individually and thus its main computation at each iteration only requires
solving two PDEs. Our target is to accelerate the primal-dual method with
either larger step sizes or operator learning techniques. For the accelerated
primal-dual method with larger step sizes, its convergence can be still proved
rigorously while it numerically accelerates the original primal-dual method in
a simple and universal way. For the operator learning acceleration, we
construct deep neural network surrogate models for the involved PDEs. Once a
neural operator is learned, solving a PDE requires only a forward pass of the
neural network, and the computational cost is thus substantially reduced. The
accelerated primal-dual method with operator learning is mesh-free, numerically
efficient, and scalable to different types of PDEs. The acceleration
effectiveness of these two techniques is promisingly validated by some
preliminary numerical results
Compressed and distributed least-squares regression: convergence rates with applications to Federated Learning
In this paper, we investigate the impact of compression on stochastic
gradient algorithms for machine learning, a technique widely used in
distributed and federated learning. We underline differences in terms of
convergence rates between several unbiased compression operators, that all
satisfy the same condition on their variance, thus going beyond the classical
worst-case analysis. To do so, we focus on the case of least-squares regression
(LSR) and analyze a general stochastic approximation algorithm for minimizing
quadratic functions relying on a random field. We consider weak assumptions on
the random field, tailored to the analysis (specifically, expected H\"older
regularity), and on the noise covariance, enabling the analysis of various
randomizing mechanisms, including compression. We then extend our results to
the case of federated learning.
More formally, we highlight the impact on the convergence of the covariance
of the additive noise induced by the algorithm.
We demonstrate despite the non-regularity of the stochastic field, that the
limit variance term scales with (where is the Hessian of the optimization problem and the
number of iterations) generalizing the rate for the vanilla LSR case where it
is (Bach and Moulines,
2013). Then, we analyze the dependency of on the
compression strategy and ultimately its impact on convergence, first in the
centralized case, then in two heterogeneous FL frameworks
Meso-scale FDM material layout design strategies under manufacturability constraints and fracture conditions
In the manufacturability-driven design (MDD) perspective, manufacturability of the product or system is the most important of the design requirements. In addition to being able to ensure that complex designs (e.g., topology optimization) are manufacturable with a given process or process family, MDD also helps mechanical designers to take advantage of unique process-material effects generated during manufacturing. One of the most recognizable examples of this comes from the scanning-type family of additive manufacturing (AM) processes; the most notable and familiar member of this family is the fused deposition modeling (FDM) or fused filament fabrication (FFF) process. This process works by selectively depositing uniform, approximately isotropic beads or elements of molten thermoplastic material (typically structural engineering plastics) in a series of pre-specified traces to build each layer of the part. There are many interesting 2-D and 3-D mechanical design problems that can be explored by designing the layout of these elements. The resulting structured, hierarchical material (which is both manufacturable and customized layer-by-layer within the limits of the process and material) can be defined as a manufacturing process-driven structured material (MPDSM). This dissertation explores several practical methods for designing these element layouts for 2-D and 3-D meso-scale mechanical problems, focusing ultimately on design-for-fracture. Three different fracture conditions are explored: (1) cases where a crack must be prevented or stopped, (2) cases where the crack must be encouraged or accelerated, and (3) cases where cracks must grow in a simple pre-determined pattern. Several new design tools, including a mapping method for the FDM manufacturability constraints, three major literature reviews, the collection, organization, and analysis of several large (qualitative and quantitative) multi-scale datasets on the fracture behavior of FDM-processed materials, some new experimental equipment, and the refinement of a fast and simple g-code generator based on commercially-available software, were developed and refined to support the design of MPDSMs under fracture conditions. The refined design method and rules were experimentally validated using a series of case studies (involving both design and physical testing of the designs) at the end of the dissertation. Finally, a simple design guide for practicing engineers who are not experts in advanced solid mechanics nor process-tailored materials was developed from the results of this project.U of I OnlyAuthor's request
Regularised variational schemes for non-gradient systems, and large deviations for a class of reflected McKean-Vlasov SDE
This thesis consists of two parts. The first part constructs entropy regularised variational schemes for a
range of evolutionary partial differential equations (PDEs), not necessarily in gradient flow form, with a
focus on kinetic models. The second part obtains Freidlin-Wentzell large deviation principles and exit times
for a class of reflected McKean-Vlasov stochastic differential equations (SDEs).
The theory ofWasserstein gradient flows in the space of probability measures has made enormous progress
over the last twenty years. It constitutes a unified and powerful framework in the study of dissipative PDEs,
providing the means to prove well-posedness, regularity, stability and quantitative convergence to the equilibrium.
The recently developed entropic regularisation technique paves the way for fast and efficient numerical
methods for solving these gradient flows. However, many PDEs of interest do not have a gradient flow
structure and, a priori, the theory is not applicable. In the first part of the thesis, we develop time-discrete
entropy regularised, (one-step and two-step), variational schemes for general classes of non-gradient PDEs.
The convergence of the schemes is proved as the time-step and regularisation strength tend to zero. For each
scheme we illustrate the breadth of the proposed framework with concrete examples.
In the second part of the thesis we study reflected McKean-Vlasov diffusions over a convex, non-bounded
domain with self-stabilizing coefficients that do not satisfy the classical Wasserstein Lipschitz condition. For
this class of problems we establish existence and uniqueness results and address the propagation of chaos.
Our results are of wider interest: without the McKean-Vlasov component they extend reflected SDE theory,
and without the reflective term they extend the McKean-Vlasov theory. Using classical tools from the theory
of Large Deviations, we prove a Freidlin-Wentzell type Large Deviation Principle for this class of problems.
Lastly, under some additional assumptions on the coefficients, we obtain an Eyring-Kramer’s law for the exit
time from subdomains contained in the interior of the reflecting domain. Our characterization of the rate
function for the exit-time distribution is explicit
Convolutional Neural Operators for robust and accurate learning of PDEs
Although very successfully used in conventional machine learning, convolution
based neural network architectures -- believed to be inconsistent in function
space -- have been largely ignored in the context of learning solution
operators of PDEs. Here, we present novel adaptations for convolutional neural
networks to demonstrate that they are indeed able to process functions as
inputs and outputs. The resulting architecture, termed as convolutional neural
operators (CNOs), is designed specifically to preserve its underlying
continuous nature, even when implemented in a discretized form on a computer.
We prove a universality theorem to show that CNOs can approximate operators
arising in PDEs to desired accuracy. CNOs are tested on a novel suite of
benchmarks, encompassing a diverse set of PDEs with possibly multi-scale
solutions and are observed to significantly outperform baselines, paving the
way for an alternative framework for robust and accurate operator learning
Splitting physics-informed neural networks for inferring the dynamics of integer- and fractional-order neuron models
We introduce a new approach for solving forward systems of differential
equations using a combination of splitting methods and physics-informed neural
networks (PINNs). The proposed method, splitting PINN, effectively addresses
the challenge of applying PINNs to forward dynamical systems and demonstrates
improved accuracy through its application to neuron models. Specifically, we
apply operator splitting to decompose the original neuron model into
sub-problems that are then solved using PINNs. Moreover, we develop an
scheme for discretizing fractional derivatives in fractional neuron models,
leading to improved accuracy and efficiency. The results of this study
highlight the potential of splitting PINNs in solving both integer- and
fractional-order neuron models, as well as other similar systems in
computational science and engineering
A general framework for modelling limit order book dynamics
We present a mathematical framework for modelling the dynamics of limit order books, built on the combination of two modelling ingredients: the order flow -modelled as a general spatial point process- and market clearing, modelled via a deterministic ‘mass transport’ operator acting on distributions of buy and sell orders. At the mathematical level, this corresponds to a natural decomposition of the infinitesimal generator describing the evolution of the limit order book into two operators: the generator of the order flow and the clearing operator. Our model provides a flexible framework for modelling and simulating order book dynamics and studying various scaling limits of discrete order book models. We show that our framework includes previous models as special cases and yields insights into the interaction between the order flow and price dynamics, the use of order book data for prediction of intraday price movements.
The framework also allows for model comparison and the study of the order flow. The modular structure of the model is well-adapted to simulation and allows the stochastic model for the order flow and the clearing mechanism to be specified independently. Then, as a simple demonstration, models with different assumptions on the order intensities are compared. The simulation result shows that orders of relatively large size also play an essential role in the evolution of the order book process.
We further investigate the asymptotic behaviour of the order book processes, including the fluid scaling and the diffusion scaling. The decomposition relation between the order flow and the order book holds when the ask and bid price do not move. In general, the price processes depend on the order flow and the current state of the order book. We prove that as the tick size becomes small, the ask price and bid price converge to the same limiting process.Open Acces
Statistical-dynamical analyses and modelling of multi-scale ocean variability
This thesis aims to provide a comprehensive analysis of multi-scale oceanic variabilities using various statistical and dynamical tools and explore the data-driven methods for correct statistical emulation of the oceans. We considered the classical, wind-driven, double-gyre ocean circulation model in quasi-geostrophic approximation and obtained its eddy-resolving solutions in terms of potential vorticity anomaly and geostrophic streamfunctions. The reference solutions possess two asymmetric gyres of opposite circulations and a strong meandering eastward jet separating them with rich eddy activities around it, such as the Gulf Stream in the North Atlantic and Kuroshio in the North Pacific.
This thesis is divided into two parts. The first part discusses a novel scale-separation method based on the local spatial correlations, called correlation-based decomposition (CBD), and provides a comprehensive analysis of mesoscale eddy forcing. In particular, we analyse the instantaneous and time-lagged interactions between the diagnosed eddy forcing and the evolving large-scale PVA using the novel `product integral' characteristics. The product integral time series uncover robust causality between two drastically different yet interacting flow quantities, termed `eddy backscatter'. We also show data-driven augmentation of non-eddy-resolving ocean models by feeding them the eddy fields to restore the missing eddy-driven features, such as the merging western boundary currents, their eastward extension and low-frequency variabilities of gyres.
In the second part, we present a systematic inter-comparison of Linear Regression (LR), stochastic and deep-learning methods to build low-cost reduced-order statistical emulators of the oceans. We obtain the forecasts on seasonal and centennial timescales and assess them for their skill, cost and complexity. We found that the multi-level linear stochastic model performs the best, followed by the ``hybrid stochastically-augmented deep learning models''. The superiority of these methods underscores the importance of incorporating core dynamics, memory effects and model errors for robust emulation of multi-scale dynamical systems, such as the oceans.Open Acces
Numerical investigation of ducted propellers for novel rotorcraft configurations
The ducted propeller is a promising propulsion or lift generator for novel rotorcraft configurations, considering the stringent restrictions on safety, efficiency, and noise/carbon emissions. However, extensive research work is still needed to further understand the aerodynamic and acoustic characteristics of ducted propellers at various conditions. This thesis aims to deliver highfidelity and systematic investigations of the aerodynamics, acoustics, and optimisation of ducted/open propellers at various conditions.
A detail survey of past works on ducted propellers was first performed to analyse the research status and challenges. Critical assessments of available data sets for validation were also carried out. Numerical validation was then performed to verify the meshing, numerical methods, and simulation strategies for ducted propellers using a test case by NASA. High-fidelity CFD methods and lower-order tools were employed and compared at a range of conditions. Detailed analyses of the aerodynamic performance of ducted/open propellers were later performed at various advance ratios, pitch angles, and crosswind angles. The near- and far-field acoustic features of the ducted/open propellers in axial flight was also computed and inspected closely.
A gradient-based design optimisation framework was also compiled to improve the ducted propeller performance at high advance ratios by varying the duct and blade shapes. The gradients of aerodynamic performance with respect to the design variables were computed using the discrete adjoint CFD methods. The ducted propeller thrust was successfully increased at high advance ratios after the optimisation. The far-field acoustics of the optimised designs was only mildly affected by the optimisation. A parametric study of the equivalent ducted/open propellers was also conducted to further evaluate the influence of different design and operating conditions. An automatic mesh generation tool chain was developed to ease the efforts required for the mesh generation.
The ducted/open propellers were then installed under a main rotor to investigate performance changes due to the aerodynamic interactions. The main rotor downwash induced imbalanced disk loadings and loading variations with complex frequency compositions. The duct was found to provide aerodynamic shielding for the blades enclosed, but it also created considerable blockage to the downwash flow. A simplified modelling approach for the rotor/propeller interactions using actuator disk models was later put forward. By introducing an inflow distortion metric quantifying the aerodynamic interactions, an optimisation framework was compiled to minimise the rotor/propeller interference by changing the propeller position, i.e. the configuration optimisation. The inflow distortion factor was used as the objective, and its gradients with respect to the propeller position were computed using the adjoint method. Gradient-based and gradient-free optimisation approaches were proposed and assessed. With constraints on the pitching and rolling moments, the optimisation managed to effectively reduce the rotor/propeller interference. The optimisation results were further verified using blade-resolved simulations
- …