2063 research outputs found
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Pointwise convergence of the Klein-Gordon flow
No full textWe consider the PDEs version of the Carleson problem in the context of the cubic nonlinear Klein-Gordon equation. This means that we aim to establish the lowest regularity class for which one has almost everywhere pointwise convergence of the solutions to the initial data, as t→0. We prove sharp results for initial data in Sobolev spaces and for their randomized counterparts
Quantitative unique continuation for non-regular perturbations of the Laplacian
No full textIn this work, we investigate the quantitative estimates of the unique continuation property for solutions
of an elliptic equation ∆u = Vu+ W1 · ∇u+ div (W2u) in an open, connected subset of Rd, where d 3.
Here, V ∈ Lq0
, W1 ∈ Lq1 , and W2 ∈ Lq2 with q0 > d/2, q1 > d, and q2 > d. Our aim is to provide an
explicit quantification of the unique continuation property with respect to the norms of the potentials. To
achieve this, we revisit the Carleman estimates established in [6] and prove a refined version of them, and
we combine them with an argument due to T. Wolff introduced in [18] for the proof of unique continuation
for solutions of equations of the form ∆u = Vu+ W1 · ∇u.In this work, we investigate the quantitative estimates of the unique continuation property for solutions
of an elliptic equation ∆u = Vu+ W1 · ∇u+ div (W2u) in an open, connected subset of Rd, where d 3.
Here, V ∈ Lq0
, W1 ∈ Lq1 , and W2 ∈ Lq2 with q0 > d/2, q1 > d, and q2 > d. Our aim is to provide an
explicit quantification of the unique continuation property with respect to the norms of the potentials. To
achieve this, we revisit the Carleman estimates established in [6] and prove a refined version of them, and
we combine them with an argument due to T. Wolff introduced in [18] for the proof of unique continuation
for solutions of equations of the form ∆u = Vu+ W1 · ∇u
Accelerating Training of Physics Informed Neural Network for 1D PDEs with Hierarchical Matrices
No full textIn this paper, we consider a training of Physics Informed Neural Networks with fully connected neural networks for approximation of solutions of one-dimensional advection-diffusion problem. In this context, the neural network is interpreted as a non-linear function of one spatial variable, approximating the solution scalar field, namely y=PINN(x)=Anσ(An-1...A2σ(A1+b1)+b2)+...+bn-1)+bn. In the standard PINN approach, the Ai denotes dense matrices, bi denotes bias vectors, and σ is the non-linear activation function (sigmoid in our case). In our paper, we consider a case when Ai are hierarchical matrices Ai=Hi. We assume a structure of our hierarchical matrices approximating the structure of finite difference matrices employed to solve analogous PDEs. In this sense, we propose a hierarchical neural network for training and approximation of PDEs using the PINN method. We verify our method on the example of a one-dimensional advection-diffusion problem
M2SKD: Multi-to-Single Knowledge Distillation of Real-Time Epileptic Seizure Detection for Low-Power Wearable Systems
No full textIntegrating low-power wearable systems into routine health monitoring is an ongoing challenge. Recent advances in the computation capabilities of wearables make it possible to target complex scenarios by exploiting multiple biosignals and using high-performance algorithms, such as Deep Neural Networks (DNNs). However, there is a tradeoff between the algorithms' performance and the low-power requirements of platforms with limited resources. Besides, physically larger and multi-biosignal-based wearables bring significant discomfort to the patients. Consequently, reducing power consumption and discomfort is necessary for patients to use wearable devices continuously during everyday life. To overcome these challenges, in the context of epileptic seizure detection, we propose the Multi-to-Single Knowledge Distillation (M2SKD) approach targeting single-biosignal processing in wearable systems. The starting point is to train a highly-accurate multi-biosignal DNN, then apply M2SKD to develop a single-biosignal DNN solution for wearable systems that achieves an accuracy comparable to the original multi-biosignal DNN. To assess the practicality of our approach to real-life scenarios, we perform a comprehensive simulation experiment analysis on several edge computing platforms.RYC2021-032853-
Probabilistic Topology Optimization Framework for Geometrically Nonlinear Structures Considering Load Position Uncertainty and Imperfections
No full textIn this manuscript, a novel approach to topology optimization is proposed which integrates considerations of uncertain load positions, thereby enhancing the reliability-based design within the context of structural engineering. Extending the conventional framework to encompass imperfect geometrically nonlinear analyses, this research discovers the intricate interplay between nonlinearity and uncertainty, shedding light on their combined effects on probabilistic analysis. A key innovation lies in treating load position as a stochastic variable, augmenting the existing parameters, such as volume fraction, material properties, and geometric imperfections, to capture the full spectrum of variability inherent in real-world conditions. To address these uncertainties, normal distributions are adopted for all relevant parameters, leveraging their computational efficacy, simplicity, and ease of implementation, which are particularly crucial in the context of complex optimization algorithms and extensive analyses. The proposed methodology undergoes rigorous validation against benchmark problems, ensuring its efficacy and reliability. Through a series of structural examples, including U-shaped plates, 3D L-shaped beams, and steel I-beams, the implications of considering imperfect geometrically nonlinear analyses within the framework of reliability-based topology optimization are explored, with a specific focus on the probabilistic aspect of load position uncertainty. The findings highlight the significant influence of probabilistic design methodologies on topology optimization, with the defined constraints serving as crucial conditions that govern the optimal topologies and their corresponding stress distributions
Modelling COVID-19 mutant dynamics: Understanding the interplay between viral evolution and disease transmission dynamics
No full textUnderstanding virus mutations is critical for shaping public health interventions. These mutations lead to complex multi-strain dynamics often under-represented in models. Aiming to understand the factors influencing variants' fitness and evolution, we explore several scenarios of virus spreading to gain qualitative insight into the factors dictating which variants ultimately predominate at the population level. To this end, we propose a two-strain stochastic model that accounts for asymptomatic transmission, mutations and the possibility of disease import. We find that variants with milder symptoms are likely to spread faster than those with severe symptoms. This is because severe variants can prompt affected individuals to seek medical help earlier, potentially leading to quicker identification and isolation of cases. However, milder or asymptomatic cases may spread more widely, making it harder to control the spread. Therefore, increased transmissibility of milder variants can still result in higher hospitalizations and fatalities due to widespread infection. The proposed model highlights the interplay between viral evolution and transmission dynamics. Offering a nuanced view of factors influencing variant spread, the model provides a foundation for further investigation into mitigating strategies and public health interventions.M.A. acknowledges the financial support by the Ministerio de Ciencia e Innovacion (MICINN) of
the Spanish Government through the Ramon y Cajal grant RYC2021-
Time-periodic weak solutions for fluid-structure interactions
Get PDFThe present doctoral thesis is devoted to the study of time-periodic
weak solutions for a class of systems which models the interaction of an in-
compressible, Newtonian fluid of Navier-Stokes type with an elastic membrane
(plate/shell) of Koiter type.
The thesis consists of three main results. In the case of elastic plates we prove
the existence of at least one periodic solution when the elastic Koiter energy is
either in general form (nonlinear) and also for a linearized Koiter model.
In the case of elastic shells we prove the same result but restricted to the linearized
Koiter energy model. Here we impose an inflow/outflow boundary condition.
The existence of solutions is guaranteed once the forces or the boundary condi-
tions that lead the dynamics of the system are of su ciently small L2 magnitude.
We provide new uniform in time estimates for the energy of the system, and we
construct suitable divergence-free extension operators adapted for moving in time
domains which might be of independent interest
Development and Validation of a Health-aware Floating Offshore Wind Farm Simulation Platform: FOWLTY
Get PDFOffshore wind energy is now an important game player towards the achievement of United Nations’ Sustainable
Goals 7 (clean and affordable energy) and 13 (climate change) due to its huge potential but, given the
harsh environment in which the turbines operate, they lead to more frequent and severe component faults
compared to onshore farms. These faults result in increased maintenance and operational costs, posing a
significant challenge to profitability. To mitigate these costs and maintain energy production during faults,
fault detection and isolation (FDI) and fault-tolerant control (FTC) strategies have gained traction. Effective
simulation tools are crucial for the development of these strategies. FOWLTY, a new simulator introduced
in this paper, which derives its name from ’floating offshore wind farm simulator’ incorporating the term
’faulty’, allows users to recreate faults in wind turbine subsystems. It simplifies simulation to facilitate the
development of control and estimation strategies, offering a balance between accuracy and computational
efficiency. FOWLTY is compared to OpenFAST considering different scenarios, demonstrating its suitability
for fault detection, diagnosis and impact mitigation through control. Additionally, a case study considering
a farm composed of eight floating wind turbines is provided to show the effects of different faults.PDC2021-121093-I0
A scalable method to model large suspensions of colloidal phoretic particles with arbitrary shapes
Get PDFPhoretic colloids self-propel thanks to surface flows generated in response to surface gradients (thermal, electrical, or chemical), that are self-induced and/or generated by other particles. Here we present a scalable and versatile framework to model chemical and hydrodynamic interactions in large suspensions of arbitrarily shaped phoretic particles, accounting for thermal fluctuations at all Damkholer numbers. Our approach, inspired by the Boundary Element Method (BEM), employs second-layer formulations, regularized kernels and a grid optimization strategy to solve the coupled Laplace-Stokes equations with reasonable accuracy at a fraction of the computational cost associated with BEM. As demonstrated by our large-scale simulations, the capabilities of our method enable the exploration of new physical phenomena that, to our knowledge, have not been previously addressed by numerical simulations.la Caixa’ Foundation fellowship LCF/BQ/PI20/11760014
European Union’s Horizon 2020 Marie Skłodowska-Curie grant agreement No 847648
French National Research Agency (ANR), under award ANR-20-CE30-000
Sedimentation dynamics of triply twisted Möbius bands: Geometry versus topology
Get PDFWe explore the sedimentation dynamics of triply twisted Möbius bands, each characterized by threefold rotational symmetry but distinguished by its construction and intrinsic geometrical properties. Three types of bands are considered: one with vanishing Gaussian curvature, constructed by isometrically deforming flat rectangular strips through bending without stretching; and two with negative Gaussian curvature, one being constructed by isometrically deforming helicoids. Experiment on these three types of bands, with a focus on varying aspect ratios, reveals a singular phenomenon: while the spin directions of bands not derived from helicoids spin in directions consistent with their inherent chirality, bands derived from helicoids exhibit an aspect-ratio-dependent spin, pointing to the existence of a critical aspect ratio at which geometric factors dominate over chiral influences. Supported by numerical simulations and a detailed analysis of the resistance tensors, we propose the unique response of bands derived from helicoids originates from a complex interaction among geometry, topology, and hydrodynamics. Two explanations are offered for the chiral transition observed in bands derived from helicoid. First, this transition may parallel the dynamics of superhelices, for which competing chiralities influence rotational behavior. Second, the unique geometrical properties of bands derived from helicoids, coupled with deviations between the rotation axis and the local symmetry axis, may underlie the observed aspect-ratio-dependent chiral transition. Our study underscores the significant role of geometrical and topological nuances in determining the behavior of chiral objects suspended in fluids. In addition to offering transformative potential across diverse fields, it promises advancements in mixing, separation processes, and innovative passive swimmers