Advection-diffusion equation with rough coefficients: weak solutions and vanishing viscosity

In the first part of the paper, we study the Cauchy problem for the advection-diffusion equation $\partial_t v + \text{div }(v\boldsymbol{b} ) = \Delta v$ associated with a merely integrable, divergence-free vector field $\boldsymbol{b}$ defined on the torus. We establish existence, regularity and uniqueness results for various notions of solutions, in different regimes of integrability both for the vector field and for the initial datum. In the second part of the paper, we use the advection-diffusion equation to build a vanishing viscosity scheme for the transport/continuity equation drifted by $\boldsymbol{b}$, i.e. $\partial_t u + \text{div }(u\boldsymbol{b} ) = 0$. Under Sobolev assumptions on $\boldsymbol{b}$, we give two independent proofs of the convergence of such scheme to the Lagrangian solution of the transport equation. One of the proofs is quantitative and yields rates of convergence. This offers a completely general selection criterion for the transport equation (even beyond the distributional regime) which compensates the wild non-uniqueness phenomenon for solutions with low integrability arising from convex integration schemes, as shown in recent works [10, 31, 32, 33], and rules out the possibility of anomalous dissipation.Comment: 28 pages, 2 figure

A regularity result for the Fokker-Planck equation with non-smooth drift and diffusion

The goal of this paper is to study weak solutions of the Fokker-Planck equation. We first discuss existence and uniqueness of weak solutions in an irregular context, providing a unified treatment of the available literature along with some extensions. Then, we prove a regularity result for distributional solutions under suitable integrability assumptions, relying on a new, simple commutator estimate in the spirit of DiPerna-Lions' theory of renormalized solutions for the transport equation. Our result is somehow transverse to Theorem 4.3 of [15]: on the diffusion matrix we relax the assumption of Lipschitz regularity in time at the price of assuming Sobolev regularity in space, and we prove the regularity (and hence the uniqueness) of distributional solutions to the Fokker-Planck equation

Weak and parabolic solutions of advection-diffusion equations with rough velocity field

We study the Cauchy problem for the advection-diffusion equation $\partial_t u + \mathrm{div} (u b ) = \Delta u$ associated with a merely integrable divergence-free vector field $b$ defined on the torus. We discuss existence, regularity and uniqueness results for distributional and parabolic solutions, in different regimes of integrability both for the vector field and for the initial datum. We offer an up-to-date picture of the available results scattered in the literature, and we include some original proofs. We also propose some open problems, motivated by very recent results which show ill-posedness of the equation in certain regimes of integrability via convex integration schemes.Comment: Part of these notes was originally contained in an preliminary version of arXiv:2107.0365

Weak and parabolic solutions of advection-diffusion equations with rough velocity field

We study the Cauchy problem for the advection-diffusion equation $\partial_t u + \mathrm{div} (u b ) = \Delta u$ associated with a merely integrable divergence-free vector field $b$ defined on the torus. We discuss existence, regularity and uniqueness results for distributional and parabolic solutions, in different regimes of integrability both for the vector field and for the initial datum. We offer an up-to-date picture of the available results scattered in the literature, and we include some original proofs. We also propose some open problems, motivated by very recent results which show ill-posedness of the equation in certain regimes of integrability via convex integration schemes

Learning HMM State Sequences from Phonemes for Speech Synthesis

AbstractThis paper presents a technique for learning hidden Markov model (HMM) state sequences from phonemes, that combined with modified discrete cosine transform (MDCT), is useful for speech synthesis. Mel-cepstral spectral parameters, currently adopted in the conventional methods as features for HMM acoustic modeling, do not ensure direct speech waveforms reconstruction. In contrast to these approaches, we use an analysis/synthesis technique based on MDCT that guarantees a perfect reconstruction of the signal frame feature vectors and allows for a 50% overlap between frames without increasing the data rate. Experimental results show that the spectrograms achieved with the suggested technique behave very closely to the original spectrograms, and the quality of synthesized speech is conveniently evaluated using the well known Itakura-Saito measure

a comparative study of machine learning algorithms for physiological signal classification

Abstract The present work aims at the evaluation of the effectiveness of different machine learning algorithms on a variety of clinical data, derived from small, medium, and large publicly available databases. To this end, several algorithms were tested, and their performance, both in terms of accuracy and time required for the training and testing phases, are here reported. Sometimes a data preprocessing phase was also deemed necessary to improve the performance of the machine learning procedures, in order to reduce the problem size. In such cases a detailed analysis of the compression strategy and results is also presented

ECG-Based Arrhythmia Classification using Recurrent Neural Networks in Embedded Systems

Cardiac arrhythmia is one of the most important cardiovascular diseases (CVDs), causing million deaths every year. Moreover it is difficult to diagnose because it occurs intermittently and as such requires the analysis of large amount of data, collected during the daily life of patients. An important tool for CVD diagnosis is the analysis of electrocardiogram (ECG), because of its non-invasive nature and simplicity of acquisition. In this work we propose a classification algorithm for arrhythmia based on recurrent neural networks (RNNs) that operate directly on ECG data, exploring the effectiveness and efficiency of several variations of the general RNN, in particular using different types of layers implementing the network memory. We use the MIT-BIH arrhythmia database and the evaluation protocol recommended by the Association for the Advancement of Medical Instrumentation (AAMI). After designing and testing the effectiveness of the different networks, we then test its porting to an embedded platform, namely the STM32 microcontroller architecture from ST, using a specific framework to port a pre-built RNN to the embedded hardware, convert it to optimized code for the platform and evaluate its performance in terms of resource usage. Both in binary and multiclass classification, the basic RNN model outperforms the other architectures in terms of memory storage (∼117 KB), number of parameters (∼5 k) and inference time (∼150 ms), while the RNN LSTM-based achieved the best accuracy (∼90%)

an acquisition system of in house parameters from wireless sensors for the identification of an environmental model

Abstract This paper presents a system for the acquisition of in-house parameters, such as temperature, pressure, humidity and so on, that can be used for the intelligent control of a building. The main objective of this work is to determine an environmental model of an in-house room using machine learning techniques. The system is based on a low data-rate network of sensing and control nodes to acquire the data, realized with a new protocol, called ToLHnet, that is able to employ both wired and wireless communication on different media. Several standard machine learning techniques, namely linear regression, classification and regression tree algorithm, support vector machine, have been used for the regression of the input-output thermal model. Additionally, a recently proposed new technique named particle-Bernstein polynomial has been successfully applied. Experimental results show that this technique outperforms the previous techniques, for both accuracy and computation time

Embedded AM-FM Signal Decomposition Algorithm for Continuous Human Activity Monitoring

AM-FM decomposition techniques have been successfully used for extracting significative features from a large variety of signals, helping realtime signal monitoring and pattern recognition, since they represent signals as a simultaneous composition of amplitude modulation and frequency modulation, where the carriers, amplitude envelopes, and the instantaneous frequencies are the features to be estimated. Human activities often involve repetitive movements, such as in running or cycling, where sinusoidal AM-FM decompositions of signals have already demonstrated to be useful to extract compact features to aid monitoring, classification, or detection. In this work we thus present the challenges and results of implementing the iterated coherent Hilbert decomposition (ICHD), a particularly effective algorithm to obtain an AM-FM decomposition, within a resource-constrained and low-power ARM Cortex-M4 microcontroller that is present in a wearable sensor we developed. We apply ICHD to the gyroscope data acquired from an inertial measurement unit (IMU) that is present in the sensor. Optimizing the implementation allowed us to achieve real-time performance using less then 16 % of the available CPU time, while consuming only about 5.4 mW of power, which results in a run-time of over 7 days using a small 250 mAh rechargeable cell