We will discuss the solution of eigenvalue problems associated with partial differential equations (PDE)s that can be written in the generalised form Ax = λMx, where the matrices A and/or M may depend on a scalar parameter. Parameter dependent matrices occur frequently when stabilised formulations are used for the numerical approximation of PDEs. With the help of classical numerical examples we will show that the presence of one (or both) parameters can produce unexpected results.

Hessian-dependent functionals play a pivotal role in a wide latitude of problems in mathematics. Arising in the context of differential geometry and probability theory, this class of problems find applications in the mechanics of deformable media (mostly in elasticity theory) and the modelling of slow viscous fluids. We study such functionals from three distinct perspectives.

We present a theoretical analysis of the Weak Adversarial Networks (WAN) method, recently proposed in [1, 2], as a method for approximating the solution of partial differential equations in high dimensions and tested in the framework of inverse problems. In a very general abstract framework.

Analogies between financial markets and critical phenomena have long been observed empirically. So far, no convincing theory has emerged that can explain these empirical observations. Here, we take a step towards such a theory by modeling financial markets as a lattice gas.

Wave propagation and scattering problems are of huge importance in many applications in science and engineering - e.g., in seismic and medical imaging and more generally in acoustics and electromagnetics.

In this short course, we will introduce some elements in deriving the hp a posteriori error estimate for a high-order unfitted finite element method for elliptic interface problems. The key ingredient is an hp domain inverse estimate, which allows us to prove a sharp lower bound of the hp a posteriori error estimator.

Pulse-shaped signal characterization is a fundamental problem in signal processing. One recently developed tool available to analyze non-stationary pulse-shaped waveforms with a suitable peak reconstruction is semiclassical signal analysis (SCSA). SCSA is a signal representation method that decomposes a real positive signal y(t) into a set of squared eigenfunctions through the discrete spectrum of the Schrödinger operator which is of particular interest. Beginning with an introduction to the young method, this dissertation discusses the relevant properties of SCSA and how they are utilized in signal denoising and biomedical application. Based on this, different frameworks and methodologies are proposed to leverage the advantages of the SCSA, especially in the pulse-shaped signal analysis field.

Cardiovascular diseases (CVDs) remain the leading cause of death worldwide. Patients at risk of evolving CVDs are assessed by evaluating a risk factor-based score that incorporates different bio-markers ranging from age and sex to arterial stiffness (AS).

Optical Wireless Communication (OWC) offers many benefits over established radio frequency-based communication links. In particular, high beam directivity generates efficient power usage and high-speed data services. Moreover, due to its ease of deployment, high transmission security, license-free operation, and insensitivity to interference, the OWC link is becoming an attractive solution for the next generation of communication systems.

Digital health solutions improve healthcare services and help achieve sustainable and higher standards of health and well-being. These solutions are mainly based on Digital Signal Processing (DSP) to record, interpret, and diagnose bio-signals such as Electrocardiogram (ECG) or Magnetoencephalography (MEG). In my thesis, a novel signal/image post-processing algorithm is proposed based on the Semi-Classical Signal Analysis method (SCSA) to enhance biomedical data quality. In addition, new feature extraction algorithms are proposed, based on the SCSA and the new Quantization-based Position Weight Matrix (QuPWM), which opens new tracks toward smart biomedical diagnosis and decision-making assistance in different fields such as predicting true Poly(A) regions in a DNA sequence, multiple hand gesture prediction.

I will present an overview of our activities around estimation problems for partial and fractional differential equations. I will present the methods and the algorithms we develop for the state, source and parameters estimation and illustrate the results with some simulations and real applications.

Abstract

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Abstract

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Model Predictive Control (MPC) is an d advanced control strategy widely used in the process industries and beyond. Therefore, industry is interested in the developments of MPC formulations that can enhance safety, reliability, and economic profitability of chemical processes. Motivated by these considerations, the first part of this talk focuses on the development of methods for integrating process operational safety and process economics within model predictive control system designs.

We will present some new methods for source and parameters estimation for partial and fractional differential equations and illustrate the results with some simulations and real applications.