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This paper focuses on the study of semilinear fractional diffusion-wave equations in the context of critical nonlinearities. Firstly, we address the issue of local well-posedness for the problem, examine spatial regularity, and the…
The main topic of this thesis is the proof of two fundamental entropic inequalities for quantum Gaussian channels. Quantum Gaussian channels model the propagation of electromagnetic waves through optical fibers and free space in the quantum…
We present a finite element implementation for the steady-state nonlocal Dirichlet problem with homogeneous volume constraints. Here, the nonlocal diffusion operator is defined as integral operator characterized by a certain kernel…
This PhD thesis lays out algebraic and topological structures relevant for the study of probabilistic graphical models. Marginal estimation algorithms are introduced as diffusion equations of the form $\dot u = \delta \varphi$. They…
The problem of communicating sensor measurements over shared networks is prevalent in many modern large-scale distributed systems such as cyber-physical systems, wireless sensor networks, and the internet of things. Due to bandwidth…
The spread of an epidemic is often modeled by an SIR random process on a social network graph. The MinINF problem for optimal social distancing involves minimizing the expected number of infections, when we are allowed to break at most $B$…
A discontinuous Galerkin (dG) method for the numerical solution of initial/boundary value multi-compartment partial differential equation (PDE) models, interconnected with interface conditions, is presented and analysed. The study of…
We study qualitative properties of initial traces of nonnegative solutions to a semilinear heat equation in a smooth domain under the Dirichlet boundary condition. Furthermore, for the corresponding Cauchy--Dirichlet problem, we obtain…
Motivated by simulations of ultrasound-enhanced drug delivery, this work presents the numerical analysis of a mathematical model that captures the influence of ultrasound waves on the diffusivity of the drug. The system under study consists…
We consider the well-posedness of a class of hyperbolic partial differential equations on a one dimensional spatial domain. This class includes in particular infinite-dimensional networks of transport, wave and beam equations, or even…
This paper studies the problem of distributed Riemannian optimization over a network of agents whose cost functions are geodesically smooth but possibly geodesically non-convex. Extending a well-known distributed optimization strategy…
Network datasets appear across a wide range of scientific fields, including biology, physics, and the social sciences. To enable data-driven discoveries from these networks, statistical inference techniques like estimation and hypothesis…
In this paper, we aim to design and analyze distributed Bayesian estimation algorithms for sensor networks. The challenges we address are to (i) derive a distributed provably-correct algorithm in the functional space of probability…
Solutions of the Dirichlet and Robin boundary value problems for the multi-term variable-distributed order diffusion equation are studied. A priori estimates for the corresponding differential and difference problems are obtained by using…
In this paper, we explore the solvability and the optimal control problem for a compartmental model based on reaction-diffusion partial differential equations describing a transmissible disease. The nonlinear model takes into account the…
Understanding the process by which a contagion disseminates throughout a network is of great importance in many real world applications. The required sophistication of the inference approach depends on the type of information we want to…
While Graph Neural Networks (GNNs) have recently become the de facto standard for modeling relational data, they impose a strong assumption on the availability of the node or edge features of the graph. In many real-world applications,…
This article studies the dynamics of the mean-field approximation of continuous random networks. These networks are stochastic integrodifferential equations driven by Gaussian noise. The kernels in the integral operators are realizations of…
We present an approach to handle Dirichlet type nonlocal boundary conditions for nonlocal diffusion models with a finite range of nonlocal interactions. Our approach utilizes a linear extrapolation of prescribed boundary data. A novelty is,…
Graphical models have been widely applied in solving distributed inference problems in sensor networks. In this paper, the problem of coordinating a network of sensors to train a unique ensemble estimator under communication constraints is…