We study the Hessian of the solutions of time-independent Schrödinger equations, aiming to obtain as large a class as possible of complete Riemannian manifolds for which the estimate (Formula Presented.) holds. For this purpose we introduce the doubly damped stochastic parallel transport equation, study them and make exponential estimates on them, deduce a second order Feynman–Kac formula and obtain the desired estimates. Our aim here is to explain the intuition, the basic techniques, and the formulas which might be useful in other studies.

We study the parabolic integral kernel for the weighted Laplacian with a potential. For manifolds with a pole we deduce formulas and estimates for the derivatives of the Feynman–Kac kernels and their logarithms, these are in terms of a ‘Gaussian’ term and the semi-classical bridge.

We show that a continuous local martingale is a strict local martingale if its supremum process is not in Lα for a positive number α smaller than 1. Using this we construct a family of strict local martingales.

We prove an integration by parts formula for the probability measure on the pinned path space induced by the Semi-classical Riemmanian Brownian Bridge, over a manifold with a pole, followed by a discussion on its equivalence with the Brownian Bridge measure.

We construct a family of SDEs with smooth coefficients whose solutions select a reflected Brownian flow as well as a corresponding stochastic damped transport process (Wt), the limiting pair gives a probabilistic representation for solutions of the heat equations on differential 1-forms with the absolute boundary conditions. The transport process evolves pathwise by the Ricci curvature in the interior, by the shape operator on the boundary where it is driven by the boundary local time, and with its normal part erased at the end of the excursions to the boundary of the reflected Brownian motion. On the half line, this construction selects the Skorohod solution (and its derivative with respect to initial points), not the Tanaka solution; on the half space it agrees with the construction of N. Ikeda and S. Watanabe [29] by Poisson point processes. The construction leads also to an approximation for the boundary local time, in the topology of uniform convergence but not in the semi-martingale topology, indicating the difficulty in proving convergence of solutions of a family of random ODE’s to the solution of a stochastic equation driven by the local time and with jumps. In addition, we obtain a differentiation formula for the heat semi-group with Neumann boundary condition and prove also that (Wt) is the weak derivative of a family of reflected Brownian motions with respect to the initial point.

Lifting nanodevices from planar to three-dimensional architectures enables new functionalities by breaking symmetries and increasing device density. Two promising material classes for beyond-CMOS nanoscale technologies are superconductors and ferromagnets. Superconductors exhibit macroscopic quantum coherence, enabling dissipationless currents and making them attractive for ultrasensitive sensors and quantum computing. In ferromagnets, low-energy spin excitations, known as spin waves or magnons, propagate without Joule heating at GHz frequencies, making them promising information carriers. In this thesis, we employ three-dimensional time-dependent Ginzburg-Landau (TDGL) and micromagnetic simulations to investigate how curvature, chirality, and topology in three-dimensional nanostructures influence superconducting and magnonic excitations. First, we investigate curved tubular type-II superconductors with a constriction. 3D TDGL simulations show that, under a magnetic field along the tube axis, this geometry stabilizes rows of vortices and anti-vortices separated by the constriction. Under a dc transport current, these vortices move azimuthally. For sufficiently large currents, periodic phase slips are stabilized at the constriction, generating voltage pulses at GHz frequencies. We further show that adding a GHz ac modulation to the transport current induces frequency mixing between the phase-slip oscillation and the modulation frequency, resulting in a superconducting frequency comb. Second, we study the influence of a normal-metal coating on vortex statics and dynamics in superconductors with tilted interfaces. By modifying the TDGL equation to model proximity-induced superconductivity in the adjacent normal metal, we show that differences in material properties bend vortices at the interface, with an angular dependency analogous to optical refraction. We derive the corresponding refraction law and verify it numerically. Furthermore, the interface can partially trap vortices due to an energy barrier. The weakening of superconductivity induced by the normal-metal coating lowers the vortex-entry barrier and, under transport current, leads to increased vortex velocities. Third, we investigate the superconducting diode effect, i.e., different critical currents for opposite polarities, in a helical superconducting tube using 3D TDGL simulations. We demonstrate that this effect originates from an inhomogeneous distribution of field-induced screening supercurrents arising from the chiral geometry. The sign of the diode efficiency reverses upon reversing either the magnetic-field direction or the structural chirality, confirming the magnetochiral origin of the effect. We provide guidelines for optimizing the diode efficiency. Finally, we use semi-analytical theory and micromagnetic simulations to study a novel class of materials-by-design ferromagnets, which we term artificial chiral magnets: ferromagnetic nanotubes with imprinted helical surface topography. Simulations show that shape anisotropy stabilizes a helical magnetic texture at zero field. This texture leads to nonreciprocal magnon transport along both the axial and azimuthal directions. Quantitative agreement between simulations and semi-analytical theory allows us to identify the relevant modes and mechanisms responsible for the nonreciprocity. We provide design rules for optimizing the nonreciprocity of the various modes toward realizing functional components such as magnon diodes.

As modern engineered systems grow increasingly complex and access to measurement data becomes ubiquitous, direct data-driven controller synthesis has emerged as a compelling alternative to conventional model-based approaches. This thesis presents a comprehensive data-driven framework for robust controller synthesis that leverages frequency-domain data, eliminating the need for explicit parametric modelling.

A key challenge in robust controller synthesis is the quantification of uncertainty, particularly in multiple-input multiple-output (MIMO) systems. Conventional approaches typically represent uncertainties in the frequency-domain as covering sets using weighting filters and are optimised for the uncertainty radius, but these approaches often yield conservative results in the MIMO context. To address this, a novel size metric for uncertainty sets is introduced, alongside a convex optimisation method to compute the nominal model and its associated uncertainty set from a set of models. Furthermore, the approach can also incorporate statistical uncertainties associated with each model.

Addressing the inherent non-convexity of frequency-domain controller synthesis, an iterative optimisation scheme is devised for fixed-structure controller synthesis for generalised plant models represented in Linear Fractional Representation (LFR) form. This framework enables simultaneous optimisation of multiple performance specifications, including H_2 and H_inf, thereby supporting complex and application-relevant specifications. This leads to a direct data-driven synthesis framework that delivers desired performance without excessive conservatism.

Furthermore, the Integral Quadratic Constraint (IQC) theory is systematically integrated to explicitly account for uncertainties within robust data-driven controller synthesis. Notably, the proposed approach supports both parametric and non-parametric IQC multipliers, allowing for flexible and precise modelling of a wide range of perturbations. In particular, novel non-parametric IQC multipliers have been developed for the representation of multimodel uncertainty sets, uncertain time delays, and slope-bounded nonlinearities, resulting in less conservative uncertainty characterisations. This allows the framework to achieve improved performance while maintaining robustness guarantees.

The efficacy and practicality of the proposed framework are demonstrated through various application case studies, which highlight enhanced robustness, reduced conservatism, and effective trade-offs between performance and uncertainty. Overall, the developed framework offers scalable and practical solutions for the next generation of high-performance control systems operating in uncertain and data-rich environments.

Explicit time integration is the gold standard for simulating the deformation of complex bodies, sometimes made up of hundreds or more trimmed patches. Its efficiency stems from mass lumping techniques that substitute the consistent mass with a (usually diagonal) approximation. Although the impact of mass lumping has been well studied for classical finite element methods based on the Lagrange basis, it remains poorly understood when combined with newer discretization techniques employing more general non-interpolatory bases, such as isogeometric analysis. This thesis partly addresses this shortcoming. Firstly, some special and practically relevant properties of lumped mass matrices are proved and later extended to (block-)banded and Kronecker product matrices for isogeometric discretizations on single-patch, multi-patch, and trimmed geometries. Secondly, we explore the possibility of constructing interpolatory spline bases for improving the accuracy of mass lumping techniques in isogeometric analysis. Although reminiscent of the spectral element method, this strategy comes with its lot of surprises and challenges, which are critically assessed.

Apart from speeding up the solution process, mass lumping techniques are also praised for increasing the critical time step of explicit time integration methods. In classical finite element analysis, the critical time step is constrained by the so-called "optical branches", representing the inaccurate high-frequency part of the spectrum. In contrast, smooth isogeometric analysis features far fewer inaccurate frequencies, which are mostly limited to a few so-called "outliers". Removing those outliers is paramount in explicit dynamics and solution techniques are often discretization-dependent. Rigorous outlier removal techniques for structured finite element discretizations give way to heuristic mass scaling techniques for unstructured ones. While the field has been flourishing over the years, it still lacks a strong theoretical basis and mostly relies on numerical experiments as the only means of assessment. This thesis thoroughly reviews existing mass scaling methods and connects them to established linear algebra results to derive rigorous eigenvalue bounds and condition number estimates. Our results cover some of the most successful instances of mass scaling, unraveling for the first time well-known numerical observations.

Outlier frequencies, however, generally have multiple origins. The high-order nature of certain partial differential equations and the slenderness of structural elements all impose a stringent constraint on the critical time step, and, unfortunately, badly cut elements in immersed finite element discretizations further aggravate the issue. For smooth isogeometric discretizations, Leidinger [1] first showed that mass lumping removed the dependency of the critical time step on the size of trimmed elements. This finding has attracted considerable attention but has unfortunately overshadowed another more subtle effect: mass lumping may disastrously impact the accuracy of lower frequencies and modes, potentially inducing spurious oscillations in the solution. In this thesis, we provide compelling evidence for this phenomenon and later propose a stabilization technique based on polynomial extensions that restores a level of accuracy comparable to boundary-fitted discretizations.

Polymeric gels are biphasic materials consisting of a crosslinked polymer network permeated by an interstitial solvent. Their macroscopic behavior is governed by the poroelastic coupling between the elastic deformation of the solid skeleton and the transport of the fluid phase. Despite the ubiquity and importance of poroelasticity in describing phenomena across natural systems and engineered applications, local characterization of solvent transport under controlled stress and transport boundary conditions remains scarce. Such characterization is experimentally challenging, yet essentialâ conventional bulk tests provide only spatial averages, whereas transport is inherently local. This limitation becomes especially pronounced under large deformations. In this thesis, we develop an experimental framework to obtain high-resolution 3D kinematic fields, combining particle tracking with a deformation gradient tensor (F) estimator. This estimator uses local least-squares to quantify F from particle trajectories, enabling subsequent calculation of various fields and volumetric changes. We analyze the estimator's error and experimentally demonstrate its superior accuracy over conventional Digital Image Correlation (DIC) in handling large deformations and displacement discontinuities.

Using this tool, we quantify silicone-oil transport with controlled viscosities in polydimethylsiloxane (PDMS) networks during free swelling and bending. In free swelling, through-thickness profiles reveal a flux-limited transport boundary condition, contradicting the commonly assumed fully drained limit and implying diffusivities ~10Ã larger than canonical bulk-uptake fits. Under bending, Under bending, we resolve tensile-side dilation and compressive-side contraction, extracting diffusivities consistent with global force-relaxation. Together, these experiments benchmark diffusivity $\simeq10^{-12}$â $10^{-10}$ m$^2$/s, decreasing with increasing solvent viscosity. Furthermore, the free swelling equilibrium is captured by a Flory--Rehner theory that requires modification to include effective finite extensibility.

We then extend this approach to Mode I fracture of brittle hydrogels. In the fast, non-poroelastic regime, near-tip kinematics and a J-integral evaluation reveal a characteristic process zone (100ÎŒm) with a "horizontal droplet" shape, where energy dissipation partitions into two contributionsâ a distributed process-zone dissipation and a tip-localized dissipation below our resolution. In the slow, poroelastic regime, we resolve the full 3D near-tip kinematic fields and directly quantify solvent migration via the determinant of F. These measurements, departing from classic linear elastic fracture mechanics predictions, reveal strong multiaxial stretch state at the tip and substantial rigid-body rotation in the wake. Pronounced swelling is observed surrounding the crack tip and intensifies towards the tip; this swelling correlates with the local stretch state and decreases at higher crack velocities.

This thesis establishes an experimental framework to reconstruct 3D deformation fields in soft polymers. By challenging canonical boundary assumptions and characterizing near-tip fields, we benchmark solvent diffusivity and elucidate fracture mechanisms in polymers. These measurements provide fundamental insights into poroelastic modeling and fracture of polymers, and are expected to shed light on broader fluid--solid coupling in geophysical settings.

Graph Neural Networks (GNNs) have become a central framework for learning on relational data such as molecules, biological systems, and social networks. Despite their success, standard message-passing architectures struggle to capture long-range dependencies. As information propagates through multiple layers, structural bottlenecks in the graph lead to phenomena such as oversquashing and oversmoothing, which severely limit the ability of GNNs to model interactions between distant nodes.

This thesis studies graph learning from the perspective of information propagation and develops methods to improve long-range reasoning in graph neural architectures. First, we propose a principled graph rewiring framework that introduces latent nodes acting as global mediators, enabling more efficient communication between distant regions of the graph and alleviating structural bottlenecks. We demonstrate that this approach improves the modeling of long-range interactions in protein structure learning tasks.

Second, we introduce a generative framework for modeling full-atom protein conformations using latent diffusion on graph embeddings. Protein structures are encoded using a Chebyshev spectral graph network, and a diffusion model is trained in a learned latent space to generate diverse conformational states that are subsequently decoded into full atomic coordinates.

Finally, we revisit Chebyshev spectral graph networks from a dynamical systems perspective and analyze the stability of high-order polynomial filtering. We introduce architectural modifications that enforce stable information propagation while preserving spectral expressivity, leading to strong performance on long-range graph benchmarks.

Together, these contributions advance the understanding and design of graph neural architectures capable of modeling long-range dependencies in complex relational systems.