Nonlinear Network description for many-body quantum systems in continuous space

We show that the recently introduced iterative backflow renormalization can be interpreted as a general neural network in continuum space with non-linear functions in the hidden units. We use this wave function within Variational Monte Carlo for liquid $^4$He in two and three dimensions, where we typically find a tenfold increase in accuracy over currently used wave functions. Furthermore, subsequent stages of the iteration procedure define a set of increasingly good wave functions, each with its own variational energy and variance of the local energy: extrapolation of these energies to zero variance gives values in close agreement with the exact values. For two dimensional $^4$He, we also show that the iterative backflow wave function can describe both the liquid and the solid phase with the same functional form -a feature shared with the Shadow Wave Function, but now joined by much higher accuracy. We also achieve significant progress for liquid $^3$He in three dimensions, improving previous variational and fixed-node energies for this very challenging fermionic system

Exploring Flow Chemistry for the Synthesis and Scale-up of Small Organic Molecules

By the late ‘90s flow chemistry had established itself as a powerful tool for organic synthesis in academia and had started to progressively attracted the interest of the industry due to the advantages that it could potentially offer compared to batch processing; among these it is worth mentioning its intrinsic ability to reduce the solvent usage and to dramatically cut the reaction time alongside providing higher purity and selectivity due to the more regulated processing conditions. In addition, it provides a safer way to handle dangerous and hazardous reagents/intermediates and simplifying the scaling up of the process. This thesis presents a series of molecular preparations involving flow chemistry to expedite the transformation to generate molecules of interest to the pharmaceutical industry. All the work disclosed has been carried out in the Baxendale’s research group at the University of Durham, under the supervision of Professor Ian R. Baxendale. The research has been partially funded and conducted in collaboration with AbbVie under the supervision of Dr Amanda W. Dombrowski and Prof. Stevan W. Djuric. Chapter 1 describes the first use of flow chemistry for performing Norrish-Yang reactions. The transformation has been exploited to synthetize a range of 3-hydroxyazetidines. The high reproducibility and short residence times of the continuous process enables easy scaling of the transformation allowing easy access to these valuable chemical entities at synthetically useful multi-gram scales. Moreover, a systematic exploration of the constituent structural components was undertaken allowing an understanding of the reactivity and functional group tolerance of the transformation. Chapter 2 details the chemistry of a novel rearrangement of the previously obtained 3-hydroxyazetidines (Chapter 1) via a Ritter initiated cascade to provide highly substituted 2-oxazolines in high yields. The reaction conditions and substrate scope of the transformation have been studied demonstrating the generality of the process. The derived products can also be functionalized in order to undergo further intramolecular cyclization leading to a new class of macrocycle. The final cyclization step was shown to be a transformation amenable to continuous flow processing allowing for a dramatic reduction in the reaction time and a simple direct scale-up. Chapter 3 deals with the nitrosation of several alkanes with tert-butyl nitrite under flow processing conditions. The continuous approach enabled a marked reduction in the reaction time compared to the analogous batch process. In addition, in order to address the necessity for large excesses of the alkane starting material a continuous recycling process was developed thus allowing the preparation of larger quantities of material in a more atom economic and cost-effective process

Quantum Monte Carlo simulations of two dimensional 3He: low-density gas-liquid coexistence on substrates and iterative backflow wave functions for strongly correlated systems

In this thesis we show results of Quantum Monte Carlo simulations for fluid 3He in the ground state. We studied 3He in a quasi two dimensional environment and we designed a new class of trial wave functions for strongly correlated fermions. 3He is a typical example of Fermi liquid, and is the focus of several theoretical and experimental studies; since two dimensional 3 He in the ground state is believed to be a homogeneous liquid up to freezing, two dimensional 3He is the ideal system to study the effects of correlation in a wide density range. In this thesis we have worked in two regimes: we studied the behaviour of low density 3He adsorbed on substrates and we designed a new class of trial wave function to be used in the strongly correlated, high density systems. While great focus has been devoted to study the strongly correlated regime at high density, some important questions about the low density behaviour of this system still have to be addressed. Recent heat capacity data were interpreted as the evidence of the presence of a self bound liquid phase for 3He adsorbed on graphite; moreover it was argued that the the appearance of a liquid phase does not depend on the substrate, but is an intrinsic property of two dimensional 3He. This is in stark contrast with theoretical studies, that exclude the presence of a self bound liquid. We performed Quantum Monte Carlo simulations, using the Variational Monte Carlo and the Fixed Node - Diffusion Monte Carlo methods, to investigate the presence of a low density liquid phase in two dimensional 3He and in 3He adsorbed on alkali, magnesium and graphite substrates. Our results exclude the formation of a self bound liquid in the strictly two dimensional environment, while in the presence of substrates the situation changes; on weakly attractive substrates the formation of a liquid phase is indeed possible, while on stronger substrates, that are closer approximations of the two dimensional system, we can\u2019t observe any liquid. We find out however that the corrugation a the substrate helps the stabilization of a liquid phase, and can lead to phase coexistence of different fluid phases even on a substrate as strong as graphite. When performing Quantum Monte Carlo simulations it is crucial to have good trial wave functions. Designing good wave functions on the other hand in a hard task, especially when we study Fermi liquids at high density. In order to study strongly correlated systems more accurate and sophisticated wave functions were designed. Including backflow transformations has proven to significantly increase the quality of trial wave functions for Fermi liquids, especially at high density, but some results still show a poor quantitative agreement with experimental data for example for the spin polarization of 3He. We introduce a new class of trial wave functions for strongly correlate Fermi systems. These wave functions are based on iterated backflow transformations and on the introduction of correlations between backflow coordinates. While exact results are usually not available in Quantum Monte Carlo simulations of Fermi system our iterative backflow procedure allows to define a set of increasingly accurate wave functions that can be used to obtain both a strict upper bound and a strict lower bound to the exact ground state energy, and have an estimate of the exact energy. We used these wave functions to study two dimensional 3 He at freezing. We could obtain variational energy estimates that are significantly lower than the ones available in literature; moreover the upper and lower bound we could obtain for 3 He allowed us to give an estimate for the ground state energy that is in good agreement with exact data obtained with the Transient Estimate technique. Having seen the good results that can be obtained using these wave functions we used them to simulate another system, three dimensional 4He; we studied this system in different conditions, at negative pressure, at equilibrium and at freezing; in all cases we could obtain variational energies that are lower than the ones obtained using Shadow Wave Functions, and the upper and lower bounds for the energy are consistent with exact Diffusion Monte Carlo data. The good results we obtained in the study of a Bose system suggest that the iterative backflow transformations could find applications that can go well beyond the simulations of strongly correlated fermions

Convergence of adaptive stochastic Galerkin FEM

We propose and analyze novel adaptive algorithms for the numerical solution of elliptic partial differential equations with parametric uncertainty. Four different marking strategies are employed for refinement of stochastic Galerkin finite element approximations. The algorithms are driven by the energy error reduction estimates derived from two-level a posteriori error indicators for spatial approximations and hierarchical a posteriori error indicators for parametric approximations. The focus of this work is on the mathematical foundation of the adaptive algorithms in the sense of rigorous convergence analysis. In particular, we prove that the proposed algorithms drive the underlying energy error estimates to zero

A decoupled, convergent and fully linear algorithm for the Landau--Lifshitz--Gilbert equation with magnetoelastic effects

We consider the coupled system of the Landau--Lifshitz--Gilbert equation and the conservation of linear momentum law to describe magnetic processes in ferromagnetic materials including magnetoelastic effects in the small-strain regime. For this nonlinear system of time-dependent partial differential equations, we present a decoupled integrator based on first-order finite elements in space and an implicit one-step method in time. We prove unconditional convergence of the sequence of discrete approximations towards a weak solution of the system as the mesh size and the time-step size go to zero. Compared to previous numerical works on this problem, for our method, we prove a discrete energy law that mimics that of the continuous problem and, passing to the limit, yields an energy inequality satisfied by weak solutions. Moreover, our method does not employ a nodal projection to impose the unit length constraint on the discrete magnetisation, so that the stability of the method does not require weakly acute meshes. Furthermore, our integrator and its analysis hold for a more general setting, including body forces and traction, as well as a more general representation of the magnetostrain. Numerical experiments underpin the theory and showcase the applicability of the scheme for the simulation of the dynamical processes involving magnetoelastic materials at submicrometer length scales.Comment: 36 pages, 7 figure

Goal-oriented adaptivity for multilevel stochastic Galerkin FEM with nonlinear goal functionals

This paper is concerned with the numerical approximation of quantities of interest associated with solutions to parametric elliptic partial differential equations (PDEs). The key novelty of this work is in its focus on the quantities of interest represented by continuously G\^ateaux differentiable nonlinear functionals. We consider a class of parametric elliptic PDEs where the underlying differential operator has affine dependence on a countably infinite number of uncertain parameters. We design a goal-oriented adaptive algorithm for approximating nonlinear functionals of solutions to this class of parametric PDEs. In the algorithm, the approximations of parametric solutions to the primal and dual problems are computed using the multilevel stochastic Galerkin finite element method (SGFEM) and the adaptive refinement process is guided by reliable spatial and parametric error indicators that identify the dominant sources of error. We prove that the proposed algorithm generates multilevel SGFEM approximations for which the estimates of the error in the goal functional converge to zero. Numerical experiments for a selection of test problems and nonlinear quantities of interest demonstrate that the proposed goal-oriented adaptive strategy yields optimal convergence rates (for both the error estimates and the reference errors in the quantities of interest) with respect to the overall dimension of the underlying multilevel approximations spaces.Comment: 26 pages, 2 figure