A weak space-time formulation for the linear stochastic heat equation

The topic covered in this thesis is the introduction of a new formulation for the linear stochastic heat equation driven by additive noise, based on the space-time variational formulation for its deterministic counterpart. Having a variational formulation allows the use of the so called inf-sup theory in order to obtain results of existence and uniqueness in a relatively simple way.The Banach-Necas-Babuska inf-sup theory is a well known functional analytic tool which has been lately used in connection to the linear heat equation. During the last years, indeed, there has been has been a renewed interest in this theory, that characterizes existence, uniqueness, and continuous dependence on data of the variational problem. The theory provides a variational formulation with different trial and test spaces ant it is therefore a natural basis for Petrov-Galerkin approximation methods. However, in most cases, its use has been limited only within a deterministic framework for the linear heat equation.In the appended paper we present a weak space-time formulation for the linear stochastic heat equation with additive noise, where both trial and test functions have a stochastic component and we give sufficient conditions on the the data and on the covariance operator associated to the noise in order to have existence and uniqueness of the solution. We show the connection between the obtained solution and the other concepts of solution present in literature and, with the further assumption that the elliptic operator in the stochastic heat equation does not have a stochastic component and is independent of time, we derive some properties regarding the spatial regularity. Finally, we present two possible semi-discrete schemes to compute an approximate solution, one in space and one in time, where, in particular, the time-stepping obtained usingpiecewise linear test functions and piecewise constant trial functions is a modification of the Crank-Nicolson scheme. In both cases we bound the errors by means of the quasi-optimality theory

A weak space-time formulation for the linear stochastic heat equation

The topic covered in this thesis is the introduction of a new formulation for the linear stochastic heat equation driven by additive noise, based on the space-time variational formulation for its deterministic counterpart. Having a variational formulation allows the use of the so called inf-sup theory in order to obtain results of existence and uniqueness in a relatively simple way.The Banach-Necas-Babuska inf-sup theory is a well known functional analytic tool which has been lately used in connection to the linear heat equation. During the last years, indeed, there has been has been a renewed interest in this theory, that characterizes existence, uniqueness, and continuous dependence on data of the variational problem. The theory provides a variational formulation with different trial and test spaces ant it is therefore a natural basis for Petrov-Galerkin approximation methods. However, in most cases, its use has been limited only within a deterministic framework for the linear heat equation.In the appended paper we present a weak space-time formulation for the linear stochastic heat equation with additive noise, where both trial and test functions have a stochastic component and we give sufficient conditions on the the data and on the covariance operator associated to the noise in order to have existence and uniqueness of the solution. We show the connection between the obtained solution and the other concepts of solution present in literature and, with the further assumption that the elliptic operator in the stochastic heat equation does not have a stochastic component and is independent of time, we derive some properties regarding the spatial regularity. Finally, we present two possible semi-discrete schemes to compute an approximate solution, one in space and one in time, where, in particular, the time-stepping obtained usingpiecewise linear test functions and piecewise constant trial functions is a modification of the Crank-Nicolson scheme. In both cases we bound the errors by means of the quasi-optimality theory

Discrete Variational Derivative Methods for the EPDiff equation

The aim of this paper is the derivation of structure preserving schemes for the solution of the EPDiff equation, with particular emphasis on the two dimensional case. We develop three different schemes based on the Discrete Variational Derivative Method (DVDM) on a rectangular domain discretized with a regular, structured, orthogonal grid. We present numerical experiments to support our claims: we investigate the preservation of energy and linear momenta, the reversibility, and the empirical convergence of the schemes. The quality of our schemes is finally tested by simulating the interaction of singular wave fronts.Comment: 41 pages, 41 figure

Hardware simulator for optical correlation spectroscopy with Gaussian statistics and arbitrary correlation functions

We present a new hardware simulator (HS) for characterization, testing and benchmarking of digital correlators used in various optical correlation spectroscopy experiments where the photon statistics is Gaussian and the corresponding time correlation function can have any arbitrary shape. Starting from the HS developed in [Rev. Sci. Instrum. 74, 4273 (2003)], and using the same I/O board (PCI-6534 National Instrument) mounted on a modern PC (Intel Core i7-CPU, 3.07GHz, 12GB RAM), we have realized an instrument capable of delivering continuous streams of TTL pulses over two channels, with a time resolution of Δt = 50ns, up to a maximum count rate of 〈I〉 ∼ 5MHz. Pulse streams, typically detected in dynamic light scattering and diffuse correlation spectroscopy experiments were generated and measured with a commercial hardware correlator obtaining measured correlation functions that match accurately the expected ones.Peer ReviewedPostprint (published version

Numerical solution of parabolic problems based on a weak space-time formulation

We investigate a weak space-time formulation of the heat equation and its use for the construction of a numerical scheme. The formulation is based on a known weak space-time formulation, with the difference that a pointwise component of the solution, which in other works is usually neglected, is now kept. We investigate the role of such a component by first using it to obtain a pointwise bound on the solution and then deploying it to construct a numerical scheme. The scheme obtained, besides being quasi-optimal in the L2 sense, is also pointwise superconvergent in the temporal nodes. We prove a priori error estimates and we present numerical experiments to empirically support our findings

Ab Initio Spectroscopic Investigation of Pharmacologically Relevant Chiral Molecules: The Cases of Avibactam, Cephems, and Idelalisib as Benchmarks for Antibiotics and Anticancer Drugs

Abstract: The ability to accurately measure or predict several physicochemical properties of molecules which play a role as active substances in drugs can be of strategic importance for pharmacological applications, in addition to its possible interest in fundamental research. Chirality is a relevant feature in the characterization of drug molecules: enantiomers can show different pharmacological activity and adverse effects. The ability to separate stereoisomers and to assign their absolute configuration can thus be crucial. Circular dichroism (CD) spectra are a useful tool to distinguish between enantiomers. In this work we apply an in-house developed code, based on an efficient DFT approach for circular dichroism, to fully characterize the molecular optical properties in the case of few selected fundamental molecules for current medical and pharmaceutical research, namely avibactam, as representative of non b-lactam inhibitors, two cephems (cefepime and cefoxitin), as examples of b-lactam antibiotics, and idelalisib, as a recent relevant anticancer active substance to treat major leukemias. For the above molecules, in addition to their optical absorption spectra, we calculate their CD spectra within state-of-the-art computational techniques. We then investigate both the conformational and chemical sensitivity of absorption and CD spectra for the chosen molecules. The outcomes of the present research could be of fundamental importance to gain additional information on molecules involved in therapeutic protocols for severe diseases or in drug design

NUMERICAL SOLUTION OF PARABOLIC PROBLEMS BASED ON A WEAK SPACE-TIME FORMULATION

Abstract. We investigate a weak space-time formulation of the heat equation and its use for the construction of a numerical scheme. The formulation is based on a known weak space-time formulation, with the difference that a pointwise component of the solution, which in other works is usually neglected, is now kept. We investigate the role of such a component by first using it to obtain a pointwise bound on the solution and then deploying it to construct a numerical scheme. The scheme obtained, besides being quasi-optimal in the L 2 sense, is also pointwise superconvergent in the temporal nodes. We prove a priori error estimates and we present numerical experiments to empirically support our findings

Investigating the quasi-liquid layer on ice surfaces: a comparison of order parameters

Ice surfaces are characterized by pre-melted quasi-liquid layers (QLLs), which mediate both crystal growth processes and interactions with external agents. Understanding QLLs at the molecular level is necessary to unravel the mechanisms of ice crystal formation. Computational studies of the QLLs heavily rely on the accuracy of the methods employed for identifying the local molecular environment and arrangements, discriminating between solid-like and liquid-like water molecules. Here we compare the results obtained using different order parameters to characterize the QLLs on hexagonal ice (Ih) and cubic ice (Ic) model surfaces investigated with molecular dynamics (MD) simulations in a range of temperatures. For the classification task, in addition to the traditional Steinhardt order parameters in different flavours, we select an entropy fingerprint and a deep learning neural network approach (DeepIce), which are conceptually different methodologies. We find that all the analysis methods give qualitatively similar trends for the behaviours of the QLLs on ice surfaces with temperature, with some subtle differences in the classification sensitivity limited to the solid-liquid interface. The thickness of QLLs on the ice surface increases gradually as the temperature increases. The trends of the QLL size and of the values of the order parameters as a function of temperature for the different facets may be linked to surface growth rates which, in turn, affect crystal morphologies at lower vapour pressure. The choice of the order parameter can be therefore informed by computational convenience except in cases where a very accurate determination of the liquid-solid interface is important