Extreme event quantification in dynamical systems with random components

A central problem in uncertainty quantification is how to characterize the impact that our incomplete knowledge about models has on the predictions we make from them. This question naturally lends itself to a probabilistic formulation, by making the unknown model parameters random with given statistics. Here this approach is used in concert with tools from large deviation theory (LDT) and optimal control to estimate the probability that some observables in a dynamical system go above a large threshold after some time, given the prior statistical information about the system's parameters and/or its initial conditions. Specifically, it is established under which conditions such extreme events occur in a predictable way, as the minimizer of the LDT action functional. It is also shown how this minimization can be numerically performed in an efficient way using tools from optimal control. These findings are illustrated on the examples of a rod with random elasticity pulled by a time-dependent force, and the nonlinear Schr\"odinger equation (NLSE) with random initial conditions

Mathematical Analysis and Applications

Investigations involving the theory and applications of mathematical analytic tools and techniques are remarkably wide-spread in many diverse areas of the mathematical, physical, chemical, engineering and statistical sciences. In this Special Issue, we invite and welcome review, expository and original research articles dealing with the recent advances in mathematical analysis and its multidisciplinary applications

A Brief Review on Mathematical Tools Applicable to Quantum Computing for Modelling and Optimization Problems in Engineering

Since its emergence, quantum computing has enabled a wide spectrum of new possibilities and advantages, including its efficiency in accelerating computational processes exponentially. This has directed much research towards completely novel ways of solving a wide variety of engineering problems, especially through describing quantum versions of many mathematical tools such as Fourier and Laplace transforms, differential equations, systems of linear equations, and optimization techniques, among others. Exploration and development in this direction will revolutionize the world of engineering. In this manuscript, we review the state of the art of these emerging techniques from the perspective of quantum computer development and performance optimization, with a focus on the most common mathematical tools that support engineering applications. This review focuses on the application of these mathematical tools to quantum computer development and performance improvement/optimization. It also identifies the challenges and limitations related to the exploitation of quantum computing and outlines the main opportunities for future contributions. This review aims at offering a valuable reference for researchers in fields of engineering that are likely to turn to quantum computing for solutions. Doi: 10.28991/ESJ-2023-07-01-020 Full Text: PD

Developing Exchange-Correlation and Kinetic Energy Functional Approximations for Density Functional Theory

A nearly endless amount of technology relies on the understanding of the properties of matter and materials. Because the properties emerge from the motion of the electrons within matter, deepest and most accurate understanding can only be achieved by measuring or simulating the electronic structure. This thesis considers the computational simulation aspect, and currently the most popular way of conducting these simulations on a computer is density functional theory (DFT). The accuracy of the DFT calculations mostly depends on a small, but very important, component of the total energy — the exchange-correlation (XC) energy. The exact form of the XC energy term is not known and therefore always has to be approximated. When calculating very big systems also the kinetic energy term has to approximated in an orbital-free manner, because computing the electronic orbitals is too expensive for the big systems. Firstly, a new gradient-level XC approximation calledQNAis presented, and it is designed for the calculation of metallic bulk alloys. QNA exploits the subsystem functional scheme to address the issue of inconsistent performance that current gradient-level approximations have with many alloys. QNA is shown to produce more accurate binary alloy formation energies, and the good accuracy of formation energies is very important in alloy theory. Secondly, a new method of computing the kinetic energy without orbitals is presented and tested in practice. This method allows one, in principle, to perform orbital-free calculations for spherically symmetric systems at the high accuracy level of orbital DFT. A succesful orbital-free solution for the electronic structure of the Be atom is presented. One of the ultimate goals in DFT research is to combine the high accuracy of orbital DFT with the excellent computational speed of orbital-free DFT, and the proofof-concept solution for the Be atom is a step in this direction.Tiheysfunktionaaliteoriaa varten kehitetyt vaihtokorrelaatio- ja liike-energia –approksimaatiot Lähes lukematon määrä teknologiaa nojautuu aineen ja materiaalien ominaisuuksien ymmärtämiseen. Koska nämä ominaisuudet kumpuavat aineen koossapitävästä elektronirakenteesta, syvällisin ja kaikista tarkin ymmärrys voidaan saavuttaa ainoastaan mittaamalla tai simuloimalla kyseistä elektronirakennetta. Tämä väitöskirja käsittelee jälkimmäistä vaihtoehtoa, eli elektronirakenteen mallintamista tietokoneella tehtävien laskujen avulla. Nykyään suosituin tällaisista laskentamenetelmistä on tiheysfunktionaalimenetelmä. Tiheysfunktionaaliteoriaan pohjautuvien laskujen tarkkuus riippuu pääasiassa yhdestä pienestä, mutta erittäin tärkeästä kokonaisenergian komponentista—vaihto-korrelaatioenergiasta. Vaihto-korrelaatioenergian tarkkaa matemaattista muotoa ei tunneta, joten sille on aina käytettävä jotakin approksimaatiota. Kun halutaan mallintaa erittäin kookkaita systeemejä, myös liike-energia on approksimoitava orbitaalivapaalla tavalla, sillä orbitaalien laskeminen kookkaille systeemeille on liian aikaavievää. Ensimmäiseksi tässä tutkielmassa esitetään uusi gradientti-tason vaihto- korrelaatioapproksimaatio QNA, joka on suunniteltu metalliseosten laskemiseen.QNA hyödyntää alisysteemifunktionaaleja (subsystem functional scheme) parantamaan laskujen tarkkuutta metalliseoksille verrattuna nykyisiin gradienttitason funktionaaleihin. Nykyiset gradienttitason funktionaalit eivät useinkaan pysty mallintamaan kaikkia seoksen komponentteja (puhtaat alkuaineet) tarkasti, minkä tässä väitöskirjassa osoitetaan johtavan epätarkkoihin tuloksiin itse seokselle. QNA-approksimaatiossa kukin seoksen komponentti mallinnetaan erillisen alisysteemifunktionaalin avulla, jolloin kukin komponentti ja itse seos voidaan laskea tarkasti. Käytännön laskuilla osoitetaan, että QNA tuottaa erittäin tarkkoja muodostumisenergioita kaksikomponenttisille metalliseoksille, mikä on erittäin tärkeä seikka metalliseosten teoriassa

Does Quantum Mechanics Breed Larger, More Intricate Quantum Theories? The Case for Experience-Centric Quantum Theory and the Interactome of Quantum Theories

We pose and address the radical question that whether quantum mechanics, known for its firm internal structure and enormous empirical success, carries in itself the genome of larger quantum theories which have higher internal intricacies and phenomenological versatilities. That is, on the basic level of closed quantum systems and regardless of interpretational aspects, whether standard quantum theory (SQT) harbors quantum theories with context-based deformed principles or structures, having definite predictive power within broader scopes. We answer the question in affirmative following complementary evidence and reasoning arising from quantum-computation-based quantum simulation and fundamental, general, abstract rationales in the frameworks of information theory, fundamental or functional emergence, and participatory agency. In this light, as we show, one is led to the recently proposed experience-centric quantum theory (ECQT), which is a larger and richer theory of quantum behaviors with drastically generalized quantum dynamics. ECQT allows the quantum information of the closed quantum system's developed state history to continually contribute to defining manybody interactions, Hamiltonians, and even internal elements and ``particles'' of the total system. Hence the unitary evolutions are continually impacted and become guidable by the agent-system's experience. The intrinsic interplay of unitarity and non-Markovianity in ECQT brings about a host of diverse behavioral phases, which concurrently infuse closed and open quantum system characteristics and even surpasses the theory of open systems in SQT. In the broader perspective, an upshot of our investigation is the existence of the quantum interactome--the interactive landscape of all coexisting, independent context-based quantum theories which emerge from inferential participatory agencies--and its predictive phenomenological utility.Comment: 54 page