Effective Integration of Ultra-Elliptic Solutions of the Focusing Nonlinear Schrödinger Equation

An effective integration method based on the classical solution of the Jacobi inversion problem, using Kleinian ultra-elliptic functions and Riemann theta functions, is presented for the quasi-periodic two-phase solutions of the focusing cubic nonlinear Schrödinger equation. Each two-phase solution with real quasi-periods forms a two-real-dimensional torus, modulo a circle of complex-phase factors, expressed as a ratio of theta functions associated with the Riemann surface of the invariant spectral curve. The initial conditions of the Dirichlet eigenvalues satisfy reality conditions which are explicitly parametrized by two physically-meaningful real variables: the squared modulus and a scalar multiple of the wavenumber. Simple new formulas for the maximum modulus and the minimum modulus are obtained in terms of the imaginary parts of the branch points of the Riemann surface

Reduced-Order Modeling based on Approximated Lax Pairs

A reduced-order model algorithm, based on approximations of Lax pairs, is proposed to solve nonlinear evolution partial differential equations. Contrary to other reduced-order methods, like Proper Orthogonal Decomposition, the space where the solution is searched for evolves according to a dynamics specific to the problem. It is therefore well-suited to solving problems with progressive waves or front propagation. Numerical examples are shown for the KdV and FKPP (nonlinear reaction diffusion) equations, in one and two dimensions

The Stochastic Toolbox User's Guide -- xSPDE3: extensible software for stochastic ordinary and partial differential equations

The xSPDE toolbox treats stochastic partial and ordinary differential equations, with applications in biology, chemistry, engineering, medicine, physics and quantum technologies. It computes statistical averages, including time-step and/or sampling error estimation. xSPDE can provide higher order convergence, Fourier spectra and probability densities. The toolbox has graphical output and $\chi^{2}$ statistics, as well as weighted, projected, or forward-backward equations. It can generate input-output quantum spectra. All equations may have independent periodic, Dirichlet, and Neumann or Robin boundary conditions in any dimension, for any vector field component, and at either end of any interval.Comment: User manual for xSPDE software on Github, at https://github.com/peterddrummond/xspde_matlab. Total of 160 pages with examples. This is a replacement with minor corrections and an updated reference list. Accepted, and will appear in Scipos

Exponential Integrator Methods for Nonlinear Fractional Reaction-diffusion Models

Nonlocality and spatial heterogeneity of many practical systems have made fractional differential equations very useful tools in Science and Engineering. However, solving these type of models is computationally demanding. In this work, we propose an exponential integrator method for nonlinear fractional reaction-diffusion equations. This scheme is based on using a real distinct poles discretization for the underlying matrix exponentials. Due to these real distinct poles, the algorithm could be easily implemented in parallel to take advantage of multiple processors for increased computational efficiency. The method is established to be second-order convergent; and proven to be robust for problems involving non-smooth/mismatched initial and boundary conditions and steep solution gradients. We examine the stability of the scheme through its amplification factor and plot the boundaries of the stability regions comparative to other second-order FETD schemes. This numerical scheme combined with fractional centered differencing is used for simulating many important nonlinear fractional models in applications. We demonstrate the superiority of our method over competing second order FETD schemes, BDF2 scheme, and IMEX schemes. Our experiments show that the proposed scheme is computationally more efficient (in terms of cpu time). Furthermore, we investigate the trade-off between using fractional centered differencing and matrix transfer technique in discretization of Riesz fractional derivatives. The generalized Mittag-Leffler function and its inverse is very useful in solving fractional differential equations and structural derivatives, respectively. However, their computational complexities have made them difficult to deal with numerically. We propose a real distinct pole rational approximation of the generalized Mittag-Leffler function. Under some mild conditions, this approximation is proven and empirically shown to be L-Acceptable. Due to the complete monotonicity property of the Mittag-Leffler function, we derive a rational approximation for the inverse generalized Mittag-Leffler function. These approximations are especially useful in developing efficient and accurate numerical schemes for partial differential equations of fractional order. Several applications are presented such as complementary error function, solution of fractional differential equations, and the ultraslow diffusion model using the structural derivative. Furthermore, we present a preliminary result of the application of the M-L RDP approximation to develop a generalized exponetial integrator scheme for time-fractional nonlinear reaction-diffusion equation

Dynamical Tunneling in Systems with a Mixed Phase Space

Tunneling is one of the most prominent features of quantum mechanics. While the tunneling process in one-dimensional integrable systems is well understood, its quantitative prediction for systems with mixed phase space is a long-standing open challenge. In such systems regions of regular and chaotic dynamics coexist in phase space, which are classically separated but quantum mechanically coupled by the process of dynamical tunneling. We derive a prediction of dynamical tunneling rates which describe the decay of states localized inside the regular region towards the so-called chaotic sea. This approach uses a fictitious integrable system which mimics the dynamics inside the regular domain and extends it into the chaotic region. Excellent agreement with numerical data is found for kicked systems, billiards, and optical microcavities, if nonlinear resonances are negligible. Semiclassically, however, such nonlinear resonance chains dominate the tunneling process. Hence, we combine our approach with an improved resonance-assisted tunneling theory and derive a unified prediction which is valid from the quantum to the semiclassical regime. We obtain results which show a drastically improved accuracy of several orders of magnitude compared to previous studies.Der Tunnelprozess ist einer der bedeutensten Effekte in der Quantenmechanik. Während das Tunneln in eindimensionalen integrablen Systemen gut verstanden ist, gestaltet sich dessen Beschreibung für Systeme mit gemischtem Phasenraum weitaus schwieriger. Solche Systeme besitzen Gebiete regulärer und chaotischer Bewegung, die klassisch getrennt sind, aber quantenmechanisch durch den Prozess des dynamischen Tunnelns gekoppelt werden. In dieser Arbeit wird eine theoretische Vorhersage für dynamische Tunnelraten abgeleitet, die den Zerfall von Zuständen, die im regulären Gebiet lokalisiert sind, in die sogenannte chaotische See beschreibt. Dazu wird ein fiktives integrables System konstruiert, das im regulären Bereich eine nahezu gleiche Dynamik aufweist und diese Dynamik in das chaotische Gebiet fortsetzt. Die Theorie zeigt eine ausgezeichnete Übereinstimmung mit numerischen Daten für gekickte Systeme, Billards und optische Mikrokavitäten, falls nichtlineare Resonanzketten vernachlässigbar sind. Semiklassisch jedoch bestimmen diese nichtlinearen Resonanzketten den Tunnelprozess. Daher kombinieren wir unseren Zugang mit einer verbesserten Theorie des Resonanz-unterstützten Tunnelns und erhalten eine Vorhersage,die vom Quanten- bis in den semiklassischen Bereich gültig ist. Ihre Resultate zeigen eine Genauigkeit, die verglichen mit früheren Theorien um mehrere Größenordnungen verbessert wurde

Statistical inference and computation in elliptic PDE models

Partial differential equations (PDE) are ubiquitous in describing real-world phenomena. In many statistical models, PDE are used to encode complex relationships between unknown quantities and the observed data. We investigate statistical and computational questions arising in such models, adopting an infinite-dimensional `nonparametric' framework and assuming the observed data are subject to random noise. The main PDE examples are of elliptic or parabolic type. Chapter 2 investigates the problem of sampling from high-dimensional Bayesian posterior distributions. The main results consist of non-asymptotic computational guarantees for Langevin-type Markov chain Monte Carlo (MCMC) algorithms which scale polynomially in key quantities such as the dimension of the model, the desired precision level, and the number of available statistical measurements. The bounds hold with high probability under the distribution of the data, assuming that certain `local geometric' assumptions are fulfilled and that a good initialiser of the algorithm is available. We study a representative non-linear PDE example where the unknown is a coefficient function in a steady-state Schr\"odinger equation, and the solution to a corresponding boundary value problem is observed. Chapter 3 studies statistical convergence rates for nonparametric Tikhonov-type estimators, which can be interpreted also as Bayesian maximum a posteriori (MAP) estimators arising from certain Gaussian process priors. The theory is derived in a general setting for non-linear inverse problems and then applied to two examples, the steady-state Schr\"odinger equation studied in Chapter \ref{sampling} and a model for the steady-state heat equation. It is shown that the rates obtained are minimax-optimal in prediction loss. The final Chapter 4 considers a model for scalar diffusion processes $(X_t:t\geq 0)$ with an unknown drift function which is modelled nonparametrically. It is shown that in the low frequency sampling case, when the sample consists of $(X_0,X_\Delta,...,X_{n\Delta})$ for some fixed sampling distance $\Delta>0$, under mild regularity assumptions, the model satisfies the local asymptotic normality (LAN) property. The key tools used are regularity estimates and spectral properties for certain parabolic and elliptic PDE related to $(X_t:t\geq 0)$