In this thesis we investigate large deviation and path properties of a few models within the Kardar-Parisi-Zhang (KPZ) universality class. The KPZ equation is the central object in the KPZ universality class. It is a stochastic PDE describing various objects in statistical mechanics such as random interface growth, directed polymers, interacting particle systems. In the first project we study one point upper tail large deviations of the KPZ equation 𝜢(t,x) started from narrow wedge initial data. We obtain precise expression of the upper tail LDP in the long time regime for the KPZ equation. We then extend our techniques and methods to obtain upper tail LDP for the asymmetric exclusion process model, which is a prelimit of the KPZ equation. In the next direction, we investigate temporal path properties of the KPZ equation. We show that the upper and lower law of iterated logarithms for the rescaled KPZ temporal process occurs at a scale (log log 𝑡)²/³ and (log log 𝑡)¹/³ respectively. We also compute the exact Hausdorff dimension of the upper level sets of the solution, i.e., the set of times when the rescaled solution exceeds 𝛼(log log 𝑡)²/³. This has relevance from the point of view of fractal geometry of the KPZ equation. We next study superdiffusivity and localization features of the (1+1)-dimensional continuum directed random polymer whose free energy is given by the KPZ equation. We show that for a point-to-point polymer of length 𝑡 and any 𝑝 ⋲ (0,1), the point on the path which is 𝑝𝑡 distance away from the origin stays within a 𝑂(1) stochastic window around a random point 𝙈_𝑝,𝑡 that depends on the environment. This provides an affirmative case of the folklore `favorite region' conjecture. Furthermore, the quenched density of the point when centered around 𝙈_𝑝,𝑡 converges in law to an explicit random density function as 𝑡 → ∞ without any scaling. The limiting random density is proportional to 𝑒^{-𝓡(x)} where 𝓡(x) is a two-sided 3D Bessel process with diffusion coefficient 2. Our proof techniques also allow us to prove properties of the KPZ equation such as ergodicity and limiting Bessel behaviors around the maximum. In a follow up project, we show that the annealed law of polymer of length 𝑡, upon 𝑡²/³ superdiffusive scaling, is tight (as 𝑡 → ∞) in the space of 𝐶([0,1]) valued random variables. On the other hand, as 𝑡 → 0, under diffusive scaling, we show that the annealed law of the polymer converges to Brownian bridge. In the final part of this thesis, we focus on an integrable discrete half-space variant of the CDRP, called half-space log-gamma polymer.We consider the point-to-point log-gamma polymer of length 2𝑁 in a half-space with i.i.d.Gamma⁻¹(2𝛳) distributed bulk weights and i.i.d. Gamma⁻¹(𝛼+𝛳) distributed boundary weights for 𝛳 > 0 and 𝛼 > -𝛳. We establish the KPZ exponents (1/3 fluctuation and 2/3 transversal) for this model when 𝛼 ≥ 0. In particular, in this regime, we show that after appropriate centering, the free energy process with spatial coordinate scaled by 𝑁²/³ and fluctuations scaled by 𝑁¹/³ is tight. The primary technical contribution of our work is to construct the half-space log-gamma Gibbsian line ensemble and develop a toolbox for extracting tightness and absolute continuity results from minimal information about the top curve of such half-space line ensembles. This is the first study of half-space line ensembles. The 𝛼 ≥ 0 regime correspond to a polymer measure which is not pinned at the boundary. In a companion work, we investigate the 𝛼 < 0 setting. We show that in this case, the endpoint of the point-to-line polymer stays within 𝑂(1) window of the diagonal. We also show that the limiting quenched endpoint distribution of the polymer around the diagonal is given by a random probability mass function proportional to the exponential of a random walk with log-gamma type increments.

No description available.

A method for trimming surfaces generated as solutions to Partial Differential Equations (PDEs) is presented. The work we present here utilises the 2D parameter space on which the trim curves are defined whose projection on the parametrically represented PDE surface is then trimmed out. To do this we define the trim curves to be a set of boundary conditions which enable us to solve a low order elliptic PDE on the parameter space. The chosen elliptic PDE is solved analytically, even in the case of a very general complex trim, allowing the design process to be carried out interactively in real time. To demonstrate the capability for this technique we discuss a series of examples where trimmed PDE surfaces may be applicable.

Wildfire modeling is a complex, computationally costly endeavor, but with droughts worsening and fires burning across the western United States, obtaining accurate wildfire predictions is more important than ever. In this paper, we present a novel approach to wildfire modeling using data assimiliation. We model wildfire spread with a modification of the partial differential equation model described by Mandel et al. in their 2008 paper. Specifically, we replace some constant parameter values with geospatial functions of fuel type. We combine deep learning and remote sensing to obtain real-time data for the model and employ the Nelder-Mead method to recover optimal model parameters with data assimilation. We demonstrate the efficacy of this approach on computer-generated fires, as well as real fire data from the 2021 Dixie Fire in California. On generated fires, this approach resulted in an average Jaccard index of 0.996 between the predicted and actual fire perimeters and an average Kulczynski measure of 0.997. On data from the Dixie Fire, the average Jaccard index achieved was 0.48, and the average Kulczynski measure was 0.66.

This thesis describes initial language extensions to the Modelica language to define a more general language called PDEModelica, with built-in support for modeling with partial differential equations (PDEs). Modelica® is a standardized modeling language for objectoriented, equation-based modeling. It also supports component-based modeling where existing components with modified parameters can be combined into new models. The aim of the language presented in this thesis is to maintain the advantages of Modelica and also add partial differential equation support. Partial differential equations can be defined using a coefficient-based approach, where a predefined PDE is modified by changing its coefficient values. Language operators to directly express PDEs in the language are also discussed. Furthermore, domain geometry description is handled and language extensions to describe geometries are presented. Boundary conditions, required for a complete PDE problem definition, are also handled. A prototype implementation is described as well. The prototype includes a translator written in the relational meta-language, RML, and interfaces to external software such as mesh generators and PDE solvers, which are needed to solve PDE problems. Finally, a few examples modeled with PDEModelica and solved using the prototype are presented. / <p>Report code: LiU-Tek-Lic-2002:63.</p>

No description available.

<p>This thesis is a numerical investigation of two-dimensional steady flows past a circular obstacle. In the fluid dynamics research there are few computational results concerning the structure of the steady wake flows at Reynolds numbers larger than 100, and the state-of-the-art results go back to the work of Fornberg (1980) Fornberg (1985). The radial velocity component approaches its asymptotic value relatively slowly if the solution is ``physically reasonable''. This presents a difficulty when using the standard approach such as domain truncation. To get around this problem, in the present research we will develop a spectral technique for the solution of the steady Navier-Stokes system. We introduce the ``bootstrap" method which is motivated by the mathematical fact that solutions of the Oseen system have the same asymptotic structure at infinity as the solutions of the steady Navier-Stokes system with the same boundary conditions. Thus, in the ``bootstrap" method, the streamfunction is calculated as a perturbation to the solution to the Oseen system. Solutions are calculated for a range of Reynolds number and we also investigate the solutions behaviour when the Reynolds number goes to infinity. The thesis compares the numerical results obtained using the proposed spectral ``bootstrap" method and a finite--difference approach for unbounded domains against previous results. For Reynolds numbers lower than 100, the wake is slender and similar to the flow hypothesized by Kirchoff (1869) and Levi-Civita (1907). For large Reynolds numbers the wake becomes wider and appears more similar to the Prandtl-Batchelor flow, see Batchelor (1956).</p> / Doctor of Science (PhD)

In this thesis we study three stochastic partial differential equations (SPDE) that arise as stochastic gradient flows via the fluctuation-dissipation principle. For the first equation we establish a finer regularity statement based on a generalized Taylor expansion which is inspired by the theory of rough paths. The second equation is the thin-film equation with thermal noise which is a singular SPDE. In order to circumvent the issue of dealing with possible renormalization, we discretize the gradient flow structure of the deterministic thin-film equation. Choosing a specific discretization of the metric tensor, we resdiscover a well-known discretization of the thin-film equation introduced by Grün and Rumpf that satisfies a discrete entropy estimate. By proving a stochastic entropy estimate in this discrete setting, we obtain positivity of the scheme in the case of no-slip boundary conditions. Moreover, we analyze the associated rate functional and perform numerical experiments which suggest that the scheme converges. The third equation is the massive $\varphi^4_2$-model on the torus which is also a singular SPDE. In the spirit of Bakry and Émery, we obtain a gradient bound on the Markov semigroup. The proof relies on an $L^2$-estimate for the linearization of the equation. Due to the required renormalization, we use a stopping time argument in order to ensure stochastic integrability of the random constant in the estimate. A postprocessing of this estimate yields an even sharper gradient bound. As a corollary, for large enough mass, we establish a local spectral gap inequality which by ergodicity yields a spectral gap inequality for the $\varphi^4_2$- measure.

We first provide a detailed background of a geometric projection methodology developed by Professor Roswitha Marz at Humboldt University in Berlin for showing uniqueness and existence of solutions for ordinary differential-algebraic equations (DAEs). Because of the geometric and operator-theoretic aspects of this particular method, it can be extended to the case of infinite-dimensional abstract DAEs. For example, partial differential equations (PDEs) are often formulated as abstract Cauchy or evolution problems which we label abstract ordinary differential equations or AODE. Using this abstract formulation, existence and uniqueness of the Cauchy problem has been studied. Similarly, we look at an AODE system with operator constraint equations to formulate an abstract differential-algebraic equation or ADAE problem. Existence and uniqueness of solutions is shown under certain conditions on the operators for both index-1 and index-2 abstract DAEs. These existence and uniqueness results are then applied to some index-1 DAEs in the area of thermodynamic modeling of a chemical vapor deposition reactor and to a structural dynamics problem. The application for the structural dynamics problem, in particular, provides a detailed construction of the model and development of the DAE framework. Existence and uniqueness are primarily demonstrated using a semigroup approach. Finally, an exploration of some issues which arise from discretizing the abstract DAE are discussed. / Ph. D.

Interval analysis is an essential tool in the construction of validated numerical solutions of Initial Value Problems (IVP) for Ordinary (ODE) and Partial (PDE) Differential Equations. A validated solution typically consists of guaranteed lower and upper bounds for the exact solution or set of exact solutions in the case of uncertain data, i.e. it is an interval function (enclosure) containing all solutions of the problem. IVP for ODE: The central point of discussion is the wrapping effect. A new concept of wrapping function is introduced and applied in studying this effect. It is proved that the wrapping function is the limit of the enclosures produced by any method of certain type (propagate and wrap type). Then, the wrapping effect can be quantified as the difference between the wrapping function and the optimal interval enclosure of the solution set (or some norm of it). The problems with no wrapping effect are characterized as problems for which the wrapping function equals the optimal interval enclosure. A sufficient condition for no wrapping effect is that there exist a linear transformation, preserving the intervals, which reduces the right-hand side of the system of ODE to a quasi-isotone function. This condition is also necessary for linear problems and "near" necessary in the general case. Hyperbolic PDE: The Initial Value Problem with periodic boundary conditions for the wave equation is considered. It is proved that under certain conditions the problem is an operator equation with an operator of monotone type. Using the established monotone properties, an interval (validated) method for numerical solution of the problem is proposed. The solution is obtained step by step in the time dimension as a Fourier series of the space variable and a polynomial of the time variable. The numerical implementation involves computations in Fourier and Taylor functoids. Propagation of discontinuo~swaves is a serious problem when a Fourier series is used (Gibbs phenomenon, etc.). We propose the combined use of periodic splines and Fourier series for representing discontinuous functions and a method for propagating discontinuous waves. The numerical implementation involves computations in a Fourier hyper functoid. / Mathematical Sciences / D. Phil. (Mathematics)