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Physics simulation computationally models physical phenomena. It is the bread-and-butter of modern-day scientific discoveries and engineering design: from plasma theory to digital twins. However, viable efficiency remains a long-standing challenge to physics simulation. Accurate, real-world-scale simulations are often computationally too expensive (e.g., excessive wall-clock time) to gain any practical usage. In this thesis, we explore two general solutions to tackle this problem. Our first proposed method is a multiscaling approach. Simulating physics at its fundamental discrete scale, e.g., the atomic-level, provides unmatched levels of detail and generality, but proves to be excessively costly when applied to large-scale systems. Alternatively, simulating physics at the continuum scale governed by partial differential equations (PDEs) is computationally tractable, but limited in applicability due to built-in modeling assumptions. We propose a multiscaling simulation technique that exploits the dual strengths of discrete and continuum treatments. In particular, we design a hybrid discrete-continuum framework for granular media. In this adaptive framework, we define an oracle to dynamically partition the domain into continuum regions where safe and discrete regions where necessary. We couple the dynamics of the discrete and continuum regions via overlapping transition zones to form one coherent simulation. Enrichment and homogenization operations convert between discrete and continuum representations, which allow the partitions to evolve over time. This approach saves the computation cost by partially employing continuum simulations and obtains up to 116X speedup over the discrete-only simulations while maintaining the same level of accuracy. To further accelerate PDE-governed continuum simulations, we propose a machine-learning-based reduced-order modeling (ROM) method. Whereas prior ROM approaches reduce the dimensionality of discretized vector fields, our continuous reduced-order modeling (CROM) approach builds a smooth, low-dimensional manifold of the continuous vector fields themselves, not their discretization. We represent this reduced manifold using neural fields, relying on their continuous and differentiable nature to efficiently solve the PDEs. CROM may train on any and all available numerical solutions of the continuous system, even when they are obtained using diverse methods or discretizations. Indeed, CROM is the first model reduction framework that can simultaneously handle data from voxels, meshes, and point clouds. After the low-dimensional manifolds are established, solving PDEs requires significantly less computational resources. Since CROM is discretization-agnostic, CROM-based PDE solvers may optimally adapt discretization resolution over time to economize computation. We validate our approach on an extensive range of PDEs from thermodynamics, image processing, solid mechanics, and fluid dynamics. Selected large-scale experiments demonstrate that our approach obtains speed, memory, and accuracy advantages over prior ROM approaches while gaining 109X wall-clock speedup over full-order models on CPUs and 89X speedup on GPUs.

We study a system of semilinear hyperbolic PDEs which arises as a continuum approximation of the discrete nonlinear dimer array model of SSH type of Hadad, Vitelli and Alú in [1]. We classify the system’s traveling waves, and study their stability properties. We focus on pulse solutions (solitons) on a nontrivial background and moving domain wall solutions (kinks and antikinks), corresponding to heteroclinic orbits for a reduced two-dimensional dynamical system. We further present analytical results on: nonlinear stability and spectral stability of supersonic pulses, and the spectral stability of moving domain walls. Our result for nonlinear stability is expressed in terms of appropriately weighted ?1-norms of the perturbation, which captures the phenomenon of convective stabilization; as time advances, the traveling wave “outruns” the growing disturbance excited by an initial perturbation. We use our analytical results to interpret phenomena observed in numerical simulations. The results for (linear) spectral stability are studied in appropriately-weighted ?2-norms.

This research investigates existing relationships between the three apparently unrelated subjects: Markov process, Semigroups and Partial difierential equations. Markov processes define semigroups through their transition functions. Conversely particular semigroups determine transition functions and can be regarded as Markov processes. We have exploited these relationships to study some Markov chains. The infnitesimal generator of a Feller semigroup on the closure of a bounded domain of Rn; (n ^ 2), is an integro-diferential operator in the interior of the domain and verifes a boundary condition. The existence of a Feller semigroup defined by a diferential operator and a boundary condition is due to the existence of solution of a bounded value problem. From this result other existence suficient conditions on the existence of Feller semigroups have been obtained and we have applied some of them to construct Feller semigroups on the unity disk of R2. / Decision Sciences / M. Sc. (Operations Research)

The thesis presents a new method of Symmetry Analysis of the Black-Scholes Merton Finance Model through modi ed Local one-parameter transformations. We determine the symmetries of both the one-dimensional and two-dimensional Black-Scholes equations through a method that involves the limit of in nitesimal ! as it approaches zero. The method is dealt with extensively in [23]. We further determine an invariant solution using one of the symmetries in each case. We determine the transformation of the Black-Scholes equation to heat equation through Lie equivalence transformations. Further applications where the method is successfully applied include working out symmetries of both a Gaussian type partial di erential equation and that of a di erential equation model of epidemiology of HIV and AIDS. We use the new method to determine the symmetries and calculate invariant solutions for operators providing them. / Mathematical Sciences / Applied Mathematics / D. Phil. (Applied Mathematics)

The thesis presents a new method of Symmetry Analysis of the Black-Scholes Merton Finance Model through modi ed Local one-parameter transformations. We determine the symmetries of both the one-dimensional and two-dimensional Black-Scholes equations through a method that involves the limit of in nitesimal ! as it approaches zero. The method is dealt with extensively in [23]. We further determine an invariant solution using one of the symmetries in each case. We determine the transformation of the Black-Scholes equation to heat equation through Lie equivalence transformations. Further applications where the method is successfully applied include working out symmetries of both a Gaussian type partial di erential equation and that of a di erential equation model of epidemiology of HIV and AIDS. We use the new method to determine the symmetries and calculate invariant solutions for operators providing them. / Mathematical Sciences / Applied Mathematics / D. Phil. (Applied Mathematics)

In general relativity, angular momentum of the gravitational field in some volume bounded by an axially symmetric sphere is well-defined as a boundary integral. The definition relies on the symmetry generating vector field, a Killing field, of the boundary. When no such symmetry exists, one defines angular momentum using an approximate Killing field. Contained in the literature are various approximations that capture certain properties of metric preserving vector fields. We explore the continuity of an angular momentum definition that employs an approximate Killing field that is an eigenvector of a particular second-order differential operator. We find that the eigenvector varies continuously in Hilbert space under smooth perturbations of a smooth boundary geometry. Furthermore, we find that not only is the approximate Killing field continuous but that the eigenvalue problem which defines it is stable in the sense that all of its eigenvalues and eigenvectors are continuous in Hilbert space. We conclude that the stability follows because the eigenvalue problem is strongly elliptic. Additionally, we provide a practical introduction to the mathematical theory of strongly elliptic operators and generalize the above stability results for a large class of such operators. / Includes bibliography. / Dissertation (Ph.D.)--Florida Atlantic University, 2014. / FAU Electronic Theses and Dissertations Collection

Singularly perturbed systems with or without delays commonly appear in mathematical modeling of physical and chemical processes, engineering applications, and increasingly, in mathematical biology. There has been intensive work for singularly perturbed systems, yet most of the work so far focused on systems without delays. In this thesis, we provide a new set of tools for the stability analysis for singularly perturbed control systems with time delays. / Includes bibliography. / Dissertation (Ph.D.)--Florida Atlantic University, 2015. / FAU Electronic Theses and Dissertations Collection

Made available in DSpace on 2015-03-04T18:57:41Z (GMT). No. of bitstreams: 1 MScCampeao.pdf: 730731 bytes, checksum: 93bde1355e901654a01dbd2a0138fe35 (MD5) Previous issue date: 2012-01-09 / Conselho Nacional de Desenvolvimento Cientifico e Tecnologico / In this work a methodology for the compliance topology design of Kirchhoff plates with volume constraint using topological derivative is presented. The topological derivative measures the sensitivity of a given shape functional with respect to an infinitesimal singular domain perturbation, such as the insertion of holes, inclusions, source-terms or even cracks. Firstly, the hypothesis associated to the Kirchhoff elastic plates bending model are presented as well as the functional that represents the total potential energy of the plate. Then, the mathematical development to obtain the topological derivative considering as singular perturbation the introduction of a small circular inclusion is presented. The total potential energy is considered as shape functional together with a volume constraint. The use of two methods for volume control is discussed. The first one is done by means of linear penalization and does not provide direct control over the required volume fraction. In this case, the penalty parameter is the coefficient of a linear term used to control the amount of material to be removed. The second approach is based on the Augmented Langrangian method which has both, linear and quadratic terms. The coefficient of the quadratic part controls the Lagrange multiplier update of the linear part. Through this last method it is possible to specify the final amount of material in the optimized structure. Next, a topology design algorithm of Kirchhoff plates is presented, which uses the information provided by the topological derivative together with a level-set domain representation method. Finally, some numerical examples are presented in the context of compliance topology optimization with volume constraint. / Neste trabalho é apresentada uma metodologia para otimização topológica de placas de Kirchhoff minimizando a flexibilidade com restrição em volume utilizando derivada topológica. A derivada topológica mede a sensibilidade de um dado funcional de forma em relação a uma perturbação singular infinitesimal no domínio, tal como a inserção de furos, inclusões, termos fonte ou trincas. Primeiramente, são apresentadas as hipóteses associadas ao modelo de flexão elástica de placas de Kirchhoff bem como o funcional que representa a energia potencial total da placa e, em seguida, o desenvolvimento matemático para a obtenção da derivada topológica considerando como perturbação singular a introdução de uma pequena inclusão circular. A energia potencial total é considerada como funcional de forma juntamente com uma restrição de volume. Discute-se ainda a utilização de dois métodos para realizar o controle da restrição de volume. O primeiro é feito por meio de penalização linear e não fornece controle direto sobre a fração de volume requerida. Nesse caso, o parâmetro de penalidade é o coeficiente de um termo linear que é usado para controlar a quantidade de material a ser removido. A segunda abordagem é baseada no método do Lagrangeano Aumentado que possui um termo linear e um quadrático. O coeficiente da parte quadrática controla a atualização do multiplicador de Lagrange da parte linear. Através desse último método é possível especificar a quantidade final de material na estrutura otimizada. Dessa forma, é apresentado um algoritmo que utiliza a informação fornecida pela derivada topológica conjuntamente com um método de representação de domínio por função level-set na otimização topológica de placas de Kirchhoff. Por fim, alguns exemplos numéricos são apresentados no contexto de otimização topológica com restrição em volume.

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