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- Anton, Rikard, 1989-
(författare)
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Exponential integrators for stochastic partial differential equations
- 2018
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Stochastic partial differential equations (SPDEs) have during the past decades become an important tool for modeling systems which are influenced by randomness. Because of the complex nature of SPDEs, knowledge of efficient numerical methods with good convergence and geometric properties is of considerable importance. Due to this, numerical analysis of SPDEs has become an important and active research field.The thesis consists of four papers, all dealing with time integration of different SPDEs using exponential integrators. We analyse exponential integrators for the stochastic wave equation, the stochastic heat equation, and the stochastic Schrödinger equation. Our primary focus is to study strong order of convergence of temporal approximations. However, occasionally, we also analyse space approximations such as finite element and finite difference approximations. In addition to this, for some SPDEs, we consider conservation properties of numerical discretizations.As seen in this thesis, exponential integrators for SPDEs have many benefits over more traditional integrators such as Euler-Maruyama schemes or the Crank-Nicolson-Maruyama scheme. They are explicit and therefore very easy to implement and use in practice. Also, they are excellent at handling stiff problems, which naturally arise from spatial discretizations of SPDEs. While many explicit integrators suffer step size restrictions due to stability issues, exponential integrators do not in general.In Paper 1 we consider a full discretization of the stochastic wave equation driven by multiplicative noise. We use a finite element method for the spatial discretization, and for the temporal discretization we use a stochastic trigonometric method. In the first part of the paper, we prove mean-square convergence of the full approximation. In the second part, we study the behavior of the total energy, or Hamiltonian, of the wave equation. It is well known that for deterministic (Hamiltonian) wave equations, the total energy remains constant in time. We prove that for stochastic wave equations with additive noise, the expected energy of the exact solution grows linearly with time. We also prove that the numerical approximation produces a small error in this linear drift.In the second paper, we study an exponential integrator applied to the time discretization of the stochastic Schrödinger equation with a multiplicative potential. We prove strong convergence order 1 and 1/2 for additive and multiplicative noise, respectively. The deterministic linear Schrödinger equation has several conserved quantities, including the energy, the mass, and the momentum. We first show that for Schrödinger equations driven by additive noise, the expected values of these quantities grow linearly with time. The exponential integrator is shown to preserve these linear drifts for all time in the case of a stochastic Schrödinger equation without potential. For the equation with a multiplicative potential, we obtain a small error in these linear drifts.The third paper is devoted to studying a full approximation of the one-dimensional stochastic heat equation. For the spatial discretization we use a finite difference method and an exponential integrator is used for the temporal approximation. We prove mean-square convergence and almost sure convergence of the approximation when the coefficients of the problem are assumed to be Lipschitz continuous. For non-Lipschitz coefficients, we prove convergence in probability.In Paper 4 we revisit the stochastic Schrödinger equation. We consider this SPDE with a power-law nonlinearity. This nonlinearity is not globally Lipschitz continuous and the exact solution is not assumed to remain bounded for all times. These difficulties are handled by considering a truncated version of the equation and by working with stopping times and random time intervals. We prove almost sure convergence and convergence in probability for the exponential integrator as well as convergence orders of ½ − ?, for all ? > 0, and 1/2, respectively.
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- Araujo-Cabarcas, Juan Carlos, 1981-
(författare)
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Reliable hp finite element computations of scattering resonances in nano optics
- 2019
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Eigenfrequencies are commonly studied in wave propagation problems, as they are important in the analysis of closed cavities such as a microwave oven. For open systems, energy leaks into infinity and therefore scattering resonances are used instead of eigenfrequencies. An interesting application where resonances take an important place is in whispering gallery mode resonators.The objective of the thesis is the reliable and accurate approximation of scattering resonances using high order finite element methods. The discussion focuses on the electromagnetic scattering resonances in metal-dielectric nano-structures using a Drude-Lorentz model for the description of the material properties. A scattering resonance pair satisfies a reduced wave equationand an outgoing wave condition. In this thesis, the outgoing wave condition is replaced by a Dirichlet-to-Neumann map, or a Perfectly Matched Layer. For electromagnetic waves and for acoustic waves, the reduced wave equation is discretized with finite elements. As a result, the scattering resonance problem is transformed into a nonlinear eigenvalue problem.In addition to the correct approximation of the true resonances, a large number of numerical solutions that are unrelated to the physical problem are also computed in the solution process. A new method based on a volume integral equation is developed to remove these false solutions.The main results of the thesis are a novel method for removing false solutions of the physical problem, efficient solutions of non-linear eigenvalue problems, and a new a-priori based refinement strategy for high order finite element methods. The overall material in the thesis translates into a reliable and accurate method to compute scattering resonances in physics and engineering.
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- Berg, André, 1990-
(författare)
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Numerical analysis and simulation of stochastic partial differential equations with white noise dispersion
- 2023
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This doctoral thesis provides a comprehensive numerical analysis and exploration of several stochastic partial differential equations (SPDEs). More specifically, this thesis investigates time integrators for SPDEs with white noise dispersion. The thesis begins by examining the stochastic nonlinear Schrödinger equation with white noise dispersion (SNLSE), see Paper 1. The investigation probes the performance of different numerical integrators for this equation, focusing on their convergences, L2-norm preservation, and computational efficiency. Further, this thesis thoroughly investigates a conjecture on the critical exponent of the SNLSE, related to a phenomenon known as blowup, through numerical means. The thesis then introduces and studies exponential integrators for the stochastic Manakov equation (SME) by presenting two new time integrators - the explicit and symmetric exponential integrators - and analyzing their convergence properties, see Paper 2. Notably, this study highlights the flexibility and efficiency of these integrators compared to traditional schemes. The narrative then turns to the Lie-Trotter splitting integrator for the SME, see Paper 3, comparing its performance to existing time integrators. Theoretical proofs for convergence in various senses, alongside extensive numerical experiments, shed light on the efficacy of the proposed numerical scheme. The thesis also deep dives into the critical exponents of the SME, proposing a conjecture regarding blowup conditions for this SPDE.Lastly, the focus shifts to the stochastic generalized Benjamin-Bona-Mahony equation, see Paper 4. The study introduces and numerically assesses four novel exponential integrators for this equation. A primary finding here is the superior performance of the symmetric exponential integrator. This thesis also offers a succinct and novel method to depict the order of convergence in probability.
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- Bodin, Mats, et al.
(författare)
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Harmonic functions representation of Besov-Lipschitz functions on nested fractals
- 2010
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- R. S. Strichartz proposes a discrete definition of Besov spaces on self-similar fractals having a regular harmonic structure. In this paper, we characterize some of these Hölder-Zygmund and Besov-Lipschitz functions on nested fractals by means of the magnitude of the coefficients of the expansion of a function in a continuous piecewise harmonic base.
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