1,721,069 research outputs found
Finite element heterogeneous multiscale methods for the wave equation
Get PDFWave phenomena appear in a wide range of applications such as full-waveform seismic inversion, medical imaging, or composite materials. Often, they are modeled by the acoustic wave equation.
It can be solved by standard numerical methods such as, e.g., the finite element (FE) or the finite difference method. However, if the wave propagation speed varies on a microscopic length scale denoted by epsilon, the computational cost becomes infeasible, since the medium must be resolved down to its finest scale. In this thesis we propose multiscale numerical methods which approximate the overall macroscopic behavior of the wave propagation with a substantially lower computational effort. We follow the design principles of the heterogeneous multiscale method (HMM), introduced in 2003 by E and Engquist. This method relies on a coarse discretization of an a priori unknown effective equation. The missing data, usually the parameters of the effective equation, are estimated on demand by solving microscale problems on small sampling domains. Hence, no precomputation of these effective parameters is needed. We choose FE methods to solve both the macroscopic and the microscopic problems.
For limited time the overall behavior of the wave is well described by the homogenized wave equation. We prove that the FE-HMM method converges to the solution of the homogenized wave equation. With increasing time, however, the true solution deviates from the classical homogenization limit, as a large secondary wave train develops. Neither the homogenized solution, nor the FE-HMM capture these dispersive effects. To capture them we need to modify the FE-HMM. Inspired by higher order homogenization techniques we additionally compute a correction term of order epsilon^2. Since its computation also relies on the solution of the same microscale problems as the original FE-HMM, the computational effort remains essentially unchanged. For this modified version we also prove convergence to the homogenized wave equation, but in contrast to the original FE-HMM the long-time dispersive behavior is recovered.
The convergence proofs for the FE-HMM follow from new Strang-type results for the wave equation. The results are general enough such that the FE-HMM with and without the long-time correction fits into the setting, even if numerical quadrature is used to evaluate the arising L^2 inner product.
In addition to these results we give alternative formulations of the FE-HMM, where the elliptic micro problems are replaced by hyperbolic ones. All the results are supported by numerical tests. The versatility of the method is demonstrated by various numerical examples
Some new results in multiphase geometrical optics
Get PDFIn order to accommodate solutions with multiple phases, corresponding to crossing rays, we formulate geometrical optics for the scalar wave equation as a kinetic transport equation set in phase space. If the maximum number of phases is finite and known a priori we can recover the exact multiphase solution from an associated system of moment equations, closed by an assumption on the form of the density function in the kinetic equation. We consider two different closure assumptions based on delta and Heaviside functions and analyze the resulting equations. They form systems of nonlinear conservation laws with source terms. In contrast to the classical eikonal equation, these equations will incorporate a finite superposition principle in the sense that while the maximum number of phases is not exceeded a sum of solutions is also a solution. We present numerical results for a variety of homogeneous and inhomogeneous problems.</p
Existence, uniqueness and a constructive solution algorithm for a class of finite Markov moment problems
No full textInternational audienceWe consider a class of finite Markov moment problems with arbitrary number of positive and negative branches. We show criteria for the existence and uniqueness of solutions, and we characterize in detail the non-unique solution families. Moreover, we present a constructive algorithm to solve the moment problems numerically and prove that the algorithm computes the right solution
Introduction to normal multiresolution approximation
No full textA multiresolution analysis of a curve is normal if each wavelet detail vector with respect to a certain subdivision scheme lies in the local normal direction. In this paper we give an introduction to the analysis of normal approximations in [3]. We define the normal approximation in its basic form and show simplified proofs of the method's convergence, approximation quality and stability. We also explain how higher order approximations can be constructed using subdivision operators and give a brief summary of the corresponding results for these more general schemes.</p
Numerical Methods for Wave Phenomena
Get PDFThis dissertation describes numerical methods for wave phenomena and is divided into two main sections. The first concerns a new time-domain approach to solving the Helmholtz equation. The second concerns numerical methods for the optimal control of closed quantum systems.
The efficient solution of the Helmholtz equation is an active area of research. Traditionally, many methods in the literature take the approach of solving the Helmholtz equation "directly''. By directly we mean solving the Helmholtz equation in the frequency domain, whether by a direct discretization of the PDE via finite differences/elements or by integral equation methods. An alternative approach is to instead solve the Helmholtz equation by seeking time-harmonic solutions in the time-domain. In this thesis we present the WaveHoltz iteration, which is a fixed-point iteration for solving the Helmholtz equation by instead solving a sequence of wave equations. We demonstrate that WaveHoltz is amenable to acceleration via Krylov subspace methods. Moreover we show that WaveHoltz is simple to implement, inherits the memory-leanness and scalability of the underlying wave equation discretization, and that it is possible to remove time-discretization errors from the WaveHoltz solution.
The second part of the thesis introduces tools for devising optimal controls to realize logic gates in closed quantum systems. We motivate a novel approximation of control functions via B-spline wavelets with carrier waves that are specifically constructed to trigger the transition frequencies of a quantum system. Using the symplectic and time reversible Störmer-Verlet scheme, we take a ``discretize-then-optimize'' approach to determine a corresponding adjoint partitioned Runge Kutta scheme. This allows the computation of exact discrete gradients for the quantum optimal control problem. Finally, we outline a submitted solution to the IBM SWAP Gate Challenge using these methods.</p
Mathematical models and numerical methods for high frequency waves
No full textThe numerical approximation of high frequency wave propagation is important in many applications. Examples include the simulation of seismic, acoustic, optical waves and microwaves. When the frequency of the waves is high, this is a difficult multiscale problem. The wavelength is short compared to the overall size of the computational domain and direct simulation using the standard wave equations is very expensive. Fortunately, there are computationally much less costly models, that are good approximations of many wave equations precisely for very high frequencies. Even for linear wave equations these models are often nonlinear. The goal of this paper is to review such mathematical models for high frequency waves, and to survey numerical methods used in simulations. We focus on the geometrical optics approximation which describes the infinite frequency limit of wave equations. We will also discuss finite frequency corrections and some other models.</p
Analysis of high order fast interface tracking methods
No full textFast high order methods for the propagation of an interface in a velocity field are constructed and analyzed. The methods are generalizations of the fast interface tracking method proposed in Runborg (Commun Math Sci 7:365-398, 2009). They are based on high order subdivision to make a multiresolution decomposition of the interface. Instead of tracking marker points on the interface the related wavelet vectors are tracked. Like the markers they satisfy ordinary differential equations (ODEs), but fine scale wavelets can be tracked with longer timesteps than coarse scale wavelets. This leads to methods with a computational cost of rather than for markers and reference timestep . These methods are proved to still have the same order of accuracy as the underlying direct ODE solver under a stability condition in terms of the order of the subdivision, the order of the ODE solver and the time step ratio between wavelet levels. In particular it is shown that with a suitable high order subdivision scheme any explicit Runge-Kutta method can be used. Numerical examples supporting the theory are also presented.</p
Going Beyond Counting First Authors in Author Co-citation Analysis
Get PDFThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
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