139 research outputs found
Adaptive Fast Interface Tracking Methods : Part II: Spatial Adaptivity
No full textIn this paper, we present a fast space-time adaptive numerical method for interface propagation in a time varying velocity field based on a multiresolution description of the interface. The interface is represented by wavelet vectors that correspond to the details of the interface on different scale levels.The method is an extension of the method proposed in "J. Popovic and O. Runborg, Adaptive fast interface tracking methods: Part I, preprint (2012)", which is only time adaptive and it is thus not suitable for problems with expanding interfaces. The method that we propose in this paper, remedies that disadvantage of the time adaptive method in an efficient way. </p
Existence, uniqueness and a constructive solution algorithm for a class of finite Markov moment problems
No full textWe 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.Inverse problems, finite Markov moment problem, exponential transform.
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
Wavelets andWavelet Based Numerical Homogenization
No full textWavelets is a tool for describing functions on different scales or level of detail. In mathematical terms, wavelets are functions that form a basis for with special properties; the basis functions are spatially localized and correspond to different scale levels. Finding the representation of a function in this basis amounts to making a multiresolution decomposition of the function. Such a wavelet representation lends itself naturally to analyzing the fine and coarse scales as well as the localization properties of a function.Wavelets have been used in many applications, from image and signal analysis to numerical methods for partial differential equations (PDEs). In this tutorial we first go through the basic wavelet theory and then show a more specific application where wavelets are used for numerical homogenization.We will mostly give references to the original sources of ideas presented. There are also a large number of books and review articles that cover the topic of wavelets, where the interested reader can find further information, e.g. [25, 51, 48, 7, 39, 26, 23], just to mention a few.</p
Finite element heterogeneous multiscale methods for the wave equation
No full textWave 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
No full textIn 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
A Multiblock Mesh Manager
No full textthis report we will describe an object-oriented technique for multiblock grid management. It is a development of earlier work at KTH described in [Oppelstrup93] and [Runborg92]. We start by showing how the grid topology can be described with a directed acyclic graph (DAG). Next, we choose what classes to use, which comes naturally given the DAG description. We proceed by giving some details on the contents of the classes and how new classes can be added. Also, we say a few words about how compound objects can solve the problem with blocks of different sizes meeting. After a short introduction to the actual implementation, which is built on the TCL/TK package, we show results exemplifying how the data structures can be utilized. We sum up our experiences in a concluding section
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
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