5,944 research outputs found
Multigrid Waveform Relaxation on Spatial Finite Element Meshes: The Discrete-Time Case
The efficiency of numerically solving time-dependent partial differential equations on parallel computers can be greatly improved by computing the solution on many time levels simultaneously. The theoretical properties of one such method, namely the discrete-time multigrid waveform relaxation method, are investigated for systems of ordinary differential equations obtained by spatial finite-element discretisation of linear parabolic initial-boundary value problems. The results are compared to the corresponding continuous-time results. The theory is illustrated for a one-dimensional and a two-dimensional model problem and checked against results obtained by numerical experiments
Compressed Passive Macromodeling
This paper presents an approach for the extraction of passive macromodels of large-scale interconnects from their frequency-domain scattering responses. Here, large scale is intended both in terms of number of electrical ports and required dynamic model order. For such structures, standard approaches based on rational approximation via vector fitting and passivity enforcement via model perturbation may fail because of excessive computational requirements, both in terms of memory size and runtime. Our approach addresses this complexity by first reducing the redundancy in the raw scattering responses through a projection and approximation process based on a truncated singular value decomposition. Then we formulate a compressed rational fitting and passivity enforcement framework which is able to obtain speedup factors up to 2 and 3 orders of magnitude with respect to standard approaches, with full control over the approximation errors. Numerical results on a large set of benchmark cases demonstrate the effectiveness of the proposed techniqu
Subtraction-noise projection in gravitational-wave detector networks
In this paper, we present a successful implementation of a subtraction-noise
projection method into a simple, simulated data analysis pipeline of a
gravitational-wave search. We investigate the problem to reveal a weak
stochastic background signal which is covered by a strong foreground of
compact-binary coalescences. The foreground which is estimated by matched
filters, has to be subtracted from the data. Even an optimal analysis of
foreground signals will leave subtraction noise due to estimation errors of
template parameters which may corrupt the measurement of the background signal.
The subtraction noise can be removed by a noise projection. We apply our
analysis pipeline to the proposed future-generation space-borne Big Bang
Observer (BBO) mission which seeks for a stochastic background of primordial
GWs in the frequency range Hz covered by a foreground of
black-hole and neutron-star binaries. Our analysis is based on a simulation
code which provides a dynamical model of a time-delay interferometer (TDI)
network. It generates the data as time series and incorporates the analysis
pipeline together with the noise projection. Our results confirm previous ad
hoc predictions which say that BBO will be sensitive to backgrounds with
fractional energy densities below Comment: 54 pages, 15 figure
Modeling and inversion of seismic data using multiple scattering, renormalization and homotopy methods
Seismic scattering theory plays an important role in seismic forward modeling and is the theoretical foundation for various seismic imaging methods. Full waveform inversion is a powerful technique for obtaining a high-resolution model of the subsurface. One objective of this thesis is to develop convergent scattering series solutions of the Lippmann-Schwinger equation in strongly scattering media using renormalization and homotopy methods. Other objectives of this thesis are to develop efficient full waveform inversion methods of time-lapse seismic data and, to investigate uncertainty quantification in full waveform inversion for anisotropic elastic media based on integral equation approaches and the iterated extended Kalman filter. The conventional Born scattering series is obtained by expanding the Lippmann-Schwinger equation in terms of an iterative solution based on perturbation theory. Such an expansion assumes weak scattering and may have the problems of convergence in strongly scattering media. This thesis presents two scattering series, referred to as convergent Born series (CBS) and homotopy analysis method (HAM) scattering series for frequency-domain seismic wave modeling. For the convergent Born series, a physical interpretation from the renormalization prospective is given. The homotopy scattering series is derived by using homotopy analysis method, which is based on a convergence control parameter and a convergence control operator that one can use to ensure convergence for strongly scattering media. The homotopy scattering scattering series solutions of the Lippmann-Schwinger equation, which is convergent in strongly scattering media. The homotopy scattering series is a kind of unified scattering series theory that includes the conventional and convergent Born series as special cases. The Fast Fourier Transform (FFT) is employed for efficient implementation of matrix-vector multiplication for the convergent Born series and the homotopy scattering series. This thesis presents homotopy methods for ray based seismic modeling in strongly anisotropic media. To overcome several limitations of small perturbations and weak anisotropy in obtaining the traveltime approximations in anisotropic media by expanding the anisotropic eikonal equation in terms of the anisotropic parameters and the elliptically anisotropic eikonal equation based on perturbation theory, this study applies the homotopy analysis method to the eikonal equation. Then this thesis presents a retrieved zero-order deformation equation that creates a map from the anisotropic eikonal equation to a linearized partial differential equation system. The new traveltime approximations are derived by using the linear and nonlinear operators in the retrieved zero-order deformation equation. Flexibility on variable anisotropy parameters is naturally incorporated into the linear differential equations, allowing a medium of arbitrarily anisotropy. This thesis investigates efficient target-oriented inversion strategies for improving full waveform inversion of time-lapse seismic data based on extending the distorted Born iterative T-matrix inverse scattering to a local inversion of a small region of interest (e. g. reservoir under production). The target-oriented approach is more efficient for inverting the monitor data. The target-oriented inversion strategy requires properly specifying the wavefield extrapolation operators in the integral equation formulation. By employing the T-matrix and the Gaussian beam based Greenās function, the wavefield extrapolation for the time-lapse inversion is performed in the baseline model from the survey surface to the target region. I demonstrate the method by presenting numerical examples illustrating the sequential and double difference strategies. To quantify the uncertainty and multiparameter trade-off in the full waveform inversion for anisotropic elastic media, this study applies the iterated extended Kalman filter to anisotropic elastic full waveform inversion based on the integral equation method. The sensitivity matrix is an explicit representation with Greenās functions based on the nonlinear inverse scattering theory. Taking the similarity of sequential strategy between the multi-scale frequency domain full waveform inversion and data assimilation with an iterated extended Kalman filter, this study applies the explicit representation of sensitivity matrix to the the framework of Bayesian inference and then estimate the uncertainties in the full waveform inversion. This thesis gives results of numerical tests with examples for anisotropic elastic media. They show that the proposed Bayesian inversion method can provide reasonable reconstruction results for the elastic coefficients of the stiffness tensor and the framework is suitable for accessing the uncertainties and analysis of parameter trade-offs
Algorithmic analysis torwards time-domain extended source waveform inversion
Full waveform inversion (FWI) updates the subsurface model from an initial
model by comparing observed and synthetic seismograms. Due to high
nonlinearity, FWI is easy to be trapped into local minima. Extended domain FWI,
including wavefield reconstruction inversion (WRI) and extended source waveform
inversion (ESI) are attractive options to mitigate this issue. This paper makes
an in-depth analysis for FWI in the extended domain, identifying key challenges
and searching for potential remedies torwards practical applications. WRI and
ESI are formulated within the same mathematical framework using
Lagrangian-based adjoint-state method with a special focus on time-domain
formulation using extended sources, while putting connections between classical
FWI, WRI and ESI: both WRI and ESI can be viewed as weighted versions of
classic FWI. Due to symmetric positive definite Hessian, the conjugate gradient
is explored to efficiently solve the normal equation in a matrix free manner,
while both time and frequency domain wave equation solvers are feasible. This
study finds that the most significant challenge comes from the huge storage
demand to store time-domain wavefields through iterations. To resolve this
challenge, two possible workaround strategies can be considered, i.e., by
extracting sparse frequencial wavefields or by considering time-domain data
instead of wavefields for reducing such challenge. We suggest that these
options should be explored more intensively for tractable workflows
Efficient Bayesian travel-time tomography with geologically-complex priors using sensitivity-informed polynomial chaos expansion and deep generative networks
Monte Carlo Markov Chain (MCMC) methods commonly confront two fundamental
challenges: the accurate characterization of the prior distribution and the
efficient evaluation of the likelihood. In the context of Bayesian studies on
tomography, principal component analysis (PCA) can in some cases facilitate the
straightforward definition of the prior distribution, while simultaneously
enabling the implementation of accurate surrogate models based on polynomial
chaos expansion (PCE) to replace computationally intensive full-physics forward
solvers. When faced with scenarios where PCA does not offer a direct means of
easily defining the prior distribution alternative methods like deep generative
models (e.g., variational autoencoders (VAEs)), can be employed as viable
options. However, accurately producing a surrogate capable of capturing the
intricate non-linear relationship between the latent parameters of a VAE and
the outputs of forward modeling presents a notable challenge. Indeed, while PCE
models provide high accuracy when the input-output relationship can be
effectively approximated by relatively low-degree multivariate polynomials,
this condition is typically unmet when utilizing latent variables derived from
deep generative models. In this contribution, we present a strategy that
combines the excellent reconstruction performances of VAE in terms of prio
representation with the accuracy of PCA-PCE surrogate modeling in the context
of Bayesian ground penetrating radar (GPR) travel-time tomography. Within the
MCMC process, the parametrization of the VAE is leveraged for prior exploration
and sample proposal. Concurrently, modeling is conducted using PCE, which
operates on either globally or locally defined principal components of the VAE
samples under examination.Comment: 25 pages, 15 figure
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