CAAM Technical Reports
http://hdl.handle.net/1911/101542
2019-10-23T09:57:09ZDiscretization of Multipole Sources in a Finite Difference Setting for Wave Propagation Problems
http://hdl.handle.net/1911/102452
Discretization of Multipole Sources in a Finite Difference Setting for Wave Propagation Problems
Bencomo, Mario J.; Symes, William
Seismic sources are commonly idealized as point-sources due to their small spatial extent relative to seismic wavelengths. The acoustic isotropic point-radiator is inadequate as a model of seismic wave generation for seismic sources that are known to exhibit directivity. Therefore, accurate modeling of seismic wavefields must include source representations generating anisotropic radiation patterns. Such seismic sources can be modeled as linear combinations of multipole point-sources. In this paper we present a method for discretizing multipole sources in a finite difference setting, an extension of the moment matching conditions developed for the Dirac delta function in other applications. We also provide the necessary analysis and numerical evidence to demonstrate the accuracy of our singular source approximations. In particular, we develop a weak convergence theory for the discretization of a family of symmetric hyperbolic systems of first-order partial differential equations, with singular source terms, solved via staggered-grid finite difference methods. Numerical experiments demonstrate a stronger result than what is presented in our convergence theory, namely, optimal convergence rates of numerical solutions are achieved point-wise in space away from the source if an appropriate source discretization is used.
2018-06-20T00:00:00ZNonlinear Waveform Inversion with Surface-Oriented Extended Modeling
http://hdl.handle.net/1911/102274
Nonlinear Waveform Inversion with Surface-Oriented Extended Modeling
Terentyev, Igor
This thesis investigates surface-oriented model extension approach to nonlinear full waveform inversion (FWI). Conventional least-squares (LS) approach is capable of reconstructing highly detailed models of subsurface. Resolution requirements of the realistic problems dictate the use of local descent methods to solve the LS optimization problem. However, in the setting of any characteristic seismic problem, LS objective functional has numerous local extrema, rendering descent methods unsuitable when initial estimate is not kinematically accurate. The aim of my work is to improve convexity properties of the objective functional. I use the extended modeling approach, and construct an extended optimization functional incorporating differential semblance condition. An important advantage of surface-oriented extensions is that they do not increase the computational complexity of the forward modeling. This approach blends FWI technique with migration velocity analysis (MVA) capability to recover long scale velocity model, producing optimization problems that combine global convergence properties of the MVA with data fitting approach of FWI. In particular, it takes into account nonlinear physical effects, such as multiple reflections. I employ variable projection approach to solve the extended optimization problem. I validate the method on synthetic models for the constant density acoustics problem.
This work was also published as a Rice University thesis/dissertation: http://hdl.handle.net/1911/102263
2017-03-01T00:00:00ZSubset Selection and Feature Identification in the Electrocardiogram
http://hdl.handle.net/1911/102271
Subset Selection and Feature Identification in the Electrocardiogram
Hendryx, Emily
Each feature in the electrocardiogram (ECG) corresponds to a different part of the cardiac cycle. Tracking changes in these features over long periods of time can offer insight regarding changes in a patientâ€™s clinical status. However, the automated identification of features in some patient populations, such as the pediatric congenital heart disease population, remains a nontrivial task that has yet to be mastered. Working toward a solution to this problem, this thesis outlines an overall framework for the identification of individual features in the ECGs of different populations. With a goal of applying part of this framework retrospectively to large sets of patient data, we focus primarily on the selection of relevant subsets of ECG beats for subsequent interpretation by clinical experts. We demonstrate the viability of the discrete empirical interpolation method (DEIM) in identifying representative subsets of beat morphologies relevant for future classification models. The success of DEIM applied to data sets from a variety of contexts is compared to results from related approaches in numerical linear algebra, as well as some more common clustering algorithms. We also present a novel extension of DEIM, called E-DEIM, in which additional representative data points can be identified as important without being limited by the rank of the corresponding data matrix. This new algorithm is evaluated on two different data sets to demonstrate its use in multiple settings, even beyond medicine. With DEIM and its related methods identifying beat-class representatives, we then propose an approach to automatically extend physician expertise on the selected beat morphologies to new and unlabeled beats. Using a fuzzy classification scheme with dynamic time warping, we are able to provide preliminary results suggesting further pursuit of this framework in application to patient data.
This work was also published as a Rice University thesis/dissertation.
2018-04-01T00:00:00ZEfficient estimation of coherent risk measures for risk-averse optimization problems governed by partial differential equations with random inputs
http://hdl.handle.net/1911/102272
Efficient estimation of coherent risk measures for risk-averse optimization problems governed by partial differential equations with random inputs
Takhtaganov, Timur
This thesis assesses and designs structure-exploiting methods for the efficient estimation of risk measures of quantities of interest in the context of optimization of partial differential equations (PDEs) with random inputs. Risk measures of the quantities of interest arise as objective functions or as constraints in the PDE-constrained optimization problems under uncertainty. A single evaluation of a risk measure requires numerical integration in a high-dimensional parameter space, which requires the solution of the PDE at many parameter samples. When the integrand is smooth in the random parameters, efficient methods, such as sparse grids, exist that substantially reduce the sample size. Unfortunately, many risk-averse formulations, such as semideviation and Conditional Value-at-Risk, introduce a non-smoothness in integrand. This work demonstrates that naive application of sparse grids and other smoothness-exploiting approaches is not beneficial in the risk-averse case. For the widely used class of coherent risk measures, this thesis proposes a new method for evaluating risk-averse objectives based on the biconjugate representation of coherent risk functions and importance sampling. The method is further enhanced by utilizing reduced order models of the PDEs under consideration. The proposed method leads to substantial reduction in the number of PDE solutions required to accurately estimate coherent risk measures. The performance of existing and of the new methods for the estimation of risk measures is demonstrated on examples of risk-averse PDE-constrained optimization problems. The resulting method can substantially reduce the number of PDE solutions required to solve optimization problems, and, therefore, enlarge the applicability of important risk measures for PDE-constrained optimization problems under uncertainty.
This work was also published as a Rice University thesis/dissertation.
2017-05-01T00:00:00Z