Convergence Analysis of a Fully Discrete Family of Iterated Deconvolution Methods for Turbulence Modeling with Time Relaxation

We present a general theory for regularization models of the Navier-Stokes equations based on the Leray deconvolution model with a general deconvolution operator designed to fit a few important key properties. We provide examples of this type of operator, such as the (modified) Tikhonov-Lavrentiev and (modified) Iterated Tikhonov-Lavrentiev operators, and study their mathematical properties. An existence theory is derived for the family of models and a rigorous convergence theory is derived for the resulting algorithms. Our theoretical results are supported by numerical testing with the Taylor-Green vortex problem, presented for the special operator cases mentioned above

Radially Symmetric Solutions of

We investigate solutions of and focus on the regime and . Our advance is to develop a technique to efficiently classify the behavior of solutions on , their maximal positive interval of existence. Our approach is to transform the nonautonomous equation into an autonomous ODE. This reduces the problem to analyzing the phase plane of the autonomous equation. We prove the existence of new families of solutions of the equation and describe their asymptotic behavior. In the subcritical case there is a well-known closed-form singular solution, , such that as and as . Our advance is to prove the existence of a family of solutions of the subcritical case which satisfies for infinitely many values . At the critical value there is a continuum of positive singular solutions, and a continuum of sign changing singular solutions. In the supercritical regime we prove the existence of a family of “super singular” sign changing singular solutions

Symptom Status Predicts Patient Outcomes in Persons with HIV and Comorbid Liver Disease

Persons living with human immunodeficiency virus (HIV) are living longer; therefore, they are more likely to suffer significant morbidity due to potentially treatable liver diseases. Clinical evidence suggests that the growing number of individuals living with HIV and liver disease may have a poorer health-related quality of life (HRQOL) than persons living with HIV who do not have comorbid liver disease. Thus, this study examined the multiple components of HRQOL by testing Wilson and Cleary’s model in a sample of 532 individuals (305 persons with HIV and 227 persons living with HIV and liver disease) using structural equation modeling. The model components include biological/physiological factors (HIV viral load, CD4 counts), symptom status (Beck Depression Inventory II and the Medical Outcomes Study HIV Health Survey (MOS-HIV) mental function), functional status (missed appointments and MOS-HIV physical function), general health perceptions (perceived burden visual analogue scale and MOS-HIV health transition), and overall quality of life (QOL) (Satisfaction with Life Scale and MOS-HIV overall QOL). The Wilson and Cleary model was found to be useful in linking clinical indicators to patient-related outcomes. The findings provide the foundation for development and future testing of targeted biobehavioral nursing interventions to improve HRQOL in persons living with HIV and liver disease

Iterated regularization methods for solving inverse problems

Typical inverse problems are ill-posed which frequently leads to difficulties in calculatingnumerical solutions. A common approximation method to solve ill-posed inverse problemsis iterated Tikhonov-Lavrentiev regularization.We examine iterated Tikhonov-Lavrentiev regularization and show that, in the casethat regularity properties are not globally satisfied, certain projections of the error converge faster than the theoretical predictions of the global error. We also explore the sensitivity of iterated Tikhonov regularization to the choice of the regularization parameter. We show that by calculating higher order sensitivities we improve the accuracy. We present a simple to implement algorithm that calculates the iterated Tikhonov updates and the sensitivities to the regularization parameter. The cost of this new algorithm is one vector addition and one scalar multiplication per step more than the standard iterated Tikhonov calculation.In considering the inverse problem of inverting the Helmholz-differential filter (with filterradius δ), we propose iterating a modification to Tikhonov-Lavrentiev regularization (withregularization parameter α and J iteration steps). We show that this modification to themethod decreases the theoretical error bounds from O(α(δ^2 +1)) for Tikhonov regularizationto O((αδ^2)^(J+1) ). We apply this modified iterated Tikhonov regularization method to theLeray deconvolution model of fluid flow. We discretize the problem with finite elements inspace and Crank-Nicolson in time and show existence, uniqueness and convergence of thissolution.We examine the combination of iterated Tikhonov regularization, the L-curve method,a new stopping criterion, and a bootstrapping algorithm as a general solution method inbrain mapping. This method is a robust method for handling the difficulties associated withbrain mapping: uncertainty quantification, co-linearity of the data, and data noise. Weuse this method to estimate correlation coefficients between brain regions and a quantified performance as well as identify regions of interest for future analysis

Time integration for complex fluid dynamics

2021 Fall.Includes bibliographical references.Efficient and accurate simulation of turbulent combusting flows in complex geometry remains a challenging and computationally expensive proposition. A significant source of computational expense is in the integration of the temporal domain, where small time steps are required for the accurate resolution of chemical reactions and long solution times are needed for many practical applications. To address the small step sizes, a fourth-order implicit-explicit additive Runge-Kutta (ARK4) method is developed to integrate the stiff chemical reactions implicitly while advancing the convective and diffusive physics explicitly in time. Applications involving complex geometry, stiff reaction mechanisms, and high-order spatial discretizations are challenged by stability issues in the numerical solution of the nonlinear problem that arises from the implicit treatment of the stiff term. Techniques for maintaining a physical thermodynamic state during the numerical solution of the nonlinear problem, such as placing constraints on the nonlinear solver and the use of a nonlinear optimizer to find valid thermodynamic states, are proposed and tested. Verification and validation are performed for the new adaptive ARK4 method using lean premixed flames burning hydrogen, showing preservation of 4th-order error convergence and recovery of literature results. ARK4 is then applied to solve lean, premixed C3H8-air combustion in a bluff-body combustor geometry. In the two-dimensional case, ARK4 provides a 70× speedup over the standard explicit four-stage Runge-Kutta method and, for the three-dimensional case, three-orders-of-magnitude-larger time step sizes are achieved. To further increase the computational scaling of the algorithms, parallel-in-time (PinT) techniques are explored. PinT has the dual benefit of providing parallelization to long temporal domains as well as taking advantage of hardware trends towards more concurrency in modern high-performance computing platforms. Specifically, the multigrid reduction-in-time (MGRIT) method is adapted and enhanced by adding adaptive mesh refinement (AMR) in time. This creates a space-time algorithm with efficient solution-adaptive grids. The new MGRIT+AMR algorithm is first verified and validated using problems dominated by diffusion or characterized by time periodicity, such as Couette flow and Stokes second problem. The adaptive space-time parallel algorithm demonstrates up to a 13.7× speedup over a time-sequential algorithm for the same solution accuracy. However, MGRIT has difficulties when applied to solve practical fluid flows, such as turbulence, governed by strong hyperbolic partial differential equations. To overcome this challenge, the multigrid operations are modified and applied in a novel way by exploiting the space-time localization of fine turbulence scales. With these new operators, the coarse-scale errors are advected out of the temporal domain while the fine-scale dynamics iterate to equilibrium. This leads to rapid convergence of the bulk flow, which is important for computing macroscopic properties useful for engineering purposes. The novel multigrid operations are applied to the compressible inviscid Taylor-Green vortex flow and the convergence of the low-frequency modes is achieved within a few iterations. Future work will be focused on a performance study for practical highly turbulent flows

HIGH ACCURACY METHODS AND REGULARIZATION TECHNIQUES FOR FLUID FLOWS AND FLUID-FLUID INTERACTION

This dissertation contains several approaches to resolve irregularity issues of CFD problems, including a decoupling of non-linearly coupled fluid-fluid interaction, due to high Reynolds number. New models present not only regularize the linear systems but also produce high accurate solutions both in space and time. To achieve this goal, methods solve a computationally attractive artificial viscosity approximation of the target problem, and then utilize a correction approach to make it high order accurate. This way, they all allow the usage of legacy code | a frequent requirement in the simulation of fluid flows in complex geometries. In addition, they all pave the way for parallelization of the correction step, which roughly halves the computational time for each method, i.e. solves at about the same time that is required for DNS with artificial viscosity. Also, methods present do not requires all over function evaluations as one can store them, and reuse for the correction steps. All of the chapters in this dissertation are self-contained, and introduce model first, and then present both theoretical and computational findings of the corresponding method

Reduced-Order Modeling of Turbulent Reacting Flows Using Inertial Manifold Theory

Turbulent flows found in aerodynamics, propulsion, and other energy conversion sys- tems pose an inherent computational challenge for extensive predictive simulations. Over the last few decades, a statistical approach for reduced-order modeling of tur- bulence has become the dominant framework for prediction. However, there exists a range of problems that the statistical approaches are ill-suited for – problems driven not only by the chaoticity in the flow, but also by uncertainty in operating, boundary, or initial conditions. Since tails of the initial flow field distribution may drive transi- tion events, there is a need to develop techniques that do not explicitly rely on the statistical representation of unresolved quantities. The uniqueness of this work lies in the development of reduced-order models that can track distinct trajectories of the dynamical behavior of reacting turbulent flows without invoking ad-hoc assumptions about underlying small-scale turbulent motions or flame structure. Treatment of turbulent flows as finite-dimensional dynamical systems opens new paths for the development of a reduced-order description of such systems. For certain types of dynamical systems, a property known as the inertial manifold (IM) is known to exist, which allows for the dynamics to be represented in a sub-space smaller than the entire state-space. The primary concept in approximate IM (AIM) is that slow dominant dynamical behavior of the system is confined to a low-dimension manifold, and fast dynamics respond to the dynamics on the IM instantaneously. Decomposition of slow/fast dynamics and formulation of an AIM is accomplished by only exploiting the governing equations. Based on this concept, a computational analysis of the use of IMs for modeling reacting turbulent flows is conducted. First, the proposed modeling ansatz has been investigated for canonical turbulent flows. An AIM is constructed for the one-dimensional Kuramoto-Sivashinsky equation and the three-dimensional Navier-Stokes equations to assess different aspects of AIM formulation. An a priori study is conducted to examine the validity of AIM assump- tions and to obtain an estimation of the inertial manifold or attractor dimension. Then a reduced-order model is developed and tested over a range of parameters. Second, the theory of IM is extended to the development of reduced-order models of turbulent combustion. Unlike pre-generated manifold-based combustion models, here the combustion trajectory is tracked in a low-dimensional manifold determined in-situ without invoking laminar flame structures or statistical assumptions about the underlying turbulent flow. The AIM performance is assessed in capturing flame behaviors with varying levels of extinction and reignition.PHDMechanical EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/169879/1/akramrym_1.pd

Image Restoration

This book represents a sample of recent contributions of researchers all around the world in the field of image restoration. The book consists of 15 chapters organized in three main sections (Theory, Applications, Interdisciplinarity). Topics cover some different aspects of the theory of image restoration, but this book is also an occasion to highlight some new topics of research related to the emergence of some original imaging devices. From this arise some real challenging problems related to image reconstruction/restoration that open the way to some new fundamental scientific questions closely related with the world we interact with