Lecture 13: A Low-Rank Factorization Framework For Building Scalable Algebraic Solvers And Preconditioners, 2021 Lawrence Berkeley National Laboratory
Lecture 13: A Low-Rank Factorization Framework For Building Scalable Algebraic Solvers And Preconditioners, X. Sherry Li
Mathematical Sciences Spring Lecture Series
Factorization based preconditioning algorithms, most notably incomplete LU (ILU) factorization, have been shown to be robust and applicable to wide ranges of problems. However, traditional ILU algorithms are not amenable to scalable implementation. In recent years, we have seen a lot of investigations using low-rank compression techniques to build approximate factorizations.
A key to achieving lower complexity is the use of hierarchical matrix algebra, stemming from the H-matrix research. In addition, the multilevel algorithm paradigm provides a good vehicle for a scalable implementation. The goal of this lecture is to give an overview of the various hierarchical matrix formats, such …
Lecture 07: Nonlinear Preconditioning Methods And Applications, 2021 University of Colorado, Boulder
Lecture 07: Nonlinear Preconditioning Methods And Applications, Xiao-Chuan Cai
Mathematical Sciences Spring Lecture Series
We consider solving system of nonlinear algebraic equations arising from the discretization of partial differential equations. Inexact Newton is a popular technique for such problems. When the nonlinearities in the system are well-balanced, Newton's method works well, but when a small number of nonlinear functions in the system are much more nonlinear than the others, Newton may converge slowly or even stagnate. In such a situation, we introduce some nonlinear preconditioners to balance the nonlinearities in the system. The preconditioners are often constructed using a combination of some domain decomposition methods and nonlinear elimination methods. For the nonlinearly preconditioned problem, …
Lecture 10: Preconditioned Iterative Methods For Linear Systems, 2021 Georgia Institute of Technology
Lecture 10: Preconditioned Iterative Methods For Linear Systems, Edmond Chow
Mathematical Sciences Spring Lecture Series
Iterative methods for the solution of linear systems of equations – such as stationary, semi-iterative, and Krylov subspace methods – are classical methods taught in numerical analysis courses, but adapting these methods to run efficiently at large-scale on high-performance computers is challenging and a constantly evolving topic. Preconditioners – necessary to aid the convergence of iterative methods – come in many forms, from algebraic to physics-based, are regularly being developed for linear systems from different classes of problems, and similarly are evolving with high-performance computers. This lecture will cover the background and some recent developments on iterative methods and preconditioning …
Analysis Of Boundary Observability Of Strongly Coupled One-Dimensional Wave Equations With Mixed Boundary Conditions, 2021 Western Kentucky University
Analysis Of Boundary Observability Of Strongly Coupled One-Dimensional Wave Equations With Mixed Boundary Conditions, Wilson Dennis Horner
Masters Theses & Specialist Projects
*see note below
In control theory, the time it takes to receive a signal after it is sent is referred to as the observation time. For certain types of materials, the observation time to receive a wave signal differs depending on a variety of factors, such as material density, flexibility, speed of the wave propagation, etc. Suppose we have a strongly coupled system of two wave equations describing the longitudinal vibrations on a piezoelectric beam of length L. These two wave equations have non-identical wave propagation speeds c1 and c2. First, we prove the exact observability inequality with the optimal …
A Direct Method For Modeling And Simulations Of Elliptic And Parabolic Interface Problems, 2021 Old Dominion University
A Direct Method For Modeling And Simulations Of Elliptic And Parabolic Interface Problems, Kumudu Gamage, Yan Peng
College of Sciences Posters
Interface problems have many applications in fluid dynamics, molecular biology, electromagnetism, material science, heat distribution in engines, and hyperthermia treatment of cancer. Mathematically, interface problems commonly lead to partial differential equations (PDE) whose in- put data are discontinuous or singular across the interfaces in the solution domain. Many standard numerical methods designed for smooth solutions poorly work for interface problems as solutions of the interface problems are mostly non-smoothness or discontinuous. Moving interface problems depends on the accuracy of the gradient of the solution at the interface. Therefore, it became essential to derive a method for interface problems that gives …
Discontinuous Galerkin Method Applied To Navier-Stokes Equations, 2021 University of Nebraska at Omaha
Discontinuous Galerkin Method Applied To Navier-Stokes Equations, Ian Deruiter, Mahboub Baccouch
UNO Student Research and Creative Activity Fair
Discontinuous Galerkin (DG) finite element methods are becoming important techniques for the computational solution of many real-world problems describe by differential equations. They combine many attractive features of the finite element and the finite volume methods. These methods have been successfully applied to many important PDEs arising from a wide range of applications. DG methods are highly accurate numerical methods and have considerable advantages over the classical numerical methods available in the literature. DG methods can easily handle meshes with hanging nodes, elements of various types and shapes, and local spaces of different orders. Furthermore, DG methods provide accurate and …
A Development Of A Polyhedron In The Galilean Space, 2021 Tashkent State Transport University
A Development Of A Polyhedron In The Galilean Space, Abdulaziz Artykbaev, Jasur Sobirov
Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences
In this paper, we study the development of a polyhedron in the Galilean space. A development of a polyhedron is an isometric mapping of a polyhedron to a plane, in which the gluing sides are indicated. Since the motion of the Galilean space differs significantly from the motion of the Euclidean space, the development of a polyhedron of the Galilean space will also differ from the development of a polyhedron of the Euclidean space. We prove that the total angle around the vertex of the polyhedral angle is preserved in the development. We also give illustrations of the developments for …
Nonlocal Problems For A Fractional Order Mixed Parabolic Equation, 2021 Fergana State University
Nonlocal Problems For A Fractional Order Mixed Parabolic Equation, Azizbek Mamanazarov
Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences
In the present work nonlocal problems with Bitsadze-Samarskii type conditions, with the first and the second kind integral conditions for mixed parabolic equation involving Riemann-Liouville fractional differential operator have been formulated and investigated. The uniqueness and the existence of the solution of the considered problems were proved. To do this, considered problems are equivalently reduced to the problems with nonlocal conditions with respect to the trace of the unknown function and its space-derivatives. Then using the representation of the solution of the second kind of Abel's integral equation, it was found integral representations of the solutions of these problems. Necessary …
Nonlocal Boundary Value Problem For A System Of Mixed Type Equations With A Line Of Degeneration, 2021 Turin Polytechnic University in Tashkent
Nonlocal Boundary Value Problem For A System Of Mixed Type Equations With A Line Of Degeneration, Kudratillo Fayazov, Ikrombek Khajiev
Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences
This work is devoted to the study of a nonlocal boundary value problem for a system of two-dimensional parabolic equations with changing direction of time. A priori estimate is obtained for the solution of the problem under consideration, and theorems on stability and conditional stability are proved depending on the parameters of the nonlocal condition. As a result, the uniqueness of the solution to the problem is presented.
Stochastic Navier-Stokes Equations With Markov Switching, 2021 Louisiana State University and Agricultural and Mechanical College
Stochastic Navier-Stokes Equations With Markov Switching, Po-Han Hsu
LSU Doctoral Dissertations
This dissertation is devoted to the study of three-dimensional (regularized) stochastic Navier-Stokes equations with Markov switching. A Markov chain is introduced into the noise term to capture the transitions from laminar to turbulent flow, and vice versa. The existence of the weak solution (in the sense of stochastic analysis) is shown by studying the martingale problem posed by it. This together with the pathwise uniqueness yields existence of the unique strong solution (in the sense of stochastic analysis). The existence and uniqueness of a stationary measure is established when the noise terms are additive and autonomous. Certain exit time estimates …
The Pencil Code, A Modular Mpi Code For Partial Differential Equations And Particles: Multipurpose And Multiuser-Maintained, 2021 University of Nevada, Las Vegas
The Pencil Code, A Modular Mpi Code For Partial Differential Equations And Particles: Multipurpose And Multiuser-Maintained, The Pencil Code Collaboration, Chao-Chin Yang
Physics & Astronomy Faculty Research
The Pencil Code is a highly modular physics-oriented simulation code that can be adapted to a wide range of applications. It is primarily designed to solve partial differential equations (PDEs) of compressible hydrodynamics and has lots of add-ons ranging from astrophysical magnetohydrodynamics (MHD) (A. Brandenburg & Dobler, 2010) to meteorological cloud microphysics (Li et al., 2017) and engineering applications in combustion (Babkovskaia et al., 2011). Nevertheless, the framework is general and can also be applied to situations not related to hydrodynamics or even PDEs, for example when just the message passing interface or input/output strategies of the code are to …
Principles For Determining The Motion Of Blood Through Arteries, 2021 University of Sao Paulo
Principles For Determining The Motion Of Blood Through Arteries, Sylvio R. Bistafa
Euleriana
Translation of Principia pro motu sanguinis per arterias determinando (E855). This work of 1775 by L. Euler is considered to be the first mathematical treatment of circulatory physiology and hemodynamics.
A Generalized Polar-Coordinate Integration Formula, Oscillatory Integral Techniques, And Applications To Convolution Powers Of Complex-Valued Functions On $\Mathbb{Z}^D$, 2021 Colby College
A Generalized Polar-Coordinate Integration Formula, Oscillatory Integral Techniques, And Applications To Convolution Powers Of Complex-Valued Functions On $\Mathbb{Z}^D$, Huan Q. Bui
Honors Theses
In this thesis, we consider a class of function on $\mathbb{R}^d$, called positive homogeneous functions, which interact well with certain continuous one-parameter groups of (generally anisotropic) dilations. Generalizing the Euclidean norm, positive homogeneous functions appear naturally in the study of convolution powers of complex-valued functions on $\mathbb{Z}^d$. As the spherical measure is a Radon measure on the unit sphere which is invariant under the symmetry group of the Euclidean norm, to each positive homogeneous function $P$, we construct a Radon measure $\sigma_P$ on $S=\{\eta \in \mathbb{R}^d:P(\eta)=1\}$ which is invariant under the symmetry group of $P$. With this measure, we prove …
Some Proofs Regarding Minami Estimates And Local Eigenvalue Statistics For Some Random Schrödinger Operator Models, 2021 University of Kentucky
Some Proofs Regarding Minami Estimates And Local Eigenvalue Statistics For Some Random Schrödinger Operator Models, Samuel Herschenfeld
Theses and Dissertations--Mathematics
We provide three proofs on different, but related models in the field of random Schrödinger operators. All three results are motivated by the desire to extend results and techniques on eigenvalue statistics or Minami estimates (an essential ingredient Poisson eigenvalue statistics).
Chapters 2 and 4 are explorations of the only two known techniques for proving Minami estimates for continuum Minami estimates. In Chapter 2, we provide an alternative and simplified proof of Klopp that holds in d = 1. Chapter 4 is an application of the techniques of Dietlein and Elgart to prove a Minami estimate for finite rank lattice …
A New Mathematical Theory For The Dynamics Of Large Tumor Populations, A Potential Mechanism For Cancer Dormancy & Recurrence And Experimental Observation Of Melanoma Progression In Zebrafish, 2021 CUNY City College
A New Mathematical Theory For The Dynamics Of Large Tumor Populations, A Potential Mechanism For Cancer Dormancy & Recurrence And Experimental Observation Of Melanoma Progression In Zebrafish, Adeyinka A. Lesi
Dissertations and Theses
Cancer, a family of over a hundred disease varieties, results in 600,000 deaths in the U.S. alone. Yet, improvements in imaging technology to detect disease earlier, pharmaceutical developments to shrink or eliminate tumors, and modeling of biological interactions to guide treatment have prevented millions of deaths. Cancer patients with initially similar disease can experience vastly different outcomes, including sustained recovery, refractory disease or, remarkably, recurrence years after apparently successful treatment. The current understanding of such recurrences is that they depend on the random occurrence of critical mutations. Clearly, these biological changes appear to be sufficient for recurrence, but are they …
Inference Of Surface Velocities From Oblique Time Lapse Photos And Terrestrial Based Lidar At The Helheim Glacier, 2021 University of Montana, Missoula
Inference Of Surface Velocities From Oblique Time Lapse Photos And Terrestrial Based Lidar At The Helheim Glacier, Franklyn T. Dunbar Ii
Graduate Student Theses, Dissertations, & Professional Papers
Using time dependent observations derived from terrestrial LiDAR and oblique
time-lapse imagery, we demonstrate that a Bayesian approach to glacial motion es-
timation provides a concise way to incorporate multiple data products into a single
motion estimation procedure effectively producing surface velocity estimates with
an associated uncertainty. This approach brings both improved computational effi-
ciency, and greater scalability across observational time-frames when compared to
existing methods. To gauge efficacy, we apply these methods to a set of observa-
tions from the Helheim Glacier, a critical actor in contemporary mass loss trends
observed in the Greenland Ice Sheet. We find that …
Improving The Temporal Accuracy Of Turbulence Models And Resolving The Implementation Issues Of Fluid Flow Modeling, 2021 Michigan Technological University
Improving The Temporal Accuracy Of Turbulence Models And Resolving The Implementation Issues Of Fluid Flow Modeling, Kyle J. Schwiebert
Dissertations, Master's Theses and Master's Reports
A sizeable proportion of the work in this thesis focuses on a new turbulence model, dubbed ADC (the approximate deconvolution model with defect correction). The ADC is improved upon using spectral deferred correction, a means of constructing a higher order ODE solver. Since both the ADC and SDC are based on a predictor-corrector approach, SDC is incorporated with essentially no additional computational cost. We will show theoretically and using numerical tests that the new scheme is indeed higher order in time than the original, and that the benefits of defect correction, on which the ADC is based, are preserved.
The …
Deterministic And Statistical Methods For Inverse Problems With Partial Data, 2021 Michigan Technological University
Deterministic And Statistical Methods For Inverse Problems With Partial Data, Yanfang Liu
Dissertations, Master's Theses and Master's Reports
Inverse problems with partial data have many applications in science and engineering. They are more challenging than the complete data cases since the lack of data increases ill-posedness and nonlinearity. The use of only deterministic or statistical methods might not provide satisfactory results. We propose to combine the deterministic and statistical methods to treat such inverse problems. The thesis is organized as follows.
In Chapter 1, we briefly introduce the inverse problems and their applications. The classical deterministic methods and Bayesian inversion are discussed. The chapter is concluded with a summary of contributions.
Chapter 2 considers the reconstruction of the …
New Numerical Approximations Of Geological Processes In Heterogeneous Systems Using Radial Basis Functions, 2021 Michigan Technological University
New Numerical Approximations Of Geological Processes In Heterogeneous Systems Using Radial Basis Functions, Nadun Lakshitha Dissanayake Kulasekera Mudiyanselage
Dissertations, Master's Theses and Master's Reports
This dissertation includes four chapters. A brief description of each chapter is organized as follows. The first chapter provides an introduction to the RBF method. The chapter follows the historical progression of the Radial Basis Function (RBF) method while outlining the method’s advantages and disadvantages. A brief introduction about RBF interpolation, the RBF-FD method, and how to use it to solve PDEs is provided. Chapter 2 introduces a novel computationally efficient RBF-FD algorithm to solve the groundwater flow equation in the presence of an active well. We show that our method analytically handles the singularities in the PDE caused by …
Multicomponent Fokas-Lenells Equations On Hermitian Symmetric Spaces, 2021 Bulgarian Academy of Sciences
Multicomponent Fokas-Lenells Equations On Hermitian Symmetric Spaces, Vladimir Gerdjikov, Rossen Ivanov
Articles
Multi-component integrable generalizations of the Fokas-Lenells equation, associated with each irreducible Hermitian symmetric space are formulated. Description of the underlying structures associated to the integrability, such as the Lax representation and the bi-Hamiltonian formulation of the equations is provided. Two reductions are considered as well, one of which leads to a nonlocal integrable model. Examples with Hermitian symmetric spaces of all classical series of types A.III, BD.I, C.I and D.III are presented in details, as well as possibilities for further reductions in a general form.