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Alternating directions implicit integration in a general linear method framework
Sarshar, Arash; Roberts, Steven; Sandu, Adrian (Elsevier, 2021-05-15)
Alternating Directions Implicit (ADI) integration is an operator splitting approach to solve parabolic and elliptic partial differential equations in multiple dimensions based on solving sequentially a set of related one-dimensional equations. Classical ADI methods have order at most two, due to the splitting errors. Moreover, when the time discretization of stiff one-dimensional problems is based on Runge–Kutta schemes, additional order reduction may occur. This work proposes a new ADI approach based on the partitioned General Linear Methods framework. This approach allows the construction of high order ADI methods. Due to their high stage order, the proposed methods can alleviate the order reduction phenomenon seen with other schemes. Numerical experiments are shown to provide further insight into the accuracy, stability, and applicability of these new methods.
Epigenetic regulation of neuronal cell specification inferred with single cell “Omics” data
Yin, Liduo; Banerjee, Sharmi; Fan, Jiayi; He, Jianlin; Lu, Xuemei; Xie, Hehuang (Elsevier, 2020-01-01)
The brain is a highly complex organ consisting of numerous types of cells with ample diversity at the epigenetic level to achieve distinct gene expression profiles. During neuronal cell specification, transcription factors (TFs) form regulatory modules with chromatin remodeling proteins to initiate the cascade of epigenetic changes. Currently, little is known about brain epigenetic regulatory modules and how they regulate gene expression in a cell-type specific manner. To infer TFs involved in neuronal specification, we applied a recursive motif search approach on the differentially methylated regions identified from single-cell methylomes. The epigenetic transcription regulatory modules (ETRM), including EGR1 and MEF2C, were predicted and the co-expression of TFs in ETRMs were examined with RNA-seq data from single or sorted brain cells using a conditional probability matrix. Lastly, computational predications were validated with EGR1 ChIP-seq data. In addition, methylome and RNA-seq data generated from Egr1 knockout mice supported the essential role of EGR1 in brain epigenome programming, in particular for excitatory neurons. In summary, we demonstrated that brain single cell methylome and RNA-seq data can be integrated to gain a better understanding of how ETRMs control cell specification. The analytical pipeline implemented in this study is freely accessible in the Github repository (https://github.com/Gavin-Yinld/brain_TF).
Linearly implicit GARK schemes
Sandu, Adrian; Guenther, Michael; Roberts, Steven (Elsevier, 2021-03-01)
Systems driven by multiple physical processes are central to many areas of science and engineering. Time discretization of multiphysics systems is challenging, since different processes have different levels of stiffness and characteristic time scales. The multimethod approach discretizes each physical process with an appropriate numerical method; the methods are coupled appropriately such that the overall solution has the desired accuracy and stability properties. The authors developed the general-structure additive Runge–Kutta (GARK) framework, which constructs multimethods based on Runge–Kutta schemes. This paper constructs the new GARK-ROS/GARK-ROW families of multimethods based on linearly implicit Rosenbrock/Rosenbrock-W schemes. For ordinary differential equation models, we develop a general order condition theory for linearly implicit methods with any number of partitions, using exact or approximate Jacobians. We generalize the order condition theory to two-way partitioned index-1 differential-algebraic equations. Applications of the framework include decoupled linearly implicit, linearly implicit/explicit, and linearly implicit/implicit methods. Practical GARK-ROS and GARK-ROW schemes of order up to four are constructed.
Multirate implicit Euler schemes for a class of differential-algebraic equations of index-1
Hachtel, Christoph; Bartel, Andreas; Guenther, Michael; Sandu, Adrian (Elsevier, 2021-05-15)
Systems of differential equations which consist of subsystems with widely different dynamical behaviour can be integrated by multirate time integration schemes to increase the efficiency. These schemes allow the usage of inherent step sizes according to the dynamical properties of the subsystem. In this paper, we extend the multirate implicit Euler method to semi-explicit differential–algebraic equations of index-1 where the algebraic constraints only occur in the slow changing subsystem. We discuss different coupling approaches and show that consistency and convergence order 1 can be reached. Numerical experiments validate the analytical results.
Multirate linearly-implicit GARK schemes
Guenther, Michael; Sandu, Adrian (Springer, 2021-12-28)
Many complex applications require the solution of initial-value problems where some components change fast, while others vary slowly. Multirate schemes apply different step sizes to resolve different components of the system, according to their dynamics, in order to achieve increased computational efficiency. The stiff components of the system, fast or slow, are best discretized with implicit base methods in order to ensure numerical stability. To this end, linearly implicit methods are particularly attractive as they solve only linear systems of equations at each step. This paper develops the Multirate GARK-ROS/ROW (MR-GARK-ROS/ROW) framework for linearly-implicit multirate time integration. The order conditions theory considers both exact and approximative Jacobians. The effectiveness of implicit multirate methods depends on the coupling between the slow and fast computations; an array of efficient coupling strategies and the resulting numerical schemes are analyzed. Multirate infinitesimal step linearly-implicit methods, that allow arbitrarily small micro-steps and offer extreme computational flexibility, are constructed. The new unifying framework includes existing multirate Rosenbrock(-W) methods as particular cases, and opens the possibility to develop new classes of highly effective linearly implicit multirate integrators.
Partitioned exponential methods for coupled multiphysics systems
Narayanamurthi, Mahesh; Sandu, Adrian (Elsevier, 2021-03-01)
Multiphysics problems involving two or more coupled physical phenomena are ubiquitous in science and engineering. This work develops a new partitioned exponential approach for the time integration of multiphysics problems. After a possible semi-discretization in space, the class of problems under consideration is modeled by a system of ordinary differential equations where the right-hand side is a summation of two component functions, each corresponding to a given set of physical processes. The partitioned-exponential methods proposed herein evolve each component of the system via an exponential integrator, and information between partitions is exchanged via coupling terms. The traditional approach to constructing exponential methods, based on the variation-of-constants formula, is not directly applicable to partitioned systems. Rather, our approach to developing new partitioned-exponential families is based on a general-structure additive formulation of the schemes. Two method formulations are considered, one based on a linear-nonlinear splitting of the right hand component functions, and another based on approximate Jacobians. The paper develops classical (non-stiff) order conditions theory for partitioned exponential schemes based on particular families of T-trees and B-series theory. Several practical methods of third order are constructed that extend the Rosenbrock-type and EPIRK families of exponential integrators. Several implementation optimizations specific to the application of these methods to reaction-diffusion systems are also discussed. Numerical experiments reveal that the new partitioned-exponential methods can perform better than traditional unpartitioned exponential methods on some problems.
Scenario-based cuts for structured two-stage stochastic and distributionally robust p-order conic mixed integer programs
Bansal, Manish; Zhang, Yingqiu (Springer, 2021-01-22)
In this paper, we derive (partial) convex hull for deterministic multi-constraint polyhedral conic mixed integer sets with multiple integer variables using conic mixed integer rounding (CMIR) cut-generation procedure of Atamtürk and Narayanan (Math Prog 122:1–20, 2008), thereby extending their result for a simple polyhedral conic mixed integer set with single constraint and one integer variable. We then introduce two-stage stochastic p-order conic mixed integer programs (denoted by TSS-CMIPs) in which the second stage problems have sum of lp-norms in the objective function along with integer variables. First, we present sufficient conditions under which the addition of scenario-based nonlinear cuts in the extensive formulation of TSS-CMIPs is sufficient to relax the integrality restrictions on the second stage integer variables without impacting the integrality of the optimal solution of the TSS-CMIP. We utilize scenario-based CMIR cuts for TSS-CMIPs and their distributionally robust generalizations with structured CMIPs in the second stage, and prove that these cuts provide conic/linear programming equivalent or approximation for the second stage CMIPs. We also perform extensive computational experiments by solving stochastic and distributionally robust capacitated facility location problem and randomly generated structured TSS-CMIPs with polyhedral CMIPs and second-order CMIPs in the second stage, i.e. p= 1 and p= 2 , respectively. We observe that there is a significant reduction in the total time taken to solve these problems after adding the scenario-based cuts.
Structural and molecular biology of hepatitis E virus
Wang, Bo; Meng, Xiang-Jin (Elsevier, 2021-01-01)
Hepatitis E virus (HEV) is one of the most common causes of acute viral hepatitis, mainly transmitted by fecal-oral route but has also been linked to fulminant hepatic failure, chronic hepatitis, and extrahepatic neurological and renal diseases. HEV is an emerging zoonotic pathogen with a broad host range, and strains of HEV from numerous animal species are known to cross species barriers and infect humans. HEV is a single-stranded, positive-sense RNA virus in the family Hepeviridae. The genome typically contains three open reading frames (ORFs): ORF1 encodes a nonstructural polyprotein for virus replication and transcription, ORF2 encodes the capsid protein that elicits neutralizing antibodies, and ORF3, which partially overlaps ORF2, encodes a multifunctional protein involved in virion morphogenesis and pathogenesis. HEV virions are non-enveloped spherical particles in feces but exist as quasi-enveloped particles in circulating blood. Two types of HEV virus-like particles (VLPs), small T = 1 (270 Å) and native virion-sized T = 3 (320–340 Å) have been reported. There exist two distinct forms of capsid protein, the secreted form (ORF2S) inhibits antibody neutralization, whereas the capsid-associated form (ORF2C) self-assembles to VLPs. Four cis-reactive elements (CREs) containing stem-loops from secondary RNA structures have been identified in the non-coding regions and are critical for virus replication. This mini-review discusses the current knowledge and gaps regarding the structural and molecular biology of HEV with emphasis on the virion structure, genomic organization, secondary RNA structures, viral proteins and their functions, and life cycle of HEV.
Subspace adaptivity in Rosenbrock-Krylov methods for the time integration of initial value problems
Tranquilli, Paul; Glandon, Ross; Sandu, Adrian (Elsevier, 2021-03-15)
The Rosenbrock–Krylov family of time integration schemes is an extension of Rosenbrock-W methods that employs a specific Krylov based approximation of the linear system solutions arising within each stage of the integrator. This work proposes an extension of Rosenbrock–Krylov methods to address stability questions which arise for methods making use of inexact linear system solution strategies. Two approaches for improving the stability and efficiency of Rosenbrock–Krylov methods are proposed, one through direct control of linear system residuals and the second through a novel extension of the underlying Krylov space to include stage right hand side vectors. Rosenbrock–Krylov methods employing the new approaches show a substantial improvement in computational efficiency relative to prior implementations.

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Browsing VTechWorks Archives by Subject "0103 Numerical and Computational Mathematics"

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