scholarly journals Chemotaxis on networks: Analysis and numerical approximation

2020 ◽  
Vol 54 (4) ◽  
pp. 1339-1372
Author(s):  
Herbert Egger ◽  
Lukas Schöbel-Kröhn
Keyword(s):  
Convergence Rates ◽  
Energy Estimates ◽  
Implicit Time ◽  
Discrete Approximations ◽  
Uniform Bounds ◽  
Time Stepping ◽  
Characterization Of Solutions ◽  
Networks Analysis ◽  
Theoretical Results

We consider the Keller–Segel model of chemotaxis on one-dimensional networks. Using a variational characterization of solutions, positivity preservation, conservation of mass, and energy estimates, we establish global existence of weak solutions and uniform bounds. This extends related results of Osaki and Yagi to the network context. We then analyze the discretization of the system by finite elements and an implicit time-stepping scheme. Mass lumping and upwinding are used to guarantee the positivity of the solutions on the discrete level. This allows us to deduce uniform bounds for the numerical approximations and to establish order optimal convergence of the discrete approximations to the continuous solution without artificial smoothness requirements. In addition, we prove convergence rates under reasonable assumptions. Some numerical tests are presented to illustrate the theoretical results.

scholarly journals Convergence Rates of First- and Higher-Order Dynamics for Solving Linear Ill-Posed Problems

2021 ◽  
Author(s):  
Radu Boţ ◽  
Guozhi Dong ◽  
Peter Elbau ◽  
Otmar Scherzer
Keyword(s):  
Linear Operator ◽  
Operator Equation ◽  
Convex Analysis ◽  
Convergence Rates ◽  
Linear Operator Equation ◽  
Optimal Convergence Rates ◽  
Ill Posed ◽  
Thresholding Algorithm ◽  
The Given ◽  
Theoretical Results

AbstractRecently, there has been a great interest in analysing dynamical flows, where the stationary limit is the minimiser of a convex energy. Particular flows of great interest have been continuous limits of Nesterov’s algorithm and the fast iterative shrinkage-thresholding algorithm, respectively. In this paper, we approach the solutions of linear ill-posed problems by dynamical flows. Because the squared norm of the residual of a linear operator equation is a convex functional, the theoretical results from convex analysis for energy minimising flows are applicable. However, in the restricted situation of this paper they can often be significantly improved. Moreover, since we show that the proposed flows for minimising the norm of the residual of a linear operator equation are optimal regularisation methods and that they provide optimal convergence rates for the regularised solutions, the given rates can be considered the benchmarks for further studies in convex analysis.

scholarly journals GWENA: gene co-expression networks analysis and extended modules characterization in a single Bioconductor package

2021 ◽  
Vol 22 (1) ◽  
Author(s):  
Gwenaëlle G. Lemoine ◽  
Marie-Pier Scott-Boyer ◽  
Bathilde Ambroise ◽  
Olivier Périn ◽  
Arnaud Droit
Keyword(s):  
Muscle Development ◽  
R Package ◽  
Topological Information ◽  
Old Patients ◽  
Network Modules ◽  
Extended Analysis ◽  
Networks Analysis ◽  
Underlying Processes

Abstract Background Network-based analysis of gene expression through co-expression networks can be used to investigate modular relationships occurring between genes performing different biological functions. An extended description of each of the network modules is therefore a critical step to understand the underlying processes contributing to a disease or a phenotype. Biological integration, topology study and conditions comparison (e.g. wild vs mutant) are the main methods to do so, but to date no tool combines them all into a single pipeline. Results Here we present GWENA, a new R package that integrates gene co-expression network construction and whole characterization of the detected modules through gene set enrichment, phenotypic association, hub genes detection, topological metric computation, and differential co-expression. To demonstrate its performance, we applied GWENA on two skeletal muscle datasets from young and old patients of GTEx study. Remarkably, we prioritized a gene whose involvement was unknown in the muscle development and growth. Moreover, new insights on the variations in patterns of co-expression were identified. The known phenomena of connectivity loss associated with aging was found coupled to a global reorganization of the relationships leading to expression of known aging related functions. Conclusion GWENA is an R package available through Bioconductor (https://bioconductor.org/packages/release/bioc/html/GWENA.html) that has been developed to perform extended analysis of gene co-expression networks. Thanks to biological and topological information as well as differential co-expression, the package helps to dissect the role of genes relationships in diseases conditions or targeted phenotypes. GWENA goes beyond existing packages that perform co-expression analysis by including new tools to fully characterize modules, such as differential co-expression, additional enrichment databases, and network visualization.

A Numerical Model of In-Channel Pebble Bed Thermal Storage for a Solar Chimney

2012 ◽  
Author(s):  
Brandon Schulte ◽  
O. A. Plumb
Keyword(s):  
Numerical Model ◽  
Present Model ◽  
Thermal Storage ◽  
Implicit Time ◽  
Heat Transfer Characteristics ◽  
Solar Chimney ◽  
Pebble Bed ◽  
One Dimensional ◽  
Time Stepping ◽  
Daily Energy

In this study, solar chimney performance is numerically modeled. Previously published models have considered water bags and natural earth as means to store daytime thermal energy for nighttime operation of the system. The present model considers in-channel pebble bed thermal storage. A one-dimensional, implicit time stepping numerical model is developed to predict solar chimney performance throughout a 24 hour period. The model is partially verified with available experimental data. The daily energy, daily efficiency and heat transfer characteristics of the solar chimney with pebble bed thermal storage are summarized. The numerical simulation showed that by introducing a pebble bed, nightly exit velocities reach 40% of the peak daytime velocity. However, the daily kinetic energy delivered by a solar chimney with pebble bed thermal storage is much less than a traditional solar chimney, suggesting pebble bed thermal storage is more practicable in building heating applications as opposed to power generation.

scholarly journals KERNEL CONVERGENCE ESTIMATES FOR DIFFUSIONS WITH CONTINUOUS COEFFICIENTS

2011 ◽  
Vol 14 (07) ◽  
pp. 979-1004
Author(s):  
CLAUDIO ALBANESE
Keyword(s):  
Convergence Rate ◽  
Convergence Rates ◽  
Fourier Transforms ◽  
Uniform Continuity ◽  
Time Step ◽  
Uniform Bounds ◽  
Kernel Convergence ◽  
Discretization Schemes ◽  
A New Technique ◽  
Derivatives Of

Bidirectional valuation models are based on numerical methods to obtain kernels of parabolic equations. Here we address the problem of robustness of kernel calculations vis a vis floating point errors from a theoretical standpoint. We are interested in kernels of one-dimensional diffusion equations with continuous coefficients as evaluated by means of explicit discretization schemes of uniform step h > 0 in the limit as h → 0. We consider both semidiscrete triangulations with continuous time and explicit Euler schemes with time step so small that the Courant condition is satisfied. We find uniform bounds for the convergence rate as a function of the degree of smoothness. We conjecture these bounds are indeed sharp. The bounds also apply to the time derivatives of the kernel and its first two space derivatives. The proof is constructive and is based on a new technique of path conditioning for Markov chains and a renormalization group argument. We make the simplifying assumption of time-independence and use longitudinal Fourier transforms in the time direction. Convergence rates depend on the degree of smoothness and Hölder differentiability of the coefficients. We find that the fastest convergence rate is of order O(h2) and is achieved if the coefficients have a bounded second derivative. Otherwise, explicit schemes still converge for any degree of Hölder differentiability except that the convergence rate is slower. Hölder continuity itself is not strictly necessary and can be relaxed by an hypothesis of uniform continuity.

An L∞-error Estimate for the h-p Version Continuous Petrov-Galerkin Method for Nonlinear Initial Value Problems

2015 ◽  
Vol 5 (4) ◽  
pp. 301-311 ◽  
Author(s):  
Lijun Yi
Keyword(s):  
Error Estimate ◽  
Galerkin Method ◽  
Error Bound ◽  
Initial Value Problems ◽  
Initial Value ◽  
Numerical Examples ◽  
Time Stepping ◽  
Theoretical Results

AbstractThe h-p version of the continuous Petrov-Galerkin time stepping method is analyzed for nonlinear initial value problems. An L∞-error bound explicit with respect to the local discretization and regularity parameters is derived. Numerical examples are provided to illustrate the theoretical results.

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