Przemyslaw Klosiewicz

Improving the arithmetic intensity of multigrid with the help of polynomial smoothers

SUMMARY The basic building blocks of a classic multigrid algorithm, which are essentially stencil computations, all have a low ratio of executed floating point operations per byte fetched from memory. This important ratio can be identified as the arithmetic intensity. Applications with a low arithmetic intensity are typically bounded by memory traffic and achieve only a small percentage of the theoretical peak performance of the underlying hardware. We propose a polynomial Chebyshev smoother, which we implement using cache-aware tiling, to increase the arithmetic intensity of a multigrid V-cycle. This tiling approach involves a trade-off between redundant computations and cache misses. Unlike common conception, we observe optimal performance for higher degrees of the smoother. The higher-degree polynomial Chebyshev smoother can be used to smooth more than just the upper half of the error frequencies, leading to better V-cycle convergence rates. Smoothing more than the upper half of the error spectrum allows a more aggressive coarsening approach where some levels in the multigrid hierarchy are skipped. Copyright

Using Pseudo-Arclength Continuation to trace the resonances of the Schrodinger Equation

Quantum bound states and resonances can be studied via the poles of the S-matrix. Using numerical continuation methods and bifurcation theory, we develop an efficient method to trace the parameter dependence of these poles. Using pseudo-arclength continuation, we minimize the numerical complexity of our algorithm. We have applied our method on a model problem.

Virtual Plant Tissue: Building Blocks for Next-Generation Plant Growth Simulation

Motivation: Computational modeling of plant developmental processes is becoming increasingly important. Cellular resolution plant tissue simulators have been developed, yet they are typically describing physiological processes in an isolated way, strongly delimited in space and time. Results: With plant systems biology moving toward an integrative perspective on development we have built the Virtual Plant Tissue (VPTissue) package to couple functional modules or models in the same framework and across different frameworks. Multiple levels of model integration and coordination enable combining existing and new models from different sources, with diverse options in terms of input/output. Besides the core simulator the toolset also comprises a tissue editor for manipulating tissue geometry and cell, wall, and node attributes in an interactive manner. A parameter exploration tool is available to study parameter dependence of simulation results by distributing calculations over multiple systems. Availability: Virtual Plant Tissue is available as open source (EUPL license) on Bitbucket (https://bitbucket.org/vptissue/vptissue). The project has a website https://vptissue.bitbucket.io.

A Short Note on Point Singularities for Robot Manipulators

We consider a specific type of singularities for kinematic chains, so-called point singularities. These were characterized in 2005 by Borcea and Streinu. We give a new proof for this result in the framework of the Exterior Algebra. As an illustration we give an exhaustive list of the point singularities of a specific robot manipulator.

Tracing the parameter dependence of quantum resonances with numerical continuation

In molecular reactions at the microscopic level the appearance of resonances has an important influence on the reactivity. To study when a bound state transitions into a resonance and how these transitions depend on various system parameters such as internuclear distances, one needs to look at the poles of the S-matrix. Using numerical continuation methods and bifurcation theory, we develop efficient and robust methods to trace the parameter dependence of the poles of the S-matrix. Using pseudo-arclength continuation, we can minimize the numerical complexity of our algorithm. As a proof-of-concept we have applied our methods on a number of model problems.

Digital Image Correlation for Full-Field High Resolution Assessment of Leaf Growth

Changes in shape and size of the leaves are driven by several transient growth parameters. Being able to assess resulting changes at a high temporal and spatial resolution is a necessary tool for studying biochemical principles of leaf development, and for construction of leaf growth models. In this short communication, a technique based on the use of 2D digital image correlation (DIC) is presented to track leaf growth. A speckle pattern with orange fluorescent paint was applied on three leaves of the sycamore maple (Acer pseudoplatanus) at an arbitrary point in growth. For ten consecutive days, images of the growing leaves were analyzed with DIC, revealing leaf growth patterns. The patterns were similar for three leaves, and results corresponded both with literature data of leaf growth and with leaf area measurements using manual segmentation of leaf images. We demonstrate that DIC can be applied for tracking leaf growth. Based on the results, a detailed macro-model for leaf growth can be developed or environmental effects on leaf growth can be assessed, amongst other possible applications.

Solving General Auxin Transport Models with a Numerical Continuation Toolbox in Python: PyNCT

Many biological processes are described with coupled non-linear systems of ordinary differential equations that contain a plethora of parameters. The goal is to understand these systems and to predict the effect of different influences. This asks for a dynamical systems approach where numerical continuation methods and bifurcation analysis are used to detect the solutions and their stability as a function of the parameters. We developed PyNCT – Python Numerical Continuation Toolbox – an open source Python package that implements numerical continuation methods and can perform bifurcation analysis based on sparse linear algebra. The software gives the user the choice of different solvers (direct and iterative) and allows the use of preconditioners to reduce the number of iterations and guarantee the convergence when working with complex non-linear models.

Numerical Analysis of Steady State Patterns in Cell-Based Auxin Transport Models

Cell-based models that describe the pattern formation and the flow of chemicals in plant organs are important building blocks in a multiscale simulation of a whole plant. An example of an important mechanism is the transport of the hormone auxin throughout the plants organs because it is closely related to the growth characteristics of roots, shoots and leaves. Based on experimental evidence, a number of cell-based auxin transport models were developed. Due to the intercellular transport of chemicals, these models are complex dynamical systems with a large set of endogenous and exogenous parameters. The models share underlying mathematicalprinciples w.r.t. steady state pattern formation which plays a central role in the growth and development of plant organs. This calls for a uniform computational approach. In our research we focus on computer simulations of general cell-based transport models and more specically we use numerical bifurcation analysis to study the steady state patterns. It indicates how the stability ofpatterns is lost or gained as the system parameters change. Bifurcation analysis of ODEs is widespread in biology and various numerical tools produce bifurcation diagrams. However, these automatic tools do not work for large scale problems, such as biological patterning. Indeed, realistic simulations of large tissues that take multiple interacting chemicals into account give rise to very large and sparse systems of coupled ODEs. We analyze recent large scale transport models with new mathematical and computational tools that enable quantitative prediction of the bifurcations that appear at the macroscopic level in these models. This allows us to predict the patterns and self-similar solutions that appear during organ growth and to see how their stability changes as endogenous parameters are modified or as externally applied changes are enforced. We use these methods to compare the model output with observed data such as the auxin distribution and venation patterns in leaves in order to get a better understanding of the processes that regulate organ development in plants.