We model stochastic chemical systems with diffusion by a reaction-diffusion master equation. On a macroscopic level, the governing equation is a reaction-diffusion equation for the averages of the chemical species. On a mesoscopic level, the master equation for a well stirred chemical system is combined with a discretized Brownian motion in space to obtain the reaction-diffusion master equation. The space is covered in our method by an unstructured mesh, and the diffusion coefficients on the mesoscale are obtained from a finite element discretization of the Laplace operator on the macroscale. The resulting method is a flexible hybrid algorithm in that the diffusion can be handled either on the meso- or on the macroscale level. The accuracy and the efficiency of the method are illustrated in three numerical examples inspired by molecular biology.

https://www.it.uu.se/research/publications/reports/2008-012/2008-012-nc.pdf

Simulation of Stochastic Reaction-Diffusion Processes on Unstructured Meshes

Definitions and basic properties of the Voronoi diagram generalizations of the Voronoi diagram algorithms for computing Voronoi diagrams poisson Voronoi diagrams spatial interpolation models of spatial processes point pattern analysis locational optimization through Voronoi diagrams.

Spatial tessellations. Concepts and Applications of Voronoi diagrams

This paper surveys some results on acute and nonobtuse simplices and associated spatial partitions. These partitions are relevant in numerical mathematics, including piecewise polynomial approximation theory and the finite element method. Special attention is paid to a basic type of nonobtuse simplices called path-simplices, the generalization of right triangles to higher dimensions. In addition to applications in numerical mathematics, we give examples of the appearance of acute and nonobtuse simplices in other areas of mathematics.

/pdf/on-nonobtuse-simplicial-partitions-2oluapqdyg.pdf

On Nonobtuse Simplicial Partitions

Meshes composed of well-centered simplices have nice orthogonal dual meshes (the dual Voronoi diagram). This is useful for certain numerical algorithms that prefer such primal-dual mesh pairs. We prove that well-centered meshes also have optimality properties and relationships to Delaunay and minmax angle triangulations. We present an iterative algorithm that seeks to transform a given triangulation in two or three dimensions into a well-centered one by minimizing a cost function and moving the interior vertices while keeping the mesh connectivity and boundary vertices fixed. The cost function is a direct result of a new characterization of well-centeredness in arbitrary dimensions that we present. Ours is the first optimization-based heuristic for well-centeredness and the first one that applies in both two and three dimensions. We show the results of applying our algorithm to small and large two-dimensional meshes, some with a complex boundary, and obtain a well-centered tetrahedralization of the cube. We also show numerical evidence that our algorithm preserves gradation and that it improves the maximum and minimum angles of acute triangulations created by the best known previous method.

Well-Centered Triangulation

A nonconforming finite element method for the Biot's consolidation model in poroelasticity

This paper surveys some interesting results on nonobtuse simplices. In particular, we recall path-simplices that generalize right triangles into higher dimensions. We also deal with partitions containing only acute or only nonobtuse simplices. Such partitions are relevant in piecewise polynomial approximation theory in general, and thus also in the finite element method. Finally, we show some applications of nonobtuse simplices in algebra, mathematical analysis, graph theory, in generating geodetical meshes, etc. AMS subject classifications: 51M20, 65N30

On acute and nonobtuse simplicial partitions

Let Ω ⊂ ℝ d , d ∈ {1,2,...}, be a bounded domain with a Lipschitz boundary ∂⊂. Consider the problem 

$$ - \Delta u = f in \Omega , $$

(1.1)



$$ u = 0 on \partial \Omega , $$

(1.2)

where f ∈ L 2(Ω). It is well known that the weak solution u of (1.1)–(1.2) satisfies the maximum principle, i.e., if f ≤ 0 then the maximum (essential supremum) of u over $ \overline \Omega $ is attained on the boundary ∂⊂. A similar maximum principle holds for a wide class of nonlinear second order elliptic problems [9].

Acute Versus Nonobtuse Tetrahedralizations

In this paper we give a sufficient condition for the validity of a discrete maximum principle (DMP) for a class of elliptic problems of the second order with a nondiagonal coefficient matrix, solved by means of linear finite elements (FEs). Numerical tests are presented.

/pdf/on-a-discrete-maximum-principle-for-linear-fe-solutions-of-4pa7j5hxmj.pdf

On a Discrete Maximum Principle for Linear FE Solutions of Elliptic Problems with a Nondiagonal Coefficient Matrix

The detection and mapping of riverscapes with Unmanned Aerial Vehicles (UAVs, drones) provide detailed, reliable, and operable spatial information in hydrological sciences, enhancing conventional field survey techniques. In this study, we present the results of long-term, optical RGB (red, green, blue) UAV monitoring of stream restoration projects to identify the positive and negative features that affect their sustainability. We determined quantitative and qualitative aspects of restoration, such as the restoration effect, the dynamics of fluvial processes, hydrological connectivity, and riparian vegetation. The study was based on six years of UAV monitoring in three restored streams in Prague, Czech Republic. The multitemporal riverscape models from the photogrammetric reconstruction served as a basis for the visual assessment, compliant with the standard hydromorphological assessment. Such a combined approach extends the potential of UAV monitoring by allowing for the use of existing classification schemes and data and the objective detection of critical features. The study pointed to the significant discrepancies in channel geometry between the planned restorations and realized restorations in all assessed projects as a general phenomenon. Multitemporal, optical RGB UAV monitoring then detected issues in qualitative aspects that limit restoration quality, such as water overuse, extensive eutrophication, or inefficient riparian shading.

/pdf/use-of-uav-monitoring-to-identify-factors-limiting-the-2k026lst.pdf

Jakub Šolc

Papers

On Nonobtuse Simplicial Partitions

On acute and nonobtuse simplicial partitions

Acute Versus Nonobtuse Tetrahedralizations

On a Discrete Maximum Principle for Linear FE Solutions of Elliptic Problems with a Nondiagonal Coefficient Matrix

Use of UAV Monitoring to Identify Factors Limiting the Sustainability of Stream Restoration Projects