Parameter Dependent Differential Equations Research Articles

Selection of a stochastic Landau-Lifshitz equation and the stochastic persistence problem

In this article, we study the persistence of properties of a given classical deterministic differential equation under a stochastic perturbation of two distinct forms: external and internal. The first case corresponds to add a noise term to a given equation using the framework of Itô or Stratonovich stochastic differential equations. The second case corresponds to consider parameter dependent differential equations and to add a stochastic dynamics on the parameters using the framework of random ordinary differential equations. Our main concerns for the preservation of properties are stability/instability of equilibrium points and symplectic/Poisson Hamiltonian structures. We formulate persistence theorem in these two cases and prove that the cases of external and internal stochastic perturbations are drastically different. We then apply our results to develop a stochastic version of the Landau-Lifshitz equation. We discuss, in particular, previous results obtained by Étoré et al. [J. Differ. Equations 257, 2115–2135 (2014)] and we finally propose a new family of stochastic Landau-Lifshitz equations.

Numerical approximation of parametrized problems in cardiac electrophysiology by a local reduced basis method

The efficient solution of coupled PDEs/ODEs problems arising in cardiac electrophysiology is of key importance whenever interested to study the electrical behavior of the tissue for several instances of relevant physical and/or geometrical parameters. This poses significant challenges to reduced order modeling (ROM) techniques –such as the reduced basis method –traditionally employed when dealing with the repeated solution of parameter dependent differential equations. Indeed, the nonlinear nature of the problem, the presence of moving fronts in the solution, and the high sensitivity of this latter to parameter variations, make the application of standard ROM techniques very problematic. In this paper we propose a local ROM built through a k-means clustering in the state space of the snapshots for both the solution and the nonlinear term. Several comparisons among alternative local ROMs on a benchmark test case show the effectivity of the proposed approach. Finally, the application to a parametrized problem set on an idealized left-ventricle geometry shows the capability of the proposed ROM to face complex problems.

The reduced basis method in all-electron calculations with finite elements

The reduced basis method successfully diminishes the required amount of degrees of freedom in the solution of a parameter dependent differential equation, and is applied in this work to the determination of the electronic structure with the Kohn-Sham equations in combination with the finite element method. It is thereby demonstrated, how the basis sizes in all-electron calculations can be essentially reduced, yielding new opportunities in the modeling of the electronic structure. In this context, a generalization of the reduced basis set construction is proposed and shown to increase the flexibility of the method.

Identification of models of heterogeneous cell populations from population snapshot data

BackgroundMost of the modeling performed in the area of systems biology aims at achieving a quantitative description of the intracellular pathways within a "typical cell". However, in many biologically important situations even clonal cell populations can show a heterogeneous response. These situations require study of cell-to-cell variability and the development of models for heterogeneous cell populations.ResultsIn this paper we consider cell populations in which the dynamics of every single cell is captured by a parameter dependent differential equation. Differences among cells are modeled by differences in parameters which are subject to a probability density. A novel Bayesian approach is presented to infer this probability density from population snapshot data, such as flow cytometric analysis, which do not provide single cell time series data. The presented approach can deal with sparse and noisy measurement data. Furthermore, it is appealing from an application point of view as in contrast to other methods the uncertainty of the resulting parameter distribution can directly be assessed.ConclusionsThe proposed method is evaluated using artificial experimental data from a model of the tumor necrosis factor signaling network. We demonstrate that the methods are computationally efficient and yield good estimation result even for sparse data sets.

Bounding the Solutions of Parameter Dependent Nonlinear Ordinary Differential Equations

This paper presents two techniques for generating rigorous bounds on the solution of parameter dependent nonlinear ordinary differential equations. The first technique is an extension of differential inequalities that enables the construction of tight time varying state bounds by utilizing prior knowledge of the solution of the differential equations. The second technique provides a method for constructing pointwise in time convex and concave relaxations of the image of the solution of a system of parameter dependent differential equations on subsets of a Euclidean space. Two examples are presented to demonstrate the construction of the bounds. The examples include a brief discussion of a computer program written to automatically generate the bounds.