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We present a new variational approach for proving exponentially slow motion in singularly-perturbed partial differential equations in one space dimension, which builds on the energy approach due to Bronsard and Kohn (Comm. Pure Appl. Math. 43 (1990), pp. 983–997) and Grant (SIAM J. Math. Anal. 26 (1995), pp. 21–34). As well as covering the known… (More)

- Marian Mrozek, Thomas Wanner
- Computers & Mathematics with Applications
- 2010

We present an algorithm for computing the homology of inclusion maps which is based on the idea of coreductions and leads to significant speed improvements over current algorithms. It is shown that this algorithm can be extended to compute both persistent homology and an extension of the persistence concept to two-sided filtrations. In addition to… (More)

This paper addresses the phenomenon of spinodal decomposition for the CahnHilliard equation. Namely, we are interested in why most solutions to the CahnHilliard equation which start near a homogeneous equilibrium u0 in the spinodal interval exhibit phase separation with a characteristic wavelength when exiting a ball of radius R. There are two mathematical… (More)

- Thomas Wanner
- 1997

This paper is the second in a series of two papers addressing the phenomenon of spinodal decomposition for the Cahn-Hilliard equation ut = ("2 u+ f(u)) in ; @u @ = @ u @ = 0 on @ ; where Rn, n 2 f1; 2; 3g, is a bounded domain with su ciently smooth boundary, and f is cubic-like, for example f(u) = u u3. Using the results of [22] the nonlinear Cahn-Hilliard… (More)

- Sarah Day, William D. Kalies, Thomas Wanner
- Multiscale Modeling & Simulation
- 2009

Homology has long been accepted as an important computational tool for quantifying complex structures. In many applications these structures arise as nodal domains of real-valued functions and are therefore amenable only to a numerical study based on suitable discretizations. Such an approach immediately raises the question of how accurately the resulting… (More)

Several models have been proposed for describing the formation of animal coat patterns. We consider reaction–diffusion models due to Murray, which rely on a Turing instability for the pattern selection. In this paper, we describe the early stages of the pattern formation process for large domain sizes. This includes the selection mechanism and the geometry… (More)

- Thomas Stephens, Thomas Wanner
- SIAM J. Applied Dynamical Systems
- 2014

Isolated invariant sets and their associated Conley indices are valuable tools for studying dynamical systems and their global invariant structures. Through their design, they aim to capture invariant behavior which is robust under small perturbations, and this in turn makes them amenable to a computational treatment. Over the years, a number of algorithms… (More)

Phase separation processes in compound materials can produce intriguing and complicated patterns. Yet, characterizing the geometry of these patterns quantitatively can be quite challenging. In this paper we use computational algebraic topology to obtain such a characterization. Our method is illustrated for the complex microstructures observed during… (More)

- Evelyn Sander, Thomas Wanner
- SIAM Journal of Applied Mathematics
- 2000

This paper gives theoretical results on spinodal decomposition for the CahnHillard equation. We prove a mechanism which explains why most solutions for the Cahn-Hilliard equation which start near a homogeneous equilibrium within the spinodal interval exhibit phase separation with a characteristic wavelength when exiting a ball of radius R. Namely, most… (More)

- Pawel Dlotko, Tomasz Kaczynski, Marian Mrozek, Thomas Wanner
- Discrete & Computational Geometry
- 2011

In this paper we present a new algorithm for computing the homology of regular CW-complexes. This algorithm is based on the coreduction algorithm due to Mrozek and Batko and consists essentially of a geometric preprocessing algorithm for the standard chain complex generated by a CW-complex. By employing the concept of S-complexes the original chain complex… (More)