Abstract

The dynamics of cross-diffusion models leads to a high computational complexity for implicit difference schemes, turning them unsuitable for tasks that require results in real-time. We propose the use of two operator splitting schemes for nonlinear cross-diffusion processes in order to lower the computational load, and establish their stability properties using discrete L2 energy methods. Furthermore, by attaining a stable factorization of the system matrix as a forward-backward pass, corresponding to the Thomas algorithm for self-diffusion processes, we show that the use of implicit cross-diffusion can be competitive in terms of execution time, widening the range of viable cross-diffusion coefficients for on-the-fly applications.

Highlights

  • Cross-diffusion models consist of evolutionary systems of diffusion type for at least two real-valued functions, where the evolution of each function is not independent of the others

  • Their use is widespread in areas like population dynamics, but has recently attracted some interest in the image processing community [ABCD17a, ABCD17b, ABCD17c, BL20] as a natural extension to the complex diffusion methods proposed by Gilboa et al [GSZ04], where the image is represented by a complex function and the filtering process is governed by a nonlinear PDE of diffusion type with a complex-valued diffusion coefficient

  • The computational complexity of traditional implicit finite difference schemes turns their use impracticable for tasks that aim for on-the-fly results, such as image processing in medical contexts

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Summary

Introduction

Cross-diffusion models consist of evolutionary systems of diffusion type for at least two real-valued functions, where the evolution of each function is not independent of the others. In order to overcome this problem, Barash and Kimmel [BIK01] added a symmetric setting by applying a multiplicative scheme in all possible orders and averaging the results The generalization of these techniques to our problem still has some drawbacks, as the numerical system to solve remains dense due to the interdependence of the evolution functions inherent to the cross-diffusion model. For general nonlinear cross-diffusion systems, a linear scheme based on the nonlinear Chernoff formula was proposed by Murakawa [Mur11] and a convergent finite volume method was introduced by Andreainov et al [ABRB11] This technique was further explored again by Murakawa in [Mur17], where explicit algebraic corrections at each time step were employed in order to achieve unconditional stability.

Nonlinear cross-diffusion system
Explicit and semi-implicit implementations and stability results
Operator splittings
Splitting models and L2 stability
Computational considerations
Generalization to three dimensional schemes
Speedup - operation count and numerical experiments
Concluding remarks

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