466 research outputs found
Primal-dual splitting algorithm for solving inclusions with mixtures of composite, Lipschitzian, and parallel-sum monotone operators
We propose a primal-dual splitting algorithm for solving monotone inclusions
involving a mixture of sums, linear compositions, and parallel sums of
set-valued and Lipschitzian operators. An important feature of the algorithm is
that the Lipschitzian operators present in the formulation can be processed
individually via explicit steps, while the set-valued operators are processed
individually via their resolvents. In addition, the algorithm is highly
parallel in that most of its steps can be executed simultaneously. This work
brings together and notably extends various types of structured monotone
inclusion problems and their solution methods. The application to convex
minimization problems is given special attention
Modelling fluid deformable surfaces with an emphasis on biological interfaces
Fluid deformable surfaces are ubiquitous in cell and tissue biology, including lipid bilayers, the actomyosin cortex or epithelial cell sheets. These interfaces exhibit a complex interplay between elasticity, low Reynolds number interfacial hydrodynamics, chemistry and geometry, and govern important biological processes such as cellular traffic, division, migration or tissue morphogenesis. To address the modelling challenges posed by this class of problems, in which interfacial phenomena tightly interact with the shape and dynamics of the surface, we develop a general continuum mechanics and computational framework for fluid deformable surfaces. The dual solid–fluid nature of fluid deformable surfaces challenges classical Lagrangian or Eulerian descriptions of deforming bodies. Here, we extend the notion of arbitrarily Lagrangian–Eulerian (ALE) formulations, well-established for bulk media, to deforming surfaces. To systematically develop models for fluid deformable surfaces, which consistently treat all couplings between fields and geometry, we follow a nonlinear Onsager formalism according to which the dynamics minimizes a Rayleighian functional where dissipation, power input and energy release rate compete. Finally, we propose new computational methods, which build on Onsager’s formalism and our ALE formulation, to deal with the resulting stiff system of higher-order partial differential equations. We apply our theoretical and computational methodology to classical models for lipid bilayers and the cell cortex. The methods developed here allow us to formulate/simulate these models in their full three-dimensional generality, accounting for finite curvatures and finite shape changes. This article has been published in a revised form in Journal of fluid mechanics
Modelling fluid deformable surfaces with an emphasis on biological interfaces
This article has been published in a revised form in Journal of fluid mechanics, http://dx.doi.org/10.1017/jfm.2019.341. This version is free to view and download for private research and study only. Not for re-distribution, re-sale or use in derivative works. © 2019Fluid deformable surfaces are ubiquitous in cell and tissue biology, including lipid bilayers, the actomyosin cortex or epithelial cell sheets. These interfaces exhibit a complex interplay between elasticity, low Reynolds number interfacial hydrodynamics, chemistry and geometry, and govern important biological processes such as cellular traffic, division, migration or tissue morphogenesis. To address the modelling challenges posed by this class of problems, in which interfacial phenomena tightly interact with the shape and dynamics of the surface, we develop a general continuum mechanics and computational framework for fluid deformable surfaces. The dual solid–fluid nature of fluid deformable surfaces challenges classical Lagrangian or Eulerian descriptions of deforming bodies. Here, we extend the notion of arbitrarily Lagrangian–Eulerian (ALE) formulations, well-established for bulk media, to deforming surfaces. To systematically develop models for fluid deformable surfaces, which consistently treat all couplings between fields and geometry, we follow a nonlinear Onsager formalism according to which the dynamics minimizes a Rayleighian functional where dissipation, power input and energy release rate compete. Finally, we propose new computational methods, which build on Onsager’s formalism and our ALE formulation, to deal with the resulting stiff system of higher-order partial differential equations. We apply our theoretical and computational methodology to classical models for lipid bilayers and the cell cortex. The methods developed here allow us to formulate/simulate these models in their full three-dimensional generality, accounting for finite curvatures and finite shape changes.Peer ReviewedPostprint (author's final draft
The Geometry of Monotone Operator Splitting Methods
We propose a geometric framework to describe and analyze a wide array of
operator splitting methods for solving monotone inclusion problems. The initial
inclusion problem, which typically involves several operators combined through
monotonicity-preserving operations, is seldom solvable in its original form. We
embed it in an auxiliary space, where it is associated with a surrogate
monotone inclusion problem with a more tractable structure and which allows for
easy recovery of solutions to the initial problem. The surrogate problem is
solved by successive projections onto half-spaces containing its solution set.
The outer approximation half-spaces are constructed by using the individual
operators present in the model separately. This geometric framework is shown to
encompass traditional methods as well as state-of-the-art asynchronous
block-iterative algorithms, and its flexible structure provides a pattern to
design new ones
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