On the rate of convergence of alternating minimization for non-smooth non-strongly convex optimization in Banach spaces

In this paper, the convergence of the fundamental alternating minimization is established for non-smooth non-strongly convex optimization problems in Banach spaces, and novel rates of convergence are provided. As objective function a composition of a smooth, and a block-separable, non-smooth part is considered, covering a large range of applications. For the former, three different relaxations of strong convexity are considered: (i) quasi-strong convexity; (ii) quadratic functional growth; and (iii) plain convexity. With new and improved rates benefiting from both separate steps of the scheme, linear convergence is proved for (i) and (ii), whereas sublinear convergence is showed for (iii).publishedVersio

Iterative quasi-newton solvers for poromechanics applied to heart perfusion

[EN] In this work, the efficient approximation of a nonlinear cardiac poromechanics model is investigated. Quasi-Newton solvers based on iterative two-way and three-way decoupling are proposed. For increased robustness and better performance, the iterative schemes are accelerated by additionally using Anderson acceleration. The solvers are tested for a numerical example simulating cardiac perfusion. The results obtained demonstrate a significant speed-up for the splitting approaches with respect to the standard monolithic Newton method.The development of this document has been supported by the following projects: “Modeling the heart across the scales: from cardiac cells to the whole organ” PRIN 2017AXL54F 003 P.I. S. Scacchi (NB), Project 250223 Research Council of Norway (JWB), and the FracFlow project funded by Equinor through Akademiaavtalen (JWB). The authors also thank Florin Radu, Paolo Zunino and Alfio Quarteroni for inspiring discussions.Barnafi, N.; Both, J. (2022). Iterative quasi-newton solvers for poromechanics applied to heart perfusion. En Proceedings of the YIC 2021 - VI ECCOMAS Young Investigators Conference. Editorial Universitat Politècnica de València. 355-363. https://doi.org/10.4995/YIC2021.2021.12324OCS35536

Iterative Coupling for Fully Dynamic Poroelasticity

We present an iterative coupling scheme for the numerical approximation of the mixed hyperbolic-parabolic system of fully dynamic poroelasticity. We prove its convergence in the Banach space setting for an abstract semi-discretization in time that allows the application of the family of diagonally implicit Runge–Kutta methods. Recasting the semi-discrete solution as the minimizer of a properly defined energy functional, the proof of convergence uses its alternating minimization. The scheme is closely related to the undrained split for the quasi-static Biot system.acceptedVersio

Well-posedness analysis of the Cahn-Hilliard-Biot model

We investigate the well-posedness of the recently proposed Cahn-Hilliard-Biot model. The model is a three-way coupled PDE of elliptic-parabolic nature, with several nonlinearities and the fourth order term known to the Cahn-Hilliard system. We show existence of weak solutions to the variational form of the equations and uniqueness under certain conditions of the material parameters and secondary consolidation, adding regularizing effects. Existence is shown by discretizing in space and applying ODE-theory (the Peano-Cauchy theorem) to prove existence of the discrete system, followed by compactness arguments to retain solutions of the continuous system. In addition, the continuous dependence of solutions on the data is established, in particular implying uniqueness. Both results build strongly on the inherent gradient flow structure of the model

The Fixed-Stress splitting scheme for Biot's equations as a modified Richardson iteration: Implications for optimal convergence

The fixed-stress splitting scheme is a popular method for iteratively solving the Biot equations. The method successively solves the flow and mechanics subproblems while adding a stabilizing term to the flow equation, which includes a parameter that can be chosen freely. However, the convergence properties of the scheme depend significantly on this parameter and choosing it carelessly might lead to a very slow, or even diverging, method. In this paper, we present a way to exploit the matrix structure arising from discretizing the equations in the regime of impermeable porous media in order to obtain a priori knowledge of the optimal choice of this tuning/stabilization parameter.acceptedVersio