Abstracts

Robust Iteration Methods for Non-linear PDEs

Florin Adrian Radu (University Bergen, Norway)
Robust iterative splitting schemes for linear or nonlinear coupled PDEs (I and II)
In this work we consider solvers for fully coupled partial differential equations (PDEs) by splitting schemes. There are plenty of relevant applications behind, e.g. water and soil pollution, CO2 storage, enhanced geothermal energy extraction or nuclear waste management. In the first part we will consider coupled linear PDEs and introduce robust iterative splitting solvers. The convergence (stabilization) and optimization will be discussed. As example we consider the Biot equations.

In the second part we will consider nonlinear, fully coupled PDEs. Splitting and linearization techniques will be discussed. Different nonlinear Biot models and reactive transport in unsaturated porous media will be used as examples.

Martin Vohralik (Inria, France)
Adaptive iterative approximation in nonlinear PDEs
Nonlinear partial differential equations (PDEs) are omnipresent in description of physical phenomena. Their numerical approximation, however, poses numerous difficulties. In this talk, we describe a holistic approach for finding a discrete (piecewise polynomial) approximation of the unknown exact solution with error below a given desired tolerance and at the expense of the minimal computational cost. The central tool are a posteriori estimates that assess the error at each stage of a numerical simulation: on each time step, on each spatial mesh, on each iterative regularization step, on each iterative linearization step, on each iterative algebraic solver step... The estimates give a guaranteed upper bound (reliability), are mathematically equivalent to the error (efficiency), and distinguish the different error components like the temporal, spatial, regularization, linearization, and algebraic solver ones. For model nonlinear problems, it is possible to obtain estimates that are provably robust, i.e., of quality independent of the strength of the nonlinearities; robustness with respect to the final time is ensured for time-dependent problems. The developed theory encompasses all standard numerical methods (finite elements, finite volumes, mixed finite elements, discontinuous Galerkin schemes, polytopal discretization schemes), standard time stepping schemes, any iterative regularization, iterative linearizations like Zarantonello, Picard, Kačanov, Newton, M-scheme, or L-scheme, and any iterative linear algebraic solver. The approach is not a novel scheme but rather consists of efficient use of existing building blocks.
We characterize the cost of our adaptive algorithms as the cumulative sum of the number of degrees of freedom of the given numerical method over all time steps, spatial meshes, and all regularization, linearization, and algebraic solver steps; by construction, this scales as the CPU time. For model steady problems, a particular theoretical effort has been dedicated to rigorously proving that the derived adaptive algorithms indeed give a convergence rate optimal wrt this computational cost, or, in other words, that the rate of decrease of the error wrt this cost cannot be made better.
The subject is an interplay between analysis of partial differential equations, numerical analysis, and numerical linear algebra. Efficient computer implementation, assessment on academic benchmarks, and applications to environmental problems with nonsmooth and degenerate nonlinearities like the geological sequestration of CO2 will be discussed.

Ben Gharbia I., Ferzly J., Vohralík M., Yousef S. Semismooth and smoothing Newton methods for nonlinear systems with complementarity constraints: Adaptivity and inexact resolution, J. Comput. Appl. Math. 420 (2023), 114765.

Ern A., Vohralík M., Adaptive inexact Newton methods with a posteriori stopping criteria for nonlinear diffusion PDEs, SIAM J. Sci. Comput. 35 (2013), A1761–A1791.

Févotte F., Rappaport A., Vohralík M., Adaptive regularization, discretization, and linearization for nonsmooth problems based on primal-dual gap estimators, Comput. Methods Appl. Mech. Engrg. 418 (2024), 116558.

Haberl A., Praetorius D., Schimanko S., Vohralík M., Convergence and quasi-optimal cost of adaptive algorithms for nonlinear operators including iterative linearization and algebraic solver, Numer. Math. 147 (2021), 679–725.

Mitra K., Vohralík M., A posteriori error estimates for the Richards equation, Math. Comp. 93 (2024), 1053–1096.

Mitra K., Vohralík M., Guaranteed, locally efficient, and robust a posteriori estimates for nonlinear elliptic problems in iteration-dependent norms. An orthogonal decomposition result based on iterative linearization, HAL Preprint 04156711, 2023.

Scientific Machine Learning

- Virginie Ehrlacher (Ecole des Ponts ParisTech, France)
- Felix Dietrich (TU Munich, Germany)

Geometric Integration

Mathieu Desbrun (Inria, France)
Talk 1:
Exploiting the "unreasonable effectiveness" of geometry in computing
It has been repeatedly noted in computational science that the quality of computing tools ultimately boils down to properties of a fundamentally geometric or topological nature. This talk will describe our approach to computing through the lens of geometry to offer a versatile and efficient toolbox for a variety of applications --- from shape processing to tangent vector field editing, to variational mechanics and non-linear dimensionality reduction, to even matrix preconditioners. Through a series of examples, we will point out how a strong grasp of classical differential geometry paired with a good understanding of the typical computational constraints in research and industry can bring forth novel theoretical and practical foundations for general-purpose computations. The importance of preserving differential geometric properties in the discrete setting will be a recurring theme throughout the talk to demonstrate the value of geometry in computations.

Talk 2:
A Discrete Exterior Calculus of Bundle-Valued Forms
Exploring novel geometry-driven discretizations of continuous equations or physical models is often ripe with unexpected difficulties. This talk discusses the development of structure-preserving discretizations of the exterior calculus of differential forms with values in a vector bundle over a combinatorial manifold equipped with a connection. Compared to their scalar-based counterparts which admit a well-established discretization via cochains, bundle-valued forms, (e.g., with values in the group of rotation matrices) present numerous difficulties when one tries to properly define a discrete counterpart to them and to the exterior covariant derivative operator acting on them. We show however that the use of specifically-selected local frame fields allows the construction of a discrete exterior covariant derivative of bundle-valued forms that not only satisfies the well-known Bianchi identities in this discrete realm, but also converges to its smooth equivalent under mesh refinement. If time allows, I will mention other ongoing projects where geometry-driven discretization is thought after.


- Elena Celledoni (NTNU, Norway)