arXiv subjects

Michael Innerberger

Publications and source records attributed to Michael Innerberger.

16 recordsLinked to original sources

arXiv2026

Multigoal-oriented adaptive finite element method with convergence rates

We formulate and analyze a goal-oriented adaptive finite element method for a symmetric linear elliptic partial differential equation (PDE) that can simultaneously deal with multiple linear goal functionals. In each step of the algorithm, only two linear finite element systems have to be solved. Moreover, all finite element solutions are computed with respect to the same discrete space, while the underlying triangulations are adapted to resolve all inherent singularities simultaneously. Unlike available results for such a setting in the literature, we give a thorough convergence analysis and verify that our algorithm guarantees, in an appropriate sense, even optimal convergence rates. Numerical experiments underline the derived theoretical results.

math.NA↗

arXiv2025

Energy minimization for skyrmions on planar thin films

We consider an energy functional that arises in micromagnetic and liquid crystal theory on thin films. In particular, our energy comprises a non-convex term that models anti-symmetric exchange as well as an anisotropy term. We devise an algorithm for energy minimization in the continuous case and show weak convergence of a subsequence towards a solution of the corresponding Euler--Lagrange equation. Furthermore, an algorithm for numerical energy minimization is presented. We show empirically that this numerical algorithm converges to the correct solutions for a benchmark problem without the need for user-supplied parameters, and present a rigorous convergence analysis for important special cases.

math.NA↗

arXiv2024

Decomposing heterogeneous dynamical systems with graph neural networks

Natural physical, chemical, and biological dynamical systems are often complex, with heterogeneous components interacting in diverse ways. We show how simple graph neural networks can be designed to jointly learn the interaction rules and the latent heterogeneity from observable dynamics. The learned latent heterogeneity and dynamics can be used to virtually decompose the complex system which is necessary to infer and parameterize the underlying governing equations. We tested the approach with simulation experiments of interacting moving particles, vector fields, and signaling networks. While our current aim is to better understand and validate the approach with simulated data, we anticipate it to become a generally applicable tool to uncover the governing rules underlying complex dynamics observed in nature.

cs.LG↗

arXiv2022

Adaptive FEM with quasi-optimal overall cost for nonsymmetric linear elliptic PDEs

We consider a general nonsymmetric second-order linear elliptic PDE in the framework of the Lax-Milgram lemma. We formulate and analyze an adaptive finite element algorithm with arbitrary polynomial degree that steers the adaptive mesh-refinement and the inexact iterative solution of the arising linear systems. More precisely, the iterative solver employs, as an outer loop, the so-called Zarantonello iteration to symmetrize the system and, as an inner loop, a uniformly contractive algebraic solver, e.g., an optimally preconditioned conjugate gradient method or an optimal geometric multigrid algorithm. We prove that the proposed inexact adaptive iteratively symmetrized finite element method (AISFEM) leads to full linear convergence and, for sufficiently small adaptivity parameters, to optimal convergence rates with respect to the overall computational cost, i.e., the total computational time. Numerical experiments underline the theory.

math.NA↗

arXiv2022

Cost-optimal adaptive iterative linearized FEM for semilinear elliptic PDEs

We consider scalar semilinear elliptic PDEs where the nonlinearity is strongly monotone, but only locally Lipschitz continuous. We formulate an adaptive iterative linearized finite element method (AILFEM) which steers the local mesh refinement as well as the iterative linearization of the arising nonlinear discrete equations. To this end, we employ a damped Zarantonello iteration so that, in each step of the algorithm, only a linear Poisson-type equation has to be solved. We prove that the proposed AILFEM strategy guarantees convergence with optimal rates, where rates are understood with respect to the overall computational complexity (i.e., the computational time). Moreover, we formulate and test an adaptive algorithm where also the damping parameter of the Zarantonello iteration is adaptively adjusted. Numerical experiments underline the theoretical findings.

math.NA↗

arXiv2022

$hp$-robust multigrid solver on locally refined meshes for FEM discretizations of symmetric elliptic PDEs

In this work, we formulate and analyze a geometric multigrid method for the iterative solution of the discrete systems arising from the finite element discretization of symmetric second-order linear elliptic diffusion problems. We show that the iterative solver contracts the algebraic error robustly with respect to the polynomial degree $p \ge 1$ and the (local) mesh size $h$. We further prove that the built-in algebraic error estimator which comes with the solver is $hp$-robustly equivalent to the algebraic error. The application of the solver within the framework of adaptive finite element methods with quasi-optimal computational cost is outlined. Numerical experiments confirm the theoretical findings.

math.NA↗

arXiv2022

Plain convergence of goal-oriented adaptive FEM

We discuss goal-oriented adaptivity in the frame of conforming finite element methods and plain convergence of the related a posteriori error estimator for different general marking strategies. We present an abstract analysis for two different settings. First, we consider problems where a local discrete efficiency estimate holds. Second, we show plain convergence in a setting that relies only on structural properties of the error estimators, namely stability on non-refined elements as well as reduction on refined elements. In particular, the second setting does not require reliability and efficiency estimates. Numerical experiments underline our theoretical findings.

math.NA↗

arXiv2022

MooAFEM: An object oriented Matlab code for higher-order adaptive FEM for (nonlinear) elliptic PDEs

We present an easily accessible, object oriented code (written exclusively in Matlab) for adaptive finite element simulations in 2D. It features various refinement routines for triangular meshes as well as fully vectorized FEM ansatz spaces of arbitrary polynomial order and allows for problems with very general coefficients. In particular, our code can handle problems typically arising from iterative linearization methods used to solve nonlinear PDEs. Due to the object oriented programming paradigm, the code can be used easily and is readily extensible. We explain the basic principles of our code and give numerical experiments that underline its flexibility as well as its efficiency.

math.NA↗

arXiv2021

Goal-oriented adaptive finite element method for semilinear elliptic PDEs

We formulate and analyze a goal-oriented adaptive finite element method (GOAFEM) for a semilinear elliptic PDE and a linear goal functional. The strategy involves the finite element solution of a linearized dual problem, where the linearization is part of the adaptive strategy. Linear convergence and optimal algebraic convergence rates are shown.

math.NA↗

arXiv2021

Adaptive FEM for parameter-errors in elliptic linear-quadratic parameter estimation problems

We consider an elliptic linear-quadratic parameter estimation problem with a finite number of parameters. A novel a priori bound for the parameter error is proved and, based on this bound, an adaptive finite element method driven by an a posteriori error estimator is presented. Unlike prior results in the literature, our estimator, which is composed of standard energy error residual estimators for the state equation and suitable co-state problems, reflects the faster convergence of the parameter error compared to the (co)-state variables. We show optimal convergence rates of our method; in particular and unlike prior works, we prove that the estimator decreases with a rate that is the sum of the best approximation rates of the state and co-state variables. Experiments confirm that our method matches the convergence rate of the parameter error.

math.NA↗

arXiv2021

Goal-oriented adaptive finite element methods with optimal computational complexity

We consider a linear symmetric and elliptic PDE and a linear goal functional. We design and analyze a goal-oriented adaptive finite element method, which steers the adaptive mesh-refinement as well as the approximate solution of the arising linear systems by means of a contractive iterative solver like the optimally preconditioned conjugate gradient method or geometric multigrid. We prove linear convergence of the proposed adaptive algorithm with optimal algebraic rates. Unlike prior work, we do not only consider rates with respect to the number of degrees of freedom but even prove optimal complexity, i.e., optimal convergence rates with respect to the total computational cost.

math.NA↗

arXiv2020

Enhancement of impact ionization in Hubbard clusters by disorder and next-nearest-neighbor hopping

We perform time-resolved exact diagonalization of the Hubbard model with time dependent hoppings on small clusters of up to $12$ sites. Here, the time dependence originates from a classic electromagnetic pulse, which mimics the impact of a photon. We investigate the behavior of the double occupation and spectral function after the pulse for different cluster geometries and on-site potentials. We find impact ionization in all studied geometries except for one-dimensional chains. Adding next-nearest neighbor hopping to the model leads to a significant enhancement of impact ionization, as does disorder and geometric frustration of a triangular lattice.

cond-mat.str-el↗

arXiv2020

Electron-light interaction in nonequilibrium -- exact diagonalization for time dependent Hubbard Hamiltonians

We present a straightforward implementation scheme for solving the time dependent Schr\"odinger equation for systems described by the Hubbard Hamiltonian with time dependent hoppings. The computations can be performed for clusters of up to 14 sites with in principle general geometry. For the time evolution, we use the exponential midpoint rule, where the exponentials are computed via a Krylov subspace method, which only uses matrix-vector multiplication. The presented implementation uses standard libraries for constructing sparse matrices and for linear algebra therefore the approach is easy to use on both desktop computer and computational cluster. We apply the method to calculate time evolution of double occupation and nonequilibrium spectral function of a photo-excited Mott-insulator. The results show that not only the double occupancy increases due to creation of electron-hole pairs but also the Mott gap becomes partially filled.

cond-mat.str-el↗

arXiv2020

Optimal convergence rates for goal-oriented FEM with quadratic goal functional

We consider a linear elliptic PDE and a quadratic goal functional. The goal-oriented adaptive FEM algorithm (GOAFEM) solves the primal as well as a dual problem, where the goal functional is always linearized around the discrete primal solution at hand. We show that the marking strategy proposed in [Feischl et al, SIAM J. Numer. Anal., 54 (2016)] for a linear goal functional is also optimal for quadratic goal functionals, i.e., GOAFEM leads to linear convergence with optimal convergence rates.

math.NA↗

arXiv2019

Weak-strong uniqueness for the Landau-Lifshitz-Gilbert equation in micromagnetics

We consider the time-dependent Landau-Lifshitz-Gilbert equation. We prove that each weak solution coincides with the (unique) strong solution, as long as the latter exists in time. Unlike available results in the literature, our analysis also includes the physically relevant lower-order terms like Zeeman contribution, anisotropy, stray field, and the Dzyaloshinskii-Moriya interaction (which accounts for the emergence of magnetic Skyrmions). Moreover, our proof gives a template on how to approach weak-strong uniqueness for even more complicated problems, where LLG is (nonlinearly) coupled to other (nonlinear) PDE systems.

math.AP↗

arXiv2019

Instance-optimal goal-oriented adaptivity

We consider an adaptive finite element method with arbitrary but fixed polynomial degree $p \ge 1$, where adaptivity is driven by an edge-based residual error estimator. Based on the modified maximum criterion from [Diening et al, Found. Comput. Math. 16, 2016], we propose a goal-oriented adaptive algorithm and prove that it is instance optimal. More precisely, the goal-error is bounded by the product of the total errors (being the sum of energy error plus data oscillations) of the primal and the dual problem, and the proposed algorithm is instance optimal with respect to this upper bound. Numerical experiments underline our theoretical findings.

math.NA↗