arXiv subjects

Maximilian Brunner

Publications and source records attributed to Maximilian Brunner.

16 recordsLinked to original sources

arXiv2026

Optimal convergence of adaptive BEM driven by functional-type error estimators

In the present work, we derive functional upper bounds for the potential error arising from boundary element discretizations of the Laplace-Dirichlet problem. These bounds are based on local auxiliary problems on patches of boundary vertices and the resulting a posteriori error estimator is shown to be locally equivalent to the well-studied residual error estimator. This equivalence result allows us to prove R-linear convergence of the functional a posteriori error estimator and, together with a suitable mesh-refining strategy, to establish that the potential error as well as the functional error estimator converge with optimal rates with respect to the number of boundary elements. Numerical experiments affirm the theoretical findings and illustrate the practical performance of the related adaptive algorithm driven by the proposed functional error estimator.

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arXiv2026

Quasi-optimal complexity of iterative Galerkin methods driven by an elliptic reconstruction error estimator

We study an iterative Galerkin method for quasilinear elliptic problems in the Browder-Minty setting. The resulting discrete nonlinear systems are solved by linearization via a (damped) Zarantonello iteration. Unlike prior work, adaptive mesh refinement is driven by an elliptic reconstruction error estimator, which is natural in the sense that the a posteriori bounds for the linearization and discretization errors are well separated. For this setting, we present the first comprehensive convergence analysis of the corresponding algorithm. We prove unconditional full R-linear convergence of a suitable quasi-error that combines linearization and discretization errors. For sufficiently small adaptivity parameters, we further establish optimal convergence rates with respect to the number of degrees of freedom and quasi-optimal complexity, i.e., optimal convergence rates with respect to the overall computational cost. Numerical experiments underpin the theoretical findings.

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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.

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arXiv2025

Newton's method in adaptive iteratively linearized FEM

This paper concerns the inclusion of Newton's method into an adaptive finite element method (FEM) for the solution of nonlinear partial differential equations (PDEs). It features an adaptive choice of the damping parameter in the Newton iteration for the discretized nonlinear problems on each level ensuring both global linear and local quadratic convergence. In contrast to energy-based arguments in the literature, a novel approach in the analysis considers the discrete dual norm of the residual as a computable measure for the linearization error. As a consequence, this paper provides the first convergence analysis with optimal rates of an adaptive iteratively linearized FEM beyond energy-minimization problems. The presented theory applies to strongly monotone operators with locally Lipschitz continuous Fr\'echet derivative. We present a class of semilinear PDEs fitting into this framework and provide numerical experiments to underline the theoretical results.

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arXiv2024

Cost-optimal adaptive FEM with linearization and algebraic solver for semilinear elliptic PDEs

We consider scalar semilinear elliptic PDEs, where the nonlinearity is strongly monotone, but only locally Lipschitz continuous. To linearize the arising discrete nonlinear problem, we employ a damped Zarantonello iteration, which leads to a linear Poisson-type equation that is symmetric and positive definite. The resulting system is solved by a contractive algebraic solver such as a multigrid method with local smoothing. We formulate a fully adaptive algorithm that equibalances the various error components coming from mesh refinement, iterative linearization, and algebraic solver. We prove that the proposed adaptive iteratively linearized finite element method (AILFEM) guarantees convergence with optimal complexity, where the rates are understood with respect to the overall computational cost (i.e., the computational time). Numerical experiments investigate the involved adaptivity parameters.

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arXiv2023

Optimal complexity of goal-oriented adaptive FEM for nonsymmetric linear elliptic PDEs

We analyze a goal-oriented adaptive algorithm that aims to efficiently compute the quantity of interest $G(u^\star)$ with a linear goal functional $G$ and the solution $u^\star$ to a general second-order nonsymmetric linear elliptic partial differential equation. The current state of the analysis of iterative algebraic solvers for nonsymmetric systems lacks the contraction property in the norms that are prescribed by the functional analytic setting. This seemingly prevents their application in the optimality analysis of goal-oriented adaptivity. As a remedy, this paper proposes a goal-oriented adaptive iteratively symmetrized finite element method (GOAISFEM). It employs a nested loop with a contractive symmetrization procedure, e.g., the Zarantonello iteration, and a contractive algebraic solver, e.g., an optimal multigrid solver. The various iterative procedures require well-designed stopping criteria such that the adaptive algorithm can effectively steer the local mesh refinement and the computation of the inexact discrete approximations. The main results consist of full linear convergence of the proposed adaptive algorithm and the proof of optimal convergence rates with respect to both degrees of freedom and total computational cost (i.e., optimal complexity). Numerical experiments confirm the theoretical results and investigate the selection of the parameters.

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arXiv2023

Design and Control of a Micro Overactuated Aerial Robot with an Origami Delta Manipulator

This work presents the mechanical design and control of a novel small-size and lightweight Micro Aerial Vehicle (MAV) for aerial manipulation. To our knowledge, with a total take-off mass of only 2.0 kg, the proposed system is the most lightweight Aerial Manipulator (AM) that has 8-DOF independently controllable: 5 for the aerial platform and 3 for the articulated arm. We designed the robot to be fully-actuated in the body forward direction. This allows independent pitching and instantaneous force generation, improving the platform's performance during physical interaction. The robotic arm is an origami delta manipulator driven by three servomotors, enabling active motion compensation at the end-effector. Its composite multimaterial links help reduce the weight, while their flexibility allow for compliant aerial interaction with the environment. In particular, the arm's stiffness can be changed according to its configuration. We provide an in depth discussion of the system design and characterize the stiffness of the delta arm. A control architecture to deal with the platform's overactuation while exploiting the delta arm is presented. Its capabilities are experimentally illustrated both in free flight and physical interaction, highlighting advantages and disadvantages of the origami's folding mechanism.

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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.

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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.

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arXiv2022

MPC with Learned Residual Dynamics with Application on Omnidirectional MAVs

The growing field of aerial manipulation often relies on fully actuated or omnidirectional micro aerial vehicles (OMAVs) which can apply arbitrary forces and torques while in contact with the environment. Control methods are usually based on model-free approaches, separating a high-level wrench controller from an actuator allocation. If necessary, disturbances are rejected by online disturbance observers. However, while being general, this approach often produces sub-optimal control commands and cannot incorporate constraints given by the platform design. We present two model-based approaches to control OMAVs for the task of trajectory tracking while rejecting disturbances. The first one optimizes wrench commands and compensates model errors by a model learned from experimental data. The second one optimizes low-level actuator commands, allowing to exploit an allocation nullspace and to consider constraints given by the actuator hardware. The efficacy and real-time feasibility of both approaches is shown and evaluated in real-world experiments.

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arXiv2022

Energy Tank-Based Policies for Robust Aerial Physical Interaction with Moving Objects

Although manipulation capabilities of aerial robots greatly improved in the last decade, only few works addressed the problem of aerial physical interaction with dynamic environments, proposing strongly model-based approaches. However, in real scenarios, modeling the environment with high accuracy is often impossible. In this work we aim at developing a control framework for OMAVs for reliable physical interaction tasks with articulated and movable objects in the presence of possibly unforeseen disturbances, and without relying on an accurate model of the environment. Inspired by previous applications of energy-based controllers for physical interaction, we propose a passivity-based impedance and wrench tracking controller in combination with a momentum-based wrench estimator. This is combined with an energy-tank framework to guarantee the stability of the system, while energy and power flow-based adaptation policies are deployed to enable safe interaction with any type of passive environment. The control framework provides formal guarantees of stability, which is validated in practice considering the challenging task of pushing a cart of unknown mass, moving on a surface of unknown friction, as well as subjected to unknown disturbances. For this scenario, we present, evaluate and discuss three different policies.

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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.

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arXiv2020

Learning dynamics for improving control of overactuated flying systems

Overactuated omnidirectional flying vehicles are capable of generating force and torque in any direction, which is important for applications such as contact-based industrial inspection. This comes at the price of an increase in model complexity. These vehicles usually have non-negligible, repetitive dynamics that are hard to model, such as the aerodynamic interference between the propellers. This makes it difficult for high-performance trajectory tracking using a model-based controller. This paper presents an approach that combines a data-driven and a first-principle model for the system actuation and uses it to improve the controller. In a first step, the first-principle model errors are learned offline using a Gaussian Process (GP) regressor. At runtime, the first-principle model and the GP regressor are used jointly to obtain control commands. This is formulated as an optimization problem, which avoids ambiguous solutions present in a standard inverse model in overactuated systems, by only using forward models. The approach is validated using a tilt-arm overactuated omnidirectional flying vehicle performing attitude trajectory tracking. The results show that with our proposed method, the attitude trajectory error is reduced by 32% on average as compared to a nominal PID controller.

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arXiv2020

Active Interaction Force Control for Contact-Based Inspection with a Fully Actuated Aerial Vehicle

This paper presents and validates active interaction force control and planning for fully actuated and omnidirectional aerial manipulation platforms, with the goal of aerial contact inspection in unstructured environments. We present a variable axis-selective impedance control which integrates direct force control for intentional interaction, using feedback from an on-board force sensor. The control approach aims to reject disturbances in free flight, while handling unintentional interaction, and actively controlling desired interaction forces. A fully actuated and omnidirectional tilt-rotor aerial system is used to show capabilities of the control and planning methods. Experiments demonstrate disturbance rejection, push-and-slide interaction, and force controlled interaction in different flight orientations. The system is validated as a tool for non-destructive testing of concrete infrastructure, and statistical results of

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arXiv2020

Design and optimal control of a tiltrotor micro aerial vehicle for efficient omnidirectional flight

Omnidirectional micro aerial vehicles are a growing field of research, with demonstrated advantages for aerial interaction and uninhibited observation. While systems with complete pose omnidirectionality and high hover efficiency have been developed independently, a robust system that combines the two has not been demonstrated to date. This paper presents the design and optimal control of a novel omnidirectional vehicle that can exert a wrench in any orientation while maintaining efficient flight configurations. The system design is motivated by the result of a morphology design optimization. A six degrees of freedom optimal controller is derived, with an actuator allocation approach that implements task prioritization, and is robust to singularities. Flight experiments demonstrate and verify the system's capabilities.

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arXiv2019

An Omnidirectional Aerial Manipulation Platform for Contact-Based Inspection

This paper presents an omnidirectional aerial manipulation platform for robust and responsive interaction with unstructured environments, toward the goal of contact-based inspection. The fully actuated tilt-rotor aerial system is equipped with a rigidly mounted end-effector, and is able to exert a 6 degree of freedom force and torque, decoupling the system's translational and rotational dynamics, and enabling precise interaction with the environment while maintaining stability. An impedance controller with selective apparent inertia is formulated to permit compliance in certain degrees of freedom while achieving precise trajectory tracking and disturbance rejection in others. Experiments demonstrate disturbance rejection, push-and-slide interaction, and on-board state estimation with depth servoing to interact with local surfaces. The system is also validated as a tool for contact-based non-destructive testing of concrete infrastructure.

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