Pascal Heid
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View article: Approximation Theory, Computing, and Deep Learning on the Wasserstein Space
Approximation Theory, Computing, and Deep Learning on the Wasserstein Space Open
The challenge of approximating functions in infinite-dimensional spaces from finite samples is widely regarded as formidable. We delve into the challenging problem of the numerical approximation of Sobolev-smooth functions defined on proba…
View article: A damped Kačanov scheme for the numerical solution of a relaxed p(x)-Poisson equation
A damped Kačanov scheme for the numerical solution of a relaxed p(x)-Poisson equation Open
The focus of the present work is the (theoretical) approximation of a solution of the p ( x )-Poisson equation. To devise an iterative solver with guaranteed convergence, we will consider a relaxation of the original problem in terms of a …
View article: A short note on an adaptive damped Newton method for strongly monotone and Lipschitz continuous operator equations
A short note on an adaptive damped Newton method for strongly monotone and Lipschitz continuous operator equations Open
We consider the damped Newton method for strongly monotone and Lipschitz continuous operator equations in a variational setting. We provide a very accessible justification why the undamped Newton method performs better than its damped coun…
View article: A damped Kačanov scheme for the numerical solution of a relaxed p(x)-Poisson equation
A damped Kačanov scheme for the numerical solution of a relaxed p(x)-Poisson equation Open
The focus of the present work is the (theoretical) approximation of a solution of the p(x)-Poisson equation. To devise an iterative solver with guaranteed convergence, we will consider a relaxation of the original problem in terms of a tru…
View article: An adaptive damped Newton method for strongly monotone and Lipschitz continuous operator equations
An adaptive damped Newton method for strongly monotone and Lipschitz continuous operator equations Open
We will consider the damped Newton method for strongly monotone and Lipschitz continuous operator equations in a variational setting. We will provide a very accessible justification why the undamped Newton method performs better than its d…
View article: Nonlinear iterative approximation of steady incompressible chemically reacting flows
Nonlinear iterative approximation of steady incompressible chemically reacting flows Open
We consider a system of nonlinear partial differential equations modelling steady flow of an incompressible chemically reacting non-Newtonian fluid, whose viscosity depends on both the shear-rate and the concentration; in particular, the v…
View article: An adaptive iterative linearised finite element method for implicitly constituted incompressible fluid flow problems and its application to Bingham fluids
An adaptive iterative linearised finite element method for implicitly constituted incompressible fluid flow problems and its application to Bingham fluids Open
In this work, we further extend the theory in [C. Kreuzer, E. Süli, Adaptive finite element approximation of steady flows of incompressible fluids with implicit power-law-like rheology, ESAIM: Math. Model. Numer. Anal. 50 (5) (2016) 1333–1…
View article: A numerical energy minimisation approach for semilinear diffusion-reaction boundary value problems based on steady state iterations
A numerical energy minimisation approach for semilinear diffusion-reaction boundary value problems based on steady state iterations Open
We present a novel energy-based numerical analysis of semilinear diffusion-reaction boundary value problems. Based on a suitable variational setting, the proposed computational scheme can be seen as an energy minimisation approach. More sp…
View article: A modified Kačanov iteration scheme with application to quasilinear diffusion models
A modified Kačanov iteration scheme with application to quasilinear diffusion models Open
The classical Kačanov scheme for the solution of nonlinear variational problems can be interpreted as a fixed point iteration method that updates a given approximation by solving a linear problem in each step. Based on this observation, we…
View article: On the convergence rate of the Kačanov scheme for shear-thinning fluids
On the convergence rate of the Kačanov scheme for shear-thinning fluids Open
We explore the convergence rate of the Kačanov iteration scheme for different models of shear-thinning fluids, including Carreau and power-law type explicit quasi-Newtonian constitutive laws. It is shown that the energy difference contract…
View article: A link between the steepest descent method and fixed-point iterations
A link between the steepest descent method and fixed-point iterations Open
We will make a link between the steepest descent method for an unconstrained minimisation problem and fixed-point iterations for its Euler-Lagrange equation. In this context, we shall rediscover the preconditioned nonlinear conjugate gradi…
View article: Adaptive iterative linearised finite element methods for implicitly constituted incompressible fluid flow problems and its application to Bingham fluids
Adaptive iterative linearised finite element methods for implicitly constituted incompressible fluid flow problems and its application to Bingham fluids Open
In this work, we introduce an iterative linearised finite element method for the solution of Bingham fluid flow problems. The proposed algorithm has the favourable property that a subsequence of the sequence of iterates generated converges…
View article: Energy Contraction and Optimal Convergence of Adaptive Iterative Linearized Finite Element Methods
Energy Contraction and Optimal Convergence of Adaptive Iterative Linearized Finite Element Methods Open
We revisit a unified methodology for the iterative solution of nonlinear equations in Hilbert spaces. Our key observation is that the general approach from [P. Heid and T. P. Wihler, Adaptive iterative linearization Galerkin methods for no…
View article: A modified Kačanov iteration scheme with application to quasilinear diffusion models
A modified Kačanov iteration scheme with application to quasilinear diffusion models Open
The classical Kačanov scheme for the solution of nonlinear variational problems can be interpreted as a fixed point iteration method that updates a given approximation by solving a linear problem in each step. Based on this observation, we…
View article: On the convergence rate of the Ka\v{c}anov scheme for shear-thinning fluids
On the convergence rate of the Ka\v{c}anov scheme for shear-thinning fluids Open
We explore the convergence rate of the Ka\v{c}anov iteration scheme for different models of shear-thinning fluids, including Carreau and power-law type explicit quasi-Newtonian constitutive laws. It is shown that the energy difference cont…
View article: On the convergence rate of the Kačanov scheme for shear-thinning fluids
On the convergence rate of the Kačanov scheme for shear-thinning fluids Open
We explore the convergence rate of the Kačanov iteration scheme for different models of shear-thinning fluids, including Carreau and power-law type explicit quasi-Newtonian constitutive laws. It is shown that the energy difference contract…
View article: Adaptive Local Minimax Galerkin Methods for Variational Problems
Adaptive Local Minimax Galerkin Methods for Variational Problems Open
In many applications of practical interest, solutions of partial differential equation models arise as critical points of an underlying (energy) functional. If such solutions are saddle points, rather than being maxima or minima, then the …
View article: Gradient flow finite element discretisations with energy-based adaptivity for excited states of Schrödingers equation
Gradient flow finite element discretisations with energy-based adaptivity for excited states of Schrödingers equation Open
We present an effective numerical procedure, which is based on the computational scheme from [Heid et al., arXiv:1906.06954], for the numerical approximation of excited states of Schrödingers equation. In particular, this procedure employs…
View article: Energy contraction and optimal convergence of adaptive iterative\n linearized finite element methods
Energy contraction and optimal convergence of adaptive iterative\n linearized finite element methods Open
We revisit a unified methodology for the iterative solution of nonlinear\nequations in Hilbert spaces. Our key observation is that the general approach\nfrom [Heid & Wihler, Math. Comp. 89 (2020), Calcolo 57 (2020)] satisfies an\nenergy co…
View article: Adaptive iterative linearization Galerkin methods for nonlinear problems
Adaptive iterative linearization Galerkin methods for nonlinear problems Open
A wide variety of (fixed-point) iterative methods for the solution of nonlinear equations (in Hilbert spaces) exists. In many cases, such schemes can be interpreted as iterative local linearization methods, which, as will be shown, can be …
View article: Gradient Flow Finite Element Discretizations with Energy-Based\n Adaptivity for the Gross-Pitaevskii Equation
Gradient Flow Finite Element Discretizations with Energy-Based\n Adaptivity for the Gross-Pitaevskii Equation Open
We present an effective adaptive procedure for the numerical approximation of\nthe steady-state Gross-Pitaevskii equation. Our approach is solely based on\nenergy minimization, and consists of a combination of gradient flow iterations\nand…