Alex Kaltenbach

Alex Kaltenbach, Marius Zeinhofer

We establish error estimates for the approximation of parametric $p$-Dirichlet problems deploying the Deep Ritz Method. Parametric dependencies include, e.g., varying geometries and exponents $p\in (1,\infty)$. Combining the derived error estimates with quantitative approximation theorems yields error decay rates and establishes that the Deep Ritz Method retains the favorable approximation capabilities of neural networks in the approximation of high dimensional functions which makes the method attractive for parametric problems. Finally, we present numerical examples to illustrate potential applications.

Sören Bartels, Alex Kaltenbach

In this paper, we devise a $\operatorname{prox}$-based semi-smooth Newton method for the non-differentiable TV-minimization problem. To this end, the primal-dual optimality conditions are reformulated as a nonlinear operator equation with Newton-(type-)differentiable structure. We investigate the question of well-posedness of the resulting semi-smooth Newton scheme in the infinite-dimensional setting and identify structural properties of the associated Newton-type derivatives. For a conforming finite element discretization, we prove that the resulting semi-smooth Newton method is globally well-posed and locally super-linearly convergent. The approach extends to a large class of convex minimization problems, coincides with established semi-smooth Newton methods for obstacle problems, satisfies a primal-dual invariance, and, under suitable additional assumptions, is well-posed in the infinite-dimensional setting. Numerical experiments indicate a robust practical performance of the proposed method, including reliable reduction of the discrete primal-dual gap estimator to machine precision, robustness with respect to the choice of proximity parameters, an improved convergence basin compared to a canonical primal semi-smooth Newton method, and effective performance even for quadratically graded meshes using only a mesh-independent initialization criterion.

Alex Kaltenbach, Julius Jeßberger

A finite element (FE) discretization for the steady, incompressible, fully inhomogeneous, generalized Navier-Stokes equations is proposed. By the method of divergence reconstruction operators, the formulation is valid for all shear stress exponents $p > \tfrac{2d}{d+2}$. The Dirichlet boundary condition is imposed strongly, using any discretization of the boundary data which converges at a sufficient rate. $\textit{A priori}$ error estimates for the velocity vector field and kinematic pressure are derived and numerical experiments are conducted. These confirm the quasi-optimality of the $\textit{a priori}$ error estimate for the velocity vector field. The $\textit{a priori}$ error estimates for the kinematic pressure are quasi-optimal if $p \leq 2$.

Harbir Antil, Alex Kaltenbach, Keegan L. A. Kirk

A finite element discretization of an optimal insulation problem with convective heat transfer is considered. The model is formulated as a non-smooth, two-variable convex minimization problem. It accounts for the temperature distribution in a thermally conducting body $Ω\subseteq\mathbb{R}^d$, with $d\in \{2,3\}$, and the distribution of a given amount of insulation material on an insulated boundary part $Γ_I\subseteq \partialΩ$. The surface integral over the insulated boundary $Γ_I$ is approximated by a mass-lumping quadrature that preserves the structure of the continuous setting and, in particular, yields discrete optimality conditions mirroring their continuous counterparts. Well-posedness, stability, and weak convergence of discrete solutions to the continuous ones are established. Furthermore, a block coordinate descent algorithm for the computation of the discrete solutions is formulated and its linear convergence is derived. Under suitable regularity assumptions, uniform $L^\infty(Γ_I)$-bounds and $\textit{a priori}$ error estimates for both the temperature distribution and the distribution of a given amount of insulation material are obtained. Numerical experiments are carried out that confirm the predicted error decay rates and demonstrate the method in a qualitative three-dimensional test on a realistic spacecraft crew module capsule geometry with idealized reentry-heating Robin data.