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Alex Kaltenbach

Publications and source records attributed to Alex Kaltenbach.

At least 19 recordsLinked to original sources
arXiv2026

QH-GEM: Quantum-Hydrodynamic Generative Modeling

In this paper, we develop a deterministic, physically constrained generative framework based on the Madelung formulation of the free-particle Schr\"odinger equation. A reference Born probability density and a controllable initial phase function serve as initial data for the free Madelung system, which couples the Born probability density and phase function through the Bohm quantum potential, while the phase function determines the hydrodynamic velocity field. Provided that the Born probability density remains positive and the hydrodynamic velocity field generates a unique global characteristic flow, samples drawn from the reference density and transported along the characteristic flow are distributed according to the evolving Born probability density at every time. As a consequence, randomness enters only through the initial sampling; the subsequent generation is deterministic and involves neither stochastic dynamics nor an independently parameterized time-dependent velocity field. We formulate terminal-time distribution matching as a PDE-constrained phase-identification problem and derive the underlying Hamiltonian and Fisher-information structure. For isotropic Gaussian wave packets, we obtain explicit dynamics and a necessary and sufficient condition for exact reachability of isotropic Gaussian targets by quadratic initial phase functions, together with the corresponding sampling map. For a smooth prescribed potential initial velocity field, we further establish that the characteristic flow approximates the associated first-order transport map with an O(T^2) error, both uniformly and in the 1- and 2-Wasserstein distances. A numerical Gaussian benchmark validates the fully discrete forward solver, while full-grid PDE-constrained phase identification is demonstrated for asymmetric bimodal targets.

math.NA↗
arXiv2026

Duality-Based $\textit{A Posteriori}$ Error Identities for Subgradient Flows Based on the Br\'ezis-Ekeland-Nayroles Principle

We derive duality-based $\textit{a posteriori}$ error identities for a broad class of subgradient flows induced by time-dependent convex integral functionals. Starting from the Br\'ezis-Ekeland-Nayroles principle, we identify an unsteady primal energy functional and derive its Fenchel dual formulation, including strong duality and the corresponding optimality system under general normal-integrand assumptions. This Fenchel duality framework is used to derive $\textit{a posteriori}$ error identities for subgradient flows. In doing so, we depart from the usual duality-based $\textit{a posteriori}$ error control framework in the unsteady setting, since the Br\'ezis-Ekeland-Nayroles formulation reveals the following unsteady feature: the minimal primal value and the maximal dual value are both prescribed by the initial datum. This allows us to pass from a combined primal-dual gap identity to separate primal and dual gap identities. These identities quantify the primal and dual errors independently and admit representations in terms of generalized Bregman divergences and, under a spatial convex conjugation formula, as non-negative time-space integral quantities suitable for localization. The abstract framework is applied to a number of variational problems of physical interest, including the unsteady heat equation, the unsteady Stokes equations, the unsteady Navier-Lam\'e equations, the unsteady Bingham flow through a pipe, the unsteady obstacle problem, and the unsteady elasto-plastic torsion problem.

math.NA↗
arXiv2026

A $\operatorname{prox}$-Based Semi-Smooth Newton Method for Convex Variational Problems

In this paper, we devise a $\operatorname{prox}$-based semi-smooth Newton method that is applicable to a finite element discretization of a broad class of nonsmooth convex variational problems, including the TV-minimization problem, the $p$-Dirichlet problem, the obstacle problem, and the elasto-plastic torsion problem. To this end, on the basis of the proximity operator, the discrete primal-dual optimality conditions are reformulated as nonlinear operator equations with Newton-differentiable structure. Under suitable assumptions on the energy densities, we establish the global well-posedness and local super-linear convergence of the resulting semi-smooth Newton method. The proposed approach coincides with established semi-smooth Newton methods for obstacle-type problems, satisfies a primal-dual invariance, and, under suitable additional assumptions, is globally well-posed in the infinite-dimensional setting.

math.OC↗
arXiv2026

A Finite Element Approximation of an Optimal Insulation Problem with Convective Heat Transfer

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 $\Omega\subseteq\mathbb{R}^d$, with $d\in \{2,3\}$, and the distribution of a given amount of insulation material on an insulated boundary part $\Gamma_I\subseteq \partial\Omega$. The surface integral over the insulated boundary $\Gamma_I$ 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 $L^\infty(\Gamma_I)$-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.

math.NA↗
arXiv2026

A $\operatorname{prox}$-Based Semi-Smooth Newton Method for TV-Minimization

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.

math.NA↗
arXiv2025

Convection Effects and Optimal Insulation: Modelling and Analysis

In this paper, we study an insulation problem that seeks to determine the optimal distribution of a given amount $m>0$ of insulating material coating an insulated boundary part $\Gamma_I\subseteq \partial\Omega$ of a thermally conducting body $\Omega\subseteq \mathbb{R}^d$, $d\in \mathbb{N}$, subject to convective heat transfer. The `$\textit{thickness}$' of the insulating layer $\Sigma_{I}^{\varepsilon}\subseteq \mathbb{R}^d$ is given locally via $\varepsilon \mathtt{d}$, where $\varepsilon>0$ denotes the (arbitrarily small) conductivity and $\mathtt{d}\colon \Gamma_{I}\to [0,+\infty)$ the (to be determined) distribution of the insulating material. Then, the physical process is modelled by the stationary heat equation in the insulated thermally conducting body $\Omega_{I}^{\varepsilon}:= \Omega\cup\Sigma_{I}^{\varepsilon}$ with Robin-type boundary conditions on the interacting insulation boundary $\Gamma_I^{\varepsilon}\subseteq \partial\Omega_{I}^{\varepsilon}$ (reflecting convective heat transfer between the thermally conducting body $\Omega$ and its surrounding medium) as well as Dirichlet and Neumann boundary conditions at the remaining boundary parts, $\textit{i.e.}$, $\partial\Omega_{I}^{\varepsilon}\setminus \Gamma_I^{\varepsilon}$. More precisely, we establish $\Gamma(L^2(\mathbb{R}^d))$-convergence of the heat loss formulation (as ${\varepsilon \to 0^+}$), in the case that the thermally conducting body $\Omega$ is a bounded Lipschitz domain having a $C^{1,1}$-regular or piece-wise flat insulated boundary $\Gamma_I$.

math.AP↗
arXiv2025

Pulsatile Flows for Simplified Smart Fluids with Variable Power-Law: Analysis and Numerics

We study the fully-developed, time-periodic motion of a shear-dependent non-Newtonian fluid with variable exponent rheology through an infinite pipe $\Omega:= \mathbb{R}\times \Sigma\subseteq \mathbb{R}^d$, $d\in \{2,3\}$, of arbitrary cross-section $\Sigma\subseteq \mathbb{R}^{d-1}$. The focus is on a generalized $p(\cdot)$-fluid model, where the power-law index is position-dependent (with respect to $\Sigma$), $\textit{i.e.}$, a function $p\colon \Sigma\to (1,+\infty)$. We prove the existence of time-periodic solutions with either assigned time-periodic flow-rate or pressure-drop, generalizing known results for the Navier-Stokes and for $p$-fluid equations. In addition, we identify explicit solutions, relevant as benchmark cases, especially for electro-rheological fluids or, more generally, $\textit{`smart fluids'}$. To support practical applications, we present a fully-constructive existence proof for variational solutions by means of a fully-discrete finite-differences/-elements discretization, consistent with our numerical experiments. Our approach, which unifies the treatment of all values of $p(\overline{x})\in (1,+\infty)$, $\overline{x}\in \Sigma$, without requiring an auxiliary Newtonian term, provides new insights even in the constant exponent case. The theoretical findings are reviewed by means of numerical experiments.

math.NA↗
arXiv2025

$\textit{A Priori}$ Error Analysis for the $p$-Stokes Equations with Slip Boundary Conditions: A Discrete Leray Projection Framework

We present an $\textit{a priori}$ error analysis for the kinematic pressure in a fully-discrete finite-differences/-elements discretization of the unsteady $p$-Stokes equations, modelling non-Newtonian fluids. This system is subject to both impermeability and perfect Navier slip boundary conditions, which are incorporated either weakly via Lagrange multipliers or strongly in the discrete velocity space. A central aspect of the $\textit{a priori}$ error analysis is the discrete Leray projection, constructed to quantitatively approximate its continuous counterpart. The discrete Leray projection enables a Helmholtz-type decomposition at the discrete level and plays a key role in deriving error decay rates for the kinematic pressure. We derive (in some cases optimal) error decay rates for both the velocity vector field and kinematic pressure, with the error for the kinematic pressure measured in an $\textit{ad hoc}$ norm informed by the projection framework. The $\textit{a priori}$ error analysis remains robust even under reduced regularity of the velocity vector field and the kinematic pressure, and illustrates how the interplay of boundary conditions and projection stability governs the accuracy of pressure approximations.

math.NA↗
arXiv2025

Duality-Based Algorithm and Numerical Analysis for Optimal Insulation Problems on Non-Smooth Domains

This article develops a numerical approximation of a convex non-local and non-smooth minimization problem. The physical problem involves determining the optimal distribution, given by $h\colon \Gamma_I\to [0,+\infty)$, of a given amount $m\in \mathbb{N}$ of insulating material attached to a boundary part $\Gamma_I\subseteq \partial\Omega$ of a thermally conducting body $\Omega \subseteq \mathbb{R}^d$, $d \in \mathbb{N}$, subject to conductive heat transfer. To tackle the non-local and non-smooth character of the problem, the article introduces a (Fenchel) duality framework: (a) At the continuous level, using (Fenchel) duality relations, we derive an a posteriori error identity that can handle arbitrary admissible approximations of the primal and dual formulations of the convex non-local and non-smooth minimization problem; (b) At the discrete level, using discrete (Fenchel) duality relations, we derive an a priori error identity that applies to a Crouzeix--Raviart discretization of the primal formulation and a Raviart--Thomas discretization of the dual formulation. The proposed framework leads to error decay rates that are optimal with respect to the specific regularity of a minimizer. In addition, we prove convergence of the numerical approximation under minimal regularity assumptions. Since the discrete dual formulation can be written as a quadratic program, it is solved using a primal-dual active set strategy interpreted as semismooth Newton method. A solution of the discrete primal formulation is reconstructed from the solution of the discrete dual formulation by means of an inverse generalized Marini formula. This is the first such formula for this class of convex non-local and non-smooth minimization problems.

math.NA↗
arXiv2025

Modeling and Analysis of an Optimal Insulation Problem on Non-Smooth Domains

In this paper, we study an insulation problem that seeks the optimal distribution of a fixed amount $m>0$ of insulating material coating an insulated boundary $\Gamma_I\subseteq \partial\Omega$ of a thermally conducting body $\Omega\subseteq \mathbb{R}^d$, $d\in \mathbb{N}$. The thickness of the thin insulating layer $\Sigma_{I}^{\varepsilon}$ is given locally via $\varepsilon \mathtt{d}$, where $\mathtt{d}\colon \Gamma_{I}\to [0,+\infty)$ specifies the (to be determined) distribution of the insulating material. We establish $\Gamma(L^2(\mathbb{R}^d))$-convergence of the problem (as $\varepsilon\to 0^+$). Different from the existing literature, which predominantly assumes that the thermally conducting body $\Omega$ has a $C^{1,1}$-boundary, we merely assume that $\Gamma_I$ is piece-wise flat. To overcome this lack of boundary regularity, we define the thin insulating boundary layer $\Sigma_{I}^{\varepsilon}$ using a Lipschitz continuous transversal vector field rather than the outward unit normal vector field. The piece-wise flatness condition on $\Gamma_I$ is only needed to prove the $\liminf$-estimate. In fact, for the $\limsup$-estimate is enough that the thermally conducting body $\Omega$ has a $C^{0,1}$-boundary.

math.AP↗
arXiv2025

Error analysis for a fully-discrete finite element approximation of the unsteady $p(\cdot,\cdot)$-Stokes equations

In this paper, we examine a fully-discrete finite element approximation of the unsteady $p(\cdot,\cdot)$-Stokes equations ($i.e.$, $p(\cdot,\cdot)$ is time- and space-dependent), employing a backward Euler step in time and conforming, discretely inf-sup stable finite elements in space. More precisely, we derive error decay rates for the vector-valued velocity field imposing fractional regularity assumptions on the velocity and the kinematic pressure. In addition, we carry out numerical experiments that confirm the optimality of the derived error decay rates in the case $p(\cdot,\cdot)\ge 2$.

math.NA↗
arXiv2024

Variational problems with gradient constraints: $\textit{A priori}$ and $\textit{a posteriori}$ error identities

In this paper, on the basis of a (Fenchel) duality theory on the continuous level, we derive an $\textit{a posteriori}$ error identity for arbitrary conforming approximations of a primal formulation and a dual formulation of variational problems involving gradient constraints. In addition, on the basis of a (Fenchel) duality theory on the discrete level, we derive an $\textit{a priori}$ error identity that applies to the approximation of the primal formulation using the Crouzeix-Raviart element and to the approximation of the dual formulation using the Raviart-Thomas element, and leads to error decay rates that are optimal with respect to the regularity of a dual solution.

math.NA↗
arXiv2024

Finite element discretization of the steady, generalized Navier-Stokes equations for small shear stress exponents

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 $p \leq 2$.

math.NA↗
arXiv2024

Energy conservation for weak solutions of incompressible Newtonian fluid equations in H\"older spaces with Dirichlet boundary conditions in the half-space

We investigate sufficient H\"older continuity conditions on Leray-Hopf (weak) solutions to the in unsteady Navier-Stokes equations in three dimensions guaranteeing energy conservation. Our focus is on the half-space case with homogeneous Dirichlet boundary conditions. This problem is more technically challenging, if compared to the Cauchy or periodic cases, and has not been previously addressed. At present are known a few sub-optimal results obtained through Morrey embedding results based on conditions for the gradient of the velocity in Sobolev spaces. Moreover, the results in this paper are obtained without any additional assumption neither on the pressure nor the flux of the velocity, near to the boundary.

math.AP↗
arXiv2024

$\textit{A priori}$ and $\textit{a posteriori}$ error identities for the scalar Signorini problem

In this paper, on the basis of a (Fenchel) duality theory on the continuous level, we derive an $\textit{a posteriori}$ error identity for arbitrary conforming approximations of the primal formulation and the dual formulation of the scalar Signorini problem. In addition, on the basis of a (Fenchel) duality theory on the discrete level, we derive an $\textit{a priori}$ error identity that applies to the approximation of the primal formulation using the Crouzeix-Raviart element and to the approximation of the dual formulation using the Raviart-Thomas element, and leads to quasi-optimal error decay rates without imposing additional assumptions on the contact set and in arbitrary space dimensions.

math.NA↗
arXiv2024

Convergence analysis for a fully-discrete finite element approximation of the unsteady $p(\cdot,\cdot)$-Navier-Stokes equations

In the present paper, we establish the well-posedness, stability, and (weak) convergence of a fully-discrete approximation of the unsteady $p(\cdot,\cdot)$-Navier-Stokes equations employing an implicit Euler step in time and a discretely inf-sup-stable finite element approximation in space. Moreover, numerical experiments are carried out that supplement the theoretical findings.

math.NA↗
arXiv2024

Exact a posteriori error control for variational problems via convex duality and explicit flux reconstruction

A posteriori error estimates are an important tool to bound discretization errors in terms of computable quantities avoiding regularity conditions that are often difficult to establish. For non-linear and non-differentiable problems, problems involving jumping coefficients, and finite element methods using anisotropic triangulations, such estimates often involve large factors, leading to sub-optimal error estimates. By making use of convex duality arguments, exact and explicit error representations are derived that avoid such effects.

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