Karsten Urban
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View article: Fractional differential equations: non-constant coefficients, simulation and model reduction
Fractional differential equations: non-constant coefficients, simulation and model reduction Open
We consider boundary value problems with Riemann-Liouville fractional derivatives of order $s\in (1, 2)$ with non-constant diffusion and reaction coefficients. A variational formulation is derived and analyzed leading to the well-posedness…
View article: The Kolmogorov N-width for linear transport: exact representation and the influence of the data
The Kolmogorov N-width for linear transport: exact representation and the influence of the data Open
The Kolmogorov N -width describes the best possible error one can achieve by elements of an N -dimensional linear space. Its decay has extensively been studied in approximation theory and for the solution of partial differential equations …
View article: A posteriori certification for neural network approximations to PDEs
A posteriori certification for neural network approximations to PDEs Open
We propose rigorous lower and upper bounds for neural network (NN) approximations to PDEs by efficiently computing the Riesz representations of suitable extension and restrictions of the NN residual towards geometrically simpler domains, w…
View article: Efficient evaluation of the Jacobian in the damped least-squares method for optical design problems using algorithmic differentiation
Efficient evaluation of the Jacobian in the damped least-squares method for optical design problems using algorithmic differentiation Open
The fast evaluation of the Jacobian is an essential part of the optimization of optical systems using the damped least-squares algorithm. While finite differences provide an intuitive way to approximate derivatives, algorithmic differentia…
View article: A parallel batch greedy algorithm in reduced basis methods: Convergence rates and numerical results
A parallel batch greedy algorithm in reduced basis methods: Convergence rates and numerical results Open
The "classical" (weak) greedy algorithm is widely used within model order reduction in order to compute a reduced basis in the offline training phase: An a posteriori error estimator is maximized and the snapshot corresponding to the maxim…
View article: An ultra-weak space-time variational formulation for the Schrödinger equation
An ultra-weak space-time variational formulation for the Schrödinger equation Open
We present a well-posed ultra-weak space-time variational formulation for the time-dependent version of the linear Schrödinger equation with an instationary Hamiltonian. We prove optimal inf-sup stability and introduce a space-time Petrov-…
View article: Differential ray tracing for an efficient computation of the Jacobian in the damped least-squares algorithm
Differential ray tracing for an efficient computation of the Jacobian in the damped least-squares algorithm Open
We present a framework for the optimization of optical systems based up on the damped least-squares method and differential ray tracing. First, a mathematical description for the differentiation of the ray-surface intersection is introduce…
View article: The Kolmogorov N-width for linear transport: Exact representation and the influence of the data
The Kolmogorov N-width for linear transport: Exact representation and the influence of the data Open
The Kolmogorov $N$-width describes the best possible error one can achieve by elements of an $N$-dimensional linear space. Its decay has extensively been studied in Approximation Theory and for the solution of Partial Differential Equation…
View article: A certified wavelet-based physics-informed neural network for the solution of parameterized partial differential equations
A certified wavelet-based physics-informed neural network for the solution of parameterized partial differential equations Open
Physics Informed Neural Networks (PINNs) have frequently been used for the numerical approximation of Partial Differential Equations (PDEs). The goal of this paper is to construct PINNs along with a computable upper bound of the error, whi…
View article: An Ultra-Weak Space-Time Variational Formulation for the Schrödinger Equation
An Ultra-Weak Space-Time Variational Formulation for the Schrödinger Equation Open
We present a well-posed ultra-weak space-time variational formulation for the time-dependent version of the linear Schrödinger equation with an instationary Hamiltonian. We prove optimal inf-sup stability and introduce a space-time Petrov-…
View article: A certified wavelet-based physics-informed neural network for the solution of parameterized partial differential equations
A certified wavelet-based physics-informed neural network for the solution of parameterized partial differential equations Open
Physics Informed Neural Networks (PINNs) have frequently been used for the numerical approximation of Partial Differential Equations (PDEs). The goal of this paper is to construct PINNs along with a computable upper bound of the error, whi…
View article: An Ultraweak Variational Method for Parameterized Linear Differential-Algebraic Equations
An Ultraweak Variational Method for Parameterized Linear Differential-Algebraic Equations Open
We investigate an ultraweak variational formulation for (parameterized) linear differential-algebraic equations with respect to the time variable which yields an optimally stable system. This is used within a Petrov-Galerkin method to deri…
View article: An ultraweak space-time variational formulation for the wave equation: Analysis and efficient numerical solution
An ultraweak space-time variational formulation for the wave equation: Analysis and efficient numerical solution Open
We introduce an ultraweak space-time variational formulation for the wave equation, prove its well-posedness (even in the case of minimal regularity) and optimal inf-sup stability. Then, we introduce a tensor product-style space-time Petro…
View article: An ultraweak variational method for parameterized linear differential-algebraic equations
An ultraweak variational method for parameterized linear differential-algebraic equations Open
We investigate an ultraweak variational formulation for (parameterized) linear differential-algebraic equations (DAEs) w.r.t. the time variable which yields an optimally stable system. This is used within a Petrov-Galerkin method to derive…
View article: Solving parametric PDEs with an enhanced model order reduction method based on Linear/Ridge expansions
Solving parametric PDEs with an enhanced model order reduction method based on Linear/Ridge expansions Open
Classical projection-based model order reduction methods, like the reduced basis method, are popular tools for getting efficiently solvable reduced order models for parametric PDEs.However, for some problems, the error-decay with respect t…
View article: Very Weak Space-Time Variational Formulation for the Wave Equation: Analysis and Efficient Numerical Solution
Very Weak Space-Time Variational Formulation for the Wave Equation: Analysis and Efficient Numerical Solution Open
We introduce a very weak space-time variational formulation for the wave equation, prove its well-posedness (even in the case of minimal regularity) and optimal inf-sup stability. Then, we introduce a tensor product-style space-time Petrov…
View article: Very Weak Space-Time Variational Formulation for the Wave Equation:\n Analysis and Efficient Numerical Solution
Very Weak Space-Time Variational Formulation for the Wave Equation:\n Analysis and Efficient Numerical Solution Open
We introduce a very weak space-time variational formulation for the wave\nequation, prove its well-posedness (even in the case of minimal regularity) and\noptimal inf-sup stability. Then, we introduce a tensor product-style space-time\nPet…
View article: Linear/Ridge expansions: Enhancing linear approximations by ridge functions
Linear/Ridge expansions: Enhancing linear approximations by ridge functions Open
We consider approximations formed by the sum of a linear combination of given functions enhanced by ridge functions -- a Linear/Ridge expansion. For an explicitly or implicitly given function, we reformulate finding a best Linear/Ridge exp…
View article: Simulating Metaphyseal Fracture Healing in the Distal Radius
Simulating Metaphyseal Fracture Healing in the Distal Radius Open
Simulating diaphyseal fracture healing via numerical models has been investigated for a long time. It is apparent from in vivo studies that metaphyseal fracture healing should follow similar biomechanical rules although the speed and heali…
View article: A Space-Time Variational Method for Optimal Control Problems: Well-posedness, stability and numerical solution
A Space-Time Variational Method for Optimal Control Problems: Well-posedness, stability and numerical solution Open
We consider an optimal control problem constrained by a parabolic partial differential equation (PDE) with Robin boundary conditions. We use a well-posed space-time variational formulation in Lebesgue--Bochner spaces with minimal regularit…
View article: A Space-Time Variational Method for Optimal Control Problems.
A Space-Time Variational Method for Optimal Control Problems. Open
We consider a space-time variational formulation of a PDE-constrained optimal control problem with box constraints on the control and a parabolic PDE with Robin boundary conditions. In this setting, the optimal control problem reduces to a…
View article: Decay of the Kolmogorov $N$-width for wave problems
Decay of the Kolmogorov $N$-width for wave problems Open
The Kolmogorov $N$-width $d_N(\mathcal{M})$ describes the rate of the worst-case error (w.r.t.\ a subset $\mathcal{M}\subset H$ of a normed space $H$) arising from a projection onto the best-possible linear subspace of $H$ of dimension $N\…
View article: Modelling the fracture-healing process as a moving-interface problem using an interface-capturing approach
Modelling the fracture-healing process as a moving-interface problem using an interface-capturing approach Open
We present a novel numerical model of the fracture-healing process using interface-capturing techniques, a well-known approach from fields like fluid dynamics, to describe tissue growth. One advantage of this method is its direct connectio…
View article: HT-AWGM: A Hierarchical Tucker-Adaptive Wavelet Galerkin Method for High\n Dimensional Elliptic Problems
HT-AWGM: A Hierarchical Tucker-Adaptive Wavelet Galerkin Method for High\n Dimensional Elliptic Problems Open
This paper is concerned with the construction, analysis and realization of a\nnumerical method to approximate the solution of high dimensional elliptic\npartial differential equations. We propose a new combination of an Adaptive\nWavelet G…