Philipp Zimmermann

Last Name

Zimmermann

First Name

Philipp

ORCID

0000-0002-6791-1779

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Publications 1 - 10 of 12

Counterexamples to uniqueness in the inverse fractional conductivity problem with partial data
Railo, Jesse; Zimmermann, Philipp (2023)
Inverse Problems and Imaging
Inverse problems for variable coefficient nonlocal operators
Zimmermann, Philipp (2023)
Well-posedness and inverse problems for semilinear nonlocal wave equations
Lin, Yi-Hsuan; Tyni, Teemu; Zimmermann, Philipp (2024)
Nonlinear Analysis
This article is devoted to forward and inverse problems associated with time-independent semilinear nonlocal wave equations. We first establish comprehensive well-posedness results for some semilinear nonlocal wave equations. The main challenge is due to the low regularity of the solutions of linear nonlocal wave equations. We then turn to an inverse problem of recovering the nonlinearity of the equation. More precisely, we show that the exterior Dirichlet-to-Neumann map uniquely determines homogeneous nonlinearities of the form f(x,u) under certain growth conditions. On the other hand, we also prove that initial data can be determined by using passive measurements under certain nonlinearity conditions. The main tools used for the inverse problem are the unique continuation principle of the fractional Laplacian and a Runge approximation property. The results hold for any spatial dimension n∈N.
Low regularity theory for the inverse fractional conductivity problem
Railo, Jesse; Zimmermann, Philipp (2024)
Nonlinear Analysis
The fractional p -biharmonic systems: optimal Poincaré constants, unique continuation and inverse problems
Kar, Manas; Railo, Jesse; Zimmermann, Philipp (2023)
Calculus of Variations and Partial Differential Equations
This article investigates nonlocal, quasilinear generalizations of the classical biharmonic operator (- Δ) 2. These fractional p -biharmonic operators appear naturally in the variational characterization of the optimal fractional Poincaré constants in Bessel potential spaces. We study the following basic questions for anisotropic fractional p -biharmonic systems: existence and uniqueness of weak solutions to the associated interior source and exterior value problems, unique continuation properties, monotonicity relations, and inverse problems for the exterior Dirichlet-to-Neumann maps. Furthermore, we show the UCP for the fractional Laplacian in all Bessel potential spaces Ht,p for any t∈ R, 1 ≤ p< ∞ and s∈ R+\ N: If u∈ Ht,p(Rn) satisfies (- Δ) su= u= 0 in a nonempty open set V, then u≡ 0 in Rn. This property of the fractional Laplacian is then used to obtain a UCP for the fractional p -biharmonic systems and plays a central role in the analysis of the associated inverse problems. Our proofs use variational methods and the Caffarelli–Silvestre extension.
Fractional Calderón problems and Poincaré inequalities on unbounded domains
Railo, Jesse; Zimmermann, Philipp (2023)
Journal of Spectral Theory
We generalize many recent uniqueness results on the fractional Calderón problem to cover the cases of all domains with nonempty exterior. The highlight of our work is the characterization of uniqueness and nonuniqueness of partial data inverse problems for the fractional conductivity equation on domains that are bounded in one direction for conductivities supported in the whole Euclidean space and decaying to a constant background conductivity at infinity. We generalize the uniqueness proof for the fractional Calderón problem by Ghosh, Salo and Uhlmann to a general abstract setting in order to use the full strength of their argument. This allows us to observe that there are also uniqueness results for many inverse problems for higher order local perturbations of a lower order fractional Laplacian. We give concrete example models to illustrate these curious situations and prove Poincaré inequalities for the fractional Laplacians of any order on domains that are bounded in one direction. We establish Runge approximation results in these general settings, improve regularity assumptions also in the cases of bounded sets and prove general exterior determination results. Counterexamples to uniqueness in the inverse fractional conductivity problem with partial data are constructed in another companion work.
Determining coefficients for a fractional p-Laplace equation from exterior measurements
Kar, Manas; Lin, Yi-Hsuan; Zimmermann, Philipp (2024)
Journal of Differential Equations
We consider an inverse problem of determining the coefficients of a fractional p -Laplace equation in the exterior domain. Assuming suitable local regularity of the coefficients in the exterior domain, we offer an explicit reconstruction formula in the region where the exterior measurements are performed. This formula is then used to establish a global uniqueness result for real-analytic coefficients. In addition, we also derive a stability estimate for the unique determination of the coefficients in the exterior measurement set.
The Calderón problem for a nonlocal diffusion equation with time-dependent coefficients
Lin, Yi-Hsuan; Railo, Jesse; Zimmermann, Philipp (2025)
Revista matemática iberoamericana
We investigate the Calderón problem for a nonlocal diffusion equation depending on a globally unknown isotropic coefficient $\gamma$(x,t). The forward problem is posed on Ω × (0,T) for a domain Ω that is bounded in one direction. We first show that the Dirichlet-to-Neumann map $\Lambda_{\gamma}$ determines $\gamma$ in the measurement set. By studying various properties of the related nonlocal Neumann derivatives N$_{\gamma}$, we prove that both quantities $\langle$$\Lambda$$_{\gamma}$f, g$\rangle$ and $\langle$N$_{\gamma}$f, g$\rangle$ carry the same information as long as f, g: R$^n$ \ $\overline{\Omega}$ $\rightarrow$ R have disjoint supports and $\gamma$ is known in supp(f) $\cup$ supp(g). We obtain the desired global uniqueness theorem using a suitable integral identity for N$_{\gamma}$ and the Runge approximation property. The results hold for any spatial dimension n $\geq$ 1. In conclusion, the main observations of this article are twofold: (1) the information of $\Lambda_{\gamma}$ is needed for exterior determination for $\gamma$, (2) the knowledge of N$_{\gamma}$ and $\gamma$ in the measurement set is enough to recover $\gamma$ in the interior.
Space-time error estimates for deep neural network approximations for differential equations
Grohs, Philipp; Hornung, Fabian; Jentzen, Arnulf; et al. (2023)
Advances in Computational Mathematics
Over the last few years deep artificial neural networks (ANNs) have very successfully been used in numerical simulations for a wide variety of computational problems including computer vision, image classification, speech recognition, natural language processing, as well as computational advertisement. In addition, it has recently been proposed to approximate solutions of high-dimensional partial differential equations (PDEs) by means of stochastic learning problems involving deep ANNs. There are now also a few rigorous mathematical results in the scientific literature which provide error estimates for such deep learning based approximation methods for PDEs. All of these articles provide spatial error estimates for ANN approximations for PDEs but do not provide error estimates for the entire space-time error for the considered ANN approximations. It is the subject of the main result of this article to provide space-time error estimates for deep ANN approximations of Euler approximations of certain perturbed differential equations. Our proof of this result is based (i) on a certain ANN calculus and (ii) on ANN approximation results for products of the form [0,T]×Rd∋(t,x)↦tx∈Rd where T∈(0,∞), d∈N, which we both develop within this article.
Switchable topological phonon channels
Süsstrunk, Roman; Zimmermann, Philipp; Huber, Sebastian D. (2017)
New Journal of Physics
Guiding energy deliberately is one of the central elements in engineering and information processing. It is often achieved by designing specific transport channels in a suitable material. Topological metamaterials offer a way to construct stable and efficient channels of unprecedented versatility. However, due to their stability it can be tricky to terminate them or to temporarily shut them off without changing the material properties massively. While a lot of effort was put into realizing mechanical topological metamaterials, almost no works deal with manipulating their edge channels in sight of applications. Here, we take a step in this direction, by taking advantage of local symmetry breaking potentials to build a switchable topological phonon channel.

Publications 1 - 10 of 12

Philipp Zimmermann

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