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Felix Otto

Publications and source records attributed to Felix Otto.

At least 37 records · Page 2Linked to original sources
arXiv2024

Lifshitz Transition and Band Structure Evolution in Alkali Metal Intercalated 1Tprime-MoTe2

In van der Waals materials, coupling between adjacent layers is weak, and consequently interlayer interactions are weakly screened. This opens the possibility to profoundly modify the electronic structure, e.g., by applying electric fields or with adsorbates. Here, we show for the case of the topologically trivial semimetal 1Tprime-MoTe2 that potassium dosing at room temperature significantly transforms its band structure. With a combination of angle-resolved photoemission spectroscopy, scanning tunneling microscopy, x-ray photoemission spectroscopy, and density functional theory we show that i) for small concentrations of K, 1Tprime-MoTe2 undergoes a Lifshitz transition with the electronic structure shifting rigidly, and ii) for larger K concentrations 1Tprime-MoTe2 undergoes significant band structure transformation. Our results demonstrate that the origin of this electronic structure change stems from alkali metal intercalation.

cond-mat.mtrl-sci↗
arXiv2024

A diagram-free approach to the stochastic estimates in regularity structures

In this paper, we explore the version of Hairer's regularity structures based on a greedier index set than trees, as introduced by Otto, Sauer, Smith and Weber. More precisely, we construct and stochastically estimate the renormalized model avoiding the use of Feynman diagrams but still in a fully automated, i. e. inductive way. This is carried out for a class of quasi-linear parabolic PDEs driven by noise in the full singular but renormalizable range. We assume a spectral gap inequality on the (not necessarily Gaussian) noise ensemble. The resulting control on the variance of the model naturally complements its vanishing expectation arising from the BPHZ-choice of renormalization. We capture the gain in regularity on the level of the Malliavin derivative of the model by describing it as a modelled distribution. Symmetry is an important guiding principle and built-in on the level of the renormalization Ansatz. Our approach is analytic and top-down rather than combinatorial and bottom-up.

math.PR↗
arXiv2024

A priori bounds for quasi-linear SPDEs in the full sub-critical regime

This paper is concerned with quasi-linear parabolic equations driven by an additive forcing $ξ\in C^{α-2}$, in the full sub-critical regime $α\in (0,1)$. We are inspired by Hairer's regularity structures, however we work with a more parsimonious model indexed by multi-indices rather than trees. This allows us to capture additional symmetries which play a crucial role in our analysis. Assuming bounds on this model, which is modified in agreement with the concept of algebraic renormalization, we prove local a priori estimates on solutions to the quasi-linear equations modified by the corresponding counter terms.

math.AP↗
arXiv2024

Optimal artificial boundary conditions based on second-order correctors for three dimensional random elliptic media

We are interested in numerical algorithms for computing the electrical field generated by a charge distribution localized on scale $\ell$ in an infinite heterogeneous medium, in a situation where the medium is only known in a box of diameter $L\gg\ell$ around the support of the charge. We propose a boundary condition that with overwhelming probability is (near) optimal with respect to scaling in terms of $\ell$ and $L$, in the setting where the medium is a sample from a stationary ensemble with a finite range of dependence (set to be unity and with the assumption that $\ell \gg 1$). The boundary condition is motivated by quantitative stochastic homogenization that allows for a multipole expansion [BGO20]. This work extends [LO21], the algorithm in which is optimal in two dimension, and thus we need to take quadrupoles, next to dipoles, into account. This in turn relies on stochastic estimates of second-order, next to first-order, correctors. These estimates are provided for finite range ensembles under consideration, based on an extension of the semi-group approach of [GO15].

math.AP↗
arXiv2023

There is no stationary $p$-cyclically monotone Poisson matching in 2D

We show that for $p>1$ there is no $p$-cyclically monotone stationary matching of two independent Poisson processes in dimension $d=2$. The proof combines the $p$-harmonic approximation result from \cite[Theorem 1.1]{koch23} with local asymptotics for the two-dimensional matching problem. Moreover, we prove a.s. local upper bounds of the correct order in the case $p>1$, which, to the best of our knowledge, are not readily available in the current literature.

math.PR↗
arXiv2023

Variational methods for a singular SPDE yielding the universality of the magnetization ripple

The magnetization ripple is a microstructure formed in thin ferromagnetic films. It can be described by minimizers of a nonconvex energy functional leading to a nonlocal and nonlinear elliptic SPDE in two dimensions driven by white noise, which is singular. We address the universal character of the magnetization ripple using variational methods based on $Γ$-convergence. Due to the infinite energy of the system, the (random) energy functional has to be renormalized. Using the topology of $Γ$-convergence, we give a sense to the law of the renormalized functional that is independent of the way white noise is approximated. More precisely, this universality holds in the class of (not necessarily Gaussian) approximations to white noise satisfying the spectral gap inequality, which allows us to obtain sharp stochastic estimates. As a corollary, we obtain the existence of minimizers with optimal regularity.

math.PR↗
arXiv2023

Lecture notes on tree-free regularity structures

These lecture notes are intended as reader's digest of recent work on a diagram-free approach to the renormalized centered model in Hairer's regularity structures. More precisely, it is about the stochastic estimates of the centered model, based on Malliavin calculus and a spectral gap assumption. We focus on a specific parabolic partial differential equation in quasi-linear form driven by (white) noise. We follow a natural renormalization strategy based on preserving symmetries, and carefully introduce Hairer's notion of a centered model, which provides the coefficients in a formal series expansion of a general solution. We explain how the Malliavin derivative in conjunction with Hairer's re-expansion map allows to reformulate this definition in a way that is stable under removing the small-scale regularization. A few exemplary proofs are provided, both of analytic and of algebraic character. The working horse of the analytic arguments is an ``annealed'' Schauder estimate and related Liouville principle, which is provided. The algebra of formal power series, in variables that play the role of coordinates of the solution manifold, and its algebra morphisms are the key algebraic objects.

math.PR↗
arXiv2023

Lecture notes on the harmonic approximation to quadratic optimal transport

These lecture notes present the quantitative harmonic approximation result for quadratic optimal transport and general measures obtained by Goldman and Otto. The aim is to give a clear presentation of the proof of the main theorem with more motivations, less PDE machinery, and a number of simplifications.

math.AP↗
arXiv2023

The structure group for quasi-linear equations via universal enveloping algebras

We consider the approach of replacing trees by multi-indices as an index set of the abstract model space $\mathsf{T}$ introduced by Otto, Sauer, Smith and Weber to tackle quasi-linear singular SPDEs. We show that this approach is consistent with the postulates of regularity structures when it comes to the structure group $\mathsf{G}$. In particular, $\mathsf{G}\subset{\rm Aut}(\mathsf{T})$ arises from a Hopf algebra $\mathsf{T}^+$ and a comodule $Δ\colon\mathsf{T}\rightarrow \mathsf{T}^+\otimes\mathsf{T}$. In fact, this approach, where the dual $\mathsf{T}^*$ of the abstract model space $\mathsf{T}$ naturally embeds into a formal power series algebra, allows to interpret $\mathsf{G}^*\subset{\rm Aut}(\mathsf{T}^*)$ as a Lie group arising from a Lie algebra $\mathsf{L} \subset{\rm End}(\mathsf{T}^*)$ consisting of derivations on this power series algebra. These derivations in turn are the infinitesimal generators of two actions on the space of pairs (nonlinearities, functions of space-time mod constants). These actions are shift of space-time and tilt by space-time polynomials. The Hopf algebra $\mathsf{T}^+$ arises from a coordinate representation of the universal enveloping algebra ${\rm U}(\mathsf{L})$ of the Lie algebra $\mathsf{L}$. The coordinates are determined by an underlying pre-Lie algebra structure of the derived algebra of $\mathsf{L}$. Strong finiteness properties, which are enforced by gradedness and the restrictive definition of $\mathsf{T}$, allow for this purely algebraic construction of $\mathsf{G}$. We also argue that there exist pre-Lie algebra and Hopf algebra morphisms between our structure and the tree-based one in the cases of branched rough paths (Grossman-Larson, Connes-Kreimer) and of the generalized parabolic Anderson model.

math-ph↗
arXiv2022

Heterostructure films of SiO$_2$ and HfO$_2$ for high power laser optics prepared by plasma-enhanced atomic layer deposition

Absorption losses and laser-induced damage threshold (LIDT) are considered as the major constraint for the development of optical coatings for high-power laser optics. Such coatings require paramount properties like low losses due to optical absorption, high mechanical stability, and enhanced damage resistance to withstand high-intensity laser pulses. In this work, heterostructure films were developed by the intermixing of SiO$_2$ and HfO$_2$ using plasma-enhanced atomic layer deposition (PEALD) technique. Thin film characterization techniques such as spectroscopic ellipsometry, spectrophotometry, substrate curvature measurements, x-ray reflectivity, and Fourier transform infrared spectroscopy were employed for extracting optical constants, spectral inter-pretation, residual stress, layer formation, and functional groups present in the heterostructures, respectively. These heterostructures demonstrate tunable refractive index, bandgap, and improved optical losses and LIDT properties. The films were incorporated into antireflection coatings (multilayer stacks and graded index coatings) and the LIDT was determined at 355 nm wavelength by the R-on-1 method. Optical absorptions at the reported wavelengths were characterized using photothermal common-path interferometry and laser-induced deflection techniques.

physics.optics↗
arXiv2022

Atomically Thin Metal-Dielectric Heterostructures by Atomic Layer Deposition

Heterostructures increasingly attracted attention over the past several years to enable various optoelectronic and photonic applications. In this work, atomically thin interfaces of Ir/Al2O3 heterostructures compatible with micro-optoelectronic technologies are reported. Their structural and optical properties were determined by spectroscopic and microscopic techniques (XRR, XPS, HRTEM, spectroscopic ellipsometry, and UV/VIS/NIR spectrophotometry). The XRR and HRTEM analyses reveal a layer-by-layer growth mechanism of Ir in atomic scale heterostructures, which is different from the typical island-type growth of metals on dielectrics. Alongside, XPS investigations imply the formation of Ir-O-Al bonding at the interfaces for lower Ir concentrations, in contrast to the nanoparticle core-shell structure formation. Precisely tuning the ratio of the constituents ensures the control of the dispersion profile along with a transition from effective dielectric to metallic heterostructures. The Ir coating thickness was varied ranging from a few Å to films of about 7 nm in the heterostructures. The transition has been observed in the structures containing individual Ir coating thicknesses of about 2-4 nm. Following this, show epsilon-near-zero metamaterials with tunable dielectric constants by precisely varying the composition of such heterostructures. Overall, a comprehensive study on structural and optical properties of the metal-dielectric interface of Ir/Al2O3 heterostructures was addressed indicating an extension of the material portfolio available for optical system design.

physics.app-ph↗
arXiv2022

A tree-free approach to regularity structures: The regular case for quasi-linear equations

We give a motivation and gentle introduction into the regularity structure and model introduced by Otto, Sauer, Smith and Weber, which fall into the framework of Hairer, but have a greedier index set than the one given by trees. We do this here for a simple quasi-linear parabolic equation and assume that the driving noise is so regular that no renormalization is needed. We introduce the abstract model space $\mathsf{T}$ and its grading, the pre-model $\mathbfΠ$, the centered model $Π_x$, the structure group $\mathsf{G}$, and the re-centering transformations $Γ_{xy}$. Using integration and reconstruction, we establish the desired estimates on $Π_x$ and $Γ_{xy}$, which here are deterministic since we deal with the regular case.

math.AP↗
arXiv2022

Bias in the representative volume element method: periodize the ensemble instead of its realizations

We study the Representative Volume Element (RVE) method, which is a method to approximately infer the effective behavior $a_{\text{hom}}$ of a stationary random medium. The latter is described by a coefficient field $a(x)$ generated from a given ensemble $\langle\cdot\rangle$ and the corresponding linear elliptic operator $-\nabla\cdot a\nabla$. In line with the theory of homogenization, the method proceeds by computing $d = 3$ correctors (d denoting the space dimension).To be numerically tractable, this computation has to be done on a finite domain: the so-called "representative" volume element, i. e. a large box with, say, periodic boundary conditions. The main message of this article is: Periodize the ensemble instead of its realizations. By this we mean that it is better to sample from a suitably periodized ensemble than to periodically extend the restriction of a realization $a(x)$ from the whole-space ensemble $\langle\cdot\rangle$. We make this point by investigating the bias (or systematic error), i. e. the difference between $a_{\text{hom}}$ and the expected value of the RVE method, in terms of its scaling w. r. t. the lateral size $L$ of the box. In case of periodizing $a(x)$, we heuristically argue that this error is generically $O(L^{-1})$. In case of a suitable periodization of $\langle\cdot\rangle$, we rigorously show that it is $O(L^{-d})$. In fact, we give a characterization of the leading-order error term for both strategies, and argue that even in the isotropic case it is generically non-degenerate. We carry out the rigorous analysis in the convenient setting of ensembles $\langle\cdot\rangle$ of Gaussian type with integrable covariance, which allow for a straightforward periodization and which make the Price theorem and the Malliavin calculus available for optimal stochastic estimates of correctors.

math.AP↗
arXiv2022

Thermodynamically consistent and positivity-preserving discretization of the thin-film equation with thermal noise

In micro-fluidics not only does capillarity dominate but also thermal fluctuations become important. On the level of the lubrication approximation, this leads to a quasi-linear fourth-order parabolic equation for the film height $h$ driven by space-time white noise. The gradient flow structure of its deterministic counterpart, the thin-film equation, which encodes the balance between driving capillary and limiting viscous forces, provides the guidance for the thermodynamically consistent introduction of fluctuations. We follow this route on the level of a spatial discretization of the gradient flow structure. Starting from an energetically conformal finite-element (FE) discretization, we point out that the numerical mobility function introduced by Grün and Rumpf can be interpreted as a discretization of the metric tensor in the sense of a mixed FE method with lumping. While this discretization was devised in order to preserve the so-called entropy estimate, we use this to show that the resulting high-dimensional stochastic differential equation (SDE) preserves pathwise and pointwise strict positivity, at least in case of the physically relevant mobility function arising from the no-slip boundary condition. As a consequence, this discretization gives rise to a consistent invariant measure, namely a discretization of the Brownian excursion (up to the volume constraint), and thus features an entropic repulsion. The price to pay over more naive discretizations is that when writing the SDE in Itô's form, which is the basis for the Euler-Mayurama time discretization, a correction term appears. To conclude, we perform various numerical experiments to compare the behavior of our discretization to that of the more naive finite difference discretization of the equation.

math.AP↗
arXiv2021

There is no stationary cyclically monotone Poisson matching in 2d

We show that there is no cyclically monotone stationary matching of two independent Poisson processes in dimension $d=2$. The proof combines the harmonic approximation result from \cite{GHO} with local asymptotics for the two-dimensional matching problem for which we give a new self-contained proof using martingale arguments.

math.PR↗
arXiv2021

Variational approach to regularity of optimal transport maps: general cost functions

We extend the variational approach to regularity for optimal transport maps initiated by Goldman and the first author to the case of general cost functions. Our main result is an $ε$-regularity result for optimal transport maps between Hölder continuous densities slightly more quantitative than the result by De Philippis-Figalli. One of the new contributions is the use of almost-minimality: if the cost is quantitatively close to the Euclidean cost function, a minimizer for the optimal transport problem with general cost is an almost-minimizer for the one with quadratic cost. This further highlights the connection between our variational approach and De Giorgi's strategy for $ε$-regularity of minimal surfaces.

math.AP↗
arXiv2021

Quantitative linearization results for the Monge-Ampère equation

This paper is about quantitative linearization results for the Monge-Ampère equation with rough data. We develop a large-scale regularity theory and prove that if a measure $μ$ is close to the Lebesgue measure in Wasserstein distance at all scales, then the displacement of the macroscopic optimal coupling is quantitatively close at all scales to the gradient of the solution of the corresponding Poisson equation. The main ingredient we use is a harmonic approximation result for the optimal transport plan between arbitrary measures. This is used in a Campanato iteration which transfers the information through the scales.

math.AP↗