Prof. Dr. Barbara Zwicknagl

Profil

• Dynamic Phenomena in Elasticity Problems

  Quelle ↗

  Förderer: Volkswagen Stiftung Zeitraum: 06/2022 - 06/2023 Projektleitung: Prof. Dr. Barbara Zwicknagl

• From Modeling and Analysis to Approximation

  Quelle ↗

  Förderer: Volkswagen Stiftung Zeitraum: 01/2020 - 12/2023 Projektleitung: Prof. Dr. Barbara Zwicknagl

• Q13 - Variational models for pattern formation in biomembranes

  Quelle ↗

  Förderer: DFG Graduiertenkolleg Zeitraum: 10/2021 - 03/2023 Projektleitung: Prof. Dr. Barbara Zwicknagl

• Qualitative properties of PDEs with non-standard growth conditions

  Quelle ↗

  Förderer: Volkswagen Stiftung Zeitraum: 02/2023 - 11/2023 Projektleitung: Prof. Dr. Barbara Zwicknagl

• SFB/TRR 109/3: Geometrische Rigidität in Spinsystemen (TP B11)

  Quelle ↗

  Förderer: DFG Sonderforschungsbereich Zeitraum: 07/2020 - 06/2024 Projektleitung: Prof. Dr. Barbara Zwicknagl

• InnovationLab GmbH

  P

  26 Treffer63.7%
  • Interfaces in opto-electronic thin film multilayer devices
    P

    63.7%

• Science & Startups Berlin

  T

  5 Treffer60.5%
  • Zuwendung im Rahmen des Programms „exist – Existenzgründungen aus der Wissenschaft“ aus dem Bundeshaushalt, Einzelplan 09, Kapitel 02, Titel 68607, Haushaltsjahr 2026, sowie aus Mitteln des Europäischen Strukturfonds (hier Euro-päischer Sozialfonds Plus – ESF Plus) Förderperiode 2021-2027 – Kofinanzierung für das Vorhaben: „exist Women“
    T

    60.5%

• Gesellschaft für angewandte Mathematik und Mechanik

  PT

  22 Treffer59.9%
  • Workshop Reliable Methods and Mathematical Modeling
    P

    59.9%

• SoftBank Robotics

  P

  7 Treffer57.1%
  • Embodied Audition for RobotS
    P

    57.1%

• Europäische Kommission

  P

  5 Treffer55.4%
  • Generalized Quatum Batalin-Vilkovisky Formalism and Graphical Calculus
    P

    55.4%

• Heraeus Battery Technology GmbH

  P

  7 Treffer55.4%
  • Optimierte Natrium-Feststoffbatterien mit neuen Anoden basierend auf Kohlenstoffgerüststrukturen
    P

    55.4%

• RISC Software GmbH

  PT

  28 Treffer55.0%
  • EU: Scattering Amplitudes: From Geometry to EXperiment (SAGEX)
    P

    55.0%

• TechOnYou Srl

  P

  20 Treffer54.6%
  • EU: Hybrid Organic/Inorganic Memory Elements for Integration of Electronic and Photonic Circuitry (HYMEC)
    P

    54.6%

• Fortiss GmbH

  P

  2 Treffer54.4%
  • EU: Context Sensitive Multisensory Object Recognition (HBP)
    P

    54.4%

• Fundação D. Anna de Sommer Champalimaud e Dr. Carlos Montez Champalimaud

  P

  2 Treffer54.4%
  • EU: Context Sensitive Multisensory Object Recognition (HBP)
    P

    54.4%

• Sampling inequalities for infinitely smooth functions, with applications to interpolation and machine learning

  2008

  Advances in Computational Mathematics · 58 Zitationen · DOI

  Sampling inequalities give a precise formulation of the fact that a differentiable function cannot attain large values if its derivatives are bounded and if it is small on a sufficiently dense discrete set. Sampling inequalities can be applied to the difference of a function and its reconstruction in order to obtain (sometimes optimal) convergence orders for very general possibly regularized recovery processes. So far, there are only sampling inequalities for finitely smooth functions, which lead to algebraic convergence orders. In this paper, the case of infinitely smooth functions is investigated, in order to derive error estimates with exponential convergence orders.

• Power Series Kernels

  2008

  Constructive Approximation · 40 Zitationen · DOI

  We introduce a class of analytic positive definite multivariate kernels which includes infinite dot product kernels as sometimes used in machine learning, certain new nonlinearly factorizable kernels, and a kernel which is closely related to the Gaussian. Each such kernel reproduces in a certain “native” Hilbert space of multivariate analytic functions. If functions from this space are interpolated in scattered locations by translates of the kernel, we prove spectral convergence rates of the interpolants and all derivatives. By truncation of the power series of the kernel-based interpolants, we constructively generalize the classical Bernstein theorem concerning polynomial approximation of analytic functions to the multivariate case. An application to machine learning algorithms is presented.

• Microstructures in Low-Hysteresis Shape Memory Alloys: Scaling Regimes and Optimal Needle Shapes

  2014

  Archive for Rational Mechanics and Analysis · 36 Zitationen · DOI

• Sampling and Stability

  2010

  Lecture notes in computer science · 28 Zitationen · DOI

• Deterministic Error Analysis of Support Vector Regression and Related Regularized Kernel Methods

  2009

  Max Planck Institute for Plasma Physics · 26 Zitationen

  We introduce a new technique for the analysis of kernel-based regression problems. The basic tools are sampling inequalities which apply to all machine learning problems involving penalty terms induced by kernels related to Sobolev spaces. They lead to explicit deterministic results concerning the worst case behaviour of e- and ν-SVRs. Using these, we show how to adjust regularization parameters to get best possible approximation orders for regression. The results are illustrated by some numerical examples.

• Multiscale Approximation and Reproducing Kernel Hilbert Space Methods

  2015

  SIAM Journal on Numerical Analysis · 22 Zitationen · DOI

  We consider reproducing kernels $K:\Omega\times \Omega \to \mathbb{R}$ in multiscale series expansion form, i.e., kernels of the form $K\left(\boldsymbol{x},\boldsymbol{y}\right)=\sum_{\ell\in\mathbb{N}}\lambda_\ell\sum_{j\in I_\ell}\phi_{\ell,j}\left(\boldsymbol{x}\right)\phi_{\ell,j}\left(\boldsymbol{y}\right)$ with weights $\lambda_\ell$ and structurally simple basis functions $\left\{\phi_{\ell,i}\right\}$. Here, we deal with basis functions such as polynomials or frame systems, where, for $\ell\in \mathbb{N}$, the index set $I_\ell$ is finite or countable. We derive relations between approximation properties of spaces based on basis functions $\{\phi_{\ell,j} :1\leq\ell\leq L, j\in I_\ell\}$ and spaces spanned by translates of the kernel span $\{K(\boldsymbol{x}_1,\cdot),\dots, K(\boldsymbol{x}_N,\cdot)\}$ with $X_{N}:=\{\boldsymbol{x}_1,\dots,\boldsymbol{x}_N\}\subset\Omega$ if the truncation index $L$ is appropriately coupled to the discrete set $X_{N}$. An analysis of a numerically feasible approximation from trial spaces span $\{K^L(\boldsymbol{x}_1,\cdot),\dots, K^L(\boldsymbol{x}_N,\cdot)\}$ based on finitely truncated series kernels of the form $K^L\left(\boldsymbol{x},\boldsymbol{y}\right):=\sum_{\ell=1}^L\lambda_\ell\sum_{j\in I_\ell}\phi_{\ell,j}(\boldsymbol{x})\phi_{\ell,j}(\boldsymbol{y})$ is provided, where the truncation index $L$ is chosen sufficiently large depending on the point set $X_{N}$. Furthermore, Bernstein-type inverse estimates and derivative-free sampling inequalities for kernel-based spaces are obtained from estimates for spaces based on the basis functions $\{\phi_{\ell,j} :1\leq\ell\leq L, j\in I_\ell\}$.

• Shapes of Epitaxially Grown Quantum Dots

  2014

  Archive for Rational Mechanics and Analysis · 22 Zitationen · DOI

• Interpolation and approximation in Taylor spaces

  2013

  Journal of Approximation Theory · 22 Zitationen · DOI

• Scaling Law and Reduced Models for Epitaxially Strained Crystalline Films

  2014

  SIAM Journal on Mathematical Analysis · 21 Zitationen · DOI

  A variational model for the epitaxial deposition of a film on a rigid substrate in the presence of a crystallographic misfit is studied. The scaling behavior of the minimal energy in terms of the volume of the film and the amplitude of the misfit is considered, and reduced models in the various regimes are derived by $\Gamma$-convergence methods. Depending on the relation between the thickness of the film and the amplitude of the misfit, the surface or the elastic energy contribution dominates, and in the critical case the two contributions balance. In particular, the formation of islands is proven if the amplitude of the misfit is large compared to the volume of the film.

• Higher Sobolev Regularity of Convex Integration Solutions in Elasticity: The Dirichlet Problem with Affine Data in int(K^{lc})

  2018

  SIAM Journal on Mathematical Analysis · 20 Zitationen · DOI

  In this article we continue our study of higher Sobolev regularity of flexible convex integration solutions to differential inclusions arising from applications in materials sciences. We present a general framework yielding higher Sobolev regularity for Dirichlet problems with affine data in ${int}(K^{lc})$. This allows us to simultaneously deal with linear and nonlinear differential inclusion problems. We show that the derived higher integrability and differentiability exponent has a lower bound, which is independent of the position of the Dirichlet boundary data in ${int}(K^{lc})$. As applications we discuss the regularity of weak isometric immersions in two and three dimensions as well as the differential inclusion problem for the geometrically linear hexagonal-to-rhombic and the cubic-to-orthorhombic phase transformations occurring in shape-memory alloys.

• Deformation concentration for martensitic microstructures in the limit of low volume fraction

  2017

  Calculus of Variations and Partial Differential Equations · 20 Zitationen · DOI

• Improved Exponential Convergence Rates by Oversampling Near the Boundary

  2013

  Constructive Approximation · 17 Zitationen · DOI

• Higher Sobolev Regularity of Convex Integration Solutions in Elasticity: The Planar Geometrically Linearized Hexagonal-to-Rhombic Phase Transformation

  2019

  Journal of Elasticity · 16 Zitationen · DOI

  In this article we discuss quantitative properties of convex integration solutions arising in problems modeling shape-memory materials. For a two-dimensional, geometrically linearized model case, the hexagonal-to-rhombic phase transformation, we prove the existence of convex integration solutions u with higher Sobolev regularity, i.e., there exists

• Piecewise affine stress-free martensitic inclusions in planar nonlinear elasticity

  2017

  Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences · 16 Zitationen · DOI

  We consider a partial differential inclusion problem which models stress-free martensitic inclusions in an austenitic matrix, based on the standard geometrically nonlinear elasticity theory. We show that for specific parameter choices there exist piecewise affine continuous solutions for the square-to-oblique and the hexagonal-to-oblique phase transitions. This suggests that for specific crystallographic parameters the hysteresis of the phase transformation will be particularly small.

• Energy Scaling Law for a Singularly Perturbed Four-Gradient Problem in Helimagnetism

  2022

  Journal of Nonlinear Science · 8 Zitationen · DOI

  Abstract We study pattern formation in magnetic compounds near the helimagnetic/ferromagnetic transition point in case of Dirichlet boundary conditions on the spin field. The energy functional is a continuum approximation of a $$J_1-J_3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>J</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>-</mml:mo> <mml:msub> <mml:mi>J</mml:mi> <mml:mn>3</mml:mn> </mml:msub> </mml:mrow> </mml:math> model and was recently derived in Cicalese et al. (SIAM J Math Anal 51: 4848–4893, 2019). It contains two parameters, one measuring the incompatibility of the boundary conditions and the other measuring the cost of changes between different chiralities. We prove the scaling law of the minimal energy in terms of these two parameters. The constructions from the upper bound indicate that in some regimes branching-type patterns form close to the boundary of the sample.

• Geometry of martensite needles in shape memory alloys

  2020

  Repository for Publications and Research Data (ETH Zurich) · 6 Zitationen · DOI

  We study the geometry of needle-shaped domains in shape-memory alloys. Needle-shaped domains are ubiquitously found in martensites around macroscopic interfaces between regions which are laminated in different directions, or close to macroscopic austenite/twinned-martensite interfaces. Their geometry results from the interplay of the local nonconvexity of the effective energy density with long-range (linear) interactions mediated by the elastic strain field, and is up to now poorly understood. We present a two-dimensional shape optimization model based on finite elasticity and discuss its numerical solution. Our results indicate that the tapering profile of the needles can be understood within finite elasticity, but not with linearized elasticity. The resulting tapering and bending reproduce the main features of experimental observations on Ni65Al35.

• Regularized Kernel-Based Reconstruction in Generalized Besov Spaces

  2017

  Foundations of Computational Mathematics · 6 Zitationen · DOI

• Microstructure of macrointerfaces in shape-memory alloys

  2023

  Journal of the Mechanics and Physics of Solids · 5 Zitationen · DOI

• Geometry of Needle-Like Microstructures in Shape-Memory Alloys

  2023

  Shape Memory and Superelasticity · 5 Zitationen · DOI

  Abstract Needle-like microstructures are often observed in shape memory alloys near macro-interfaces that separate regions with different laminate orientation. We study their shape with a two-dimensional model based on nonlinear elasticity, that contains an explicit parametrization of the needle profiles. Energy minimization leads to specific predictions for the geometry of needle-like domains. Our simulations are based on shape optimization of the needle interfaces, using a polyconvex energy density with cubic symmetry for the elastic problem, and a numerical implementation via finite elements on a dynamically changing grid.

• Asymptotic Self-Similarity of Minimizers and Local Bounds in a Model of Shape-Memory Alloys

  2021

  Journal of Elasticity · 5 Zitationen · DOI

  Abstract We prove that microstructures in shape-memory alloys have a self-similar refinement pattern close to austenite-martensite interfaces, working within the scalar Kohn-Müller model. The latter is based on nonlinear elasticity and includes a singular perturbation representing the energy of the interfaces between martensitic variants. Our results include the case of low-hysteresis materials in which one variant has a small volume fraction. Precisely, we prove asymptotic self-similarity in the sense of strong convergence of blow-ups around points at the austenite-martensite interface. Key ingredients in the proof are pointwise estimates and local energy bounds. This generalizes previous results by one of us to various boundary conditions, arbitrary rectangular domains, and arbitrary volume fractions of the martensitic variants, including the regime in which the energy scales as $\varepsilon ^{2/3}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>ε</mml:mi> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo>/</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:msup> </mml:math> as well as the one where the energy scales as $\varepsilon ^{1/2}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>ε</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:math> .

• A few remarks on variational models for denoising

  2014

  Communications in Mathematical Sciences · 5 Zitationen · DOI

  Variational models for image and signal denoising are based on the minimization of energy functionals consisting of a fidelity term together with higher-order regularization. In addition to the choices of function spaces to measure fidelity and impose regularization, different scaling exponents appear. In this note we present a few simple remarks on (i) the stability with respect to deterministic noise perturbations, captured via oscillatory sequences converging weakly to zero, and (ii) exact reconstruction.

• A Sharp Interface Limit of a Nonlocal Variational Model for Pattern Formation in Biomembranes

  2024

  SIAM Journal on Mathematical Analysis · 3 Zitationen · DOI

• The Tapering Length of Needles in Martensite/Martensite Macrotwins

  2023

  Archive for Rational Mechanics and Analysis · 3 Zitationen · DOI

  Abstract We study needle formation at martensite/martensite macro interfaces in shape-memory alloys. We characterize the scaling of the energy in terms of the needle tapering length and the transformation strain, both in geometrically linear and in finite elasticity. We find that linearized elasticity is unable to predict the value of the tapering length, as the energy tends to zero with needle length tending to infinity. Finite elasticity shows that the optimal tapering length is inversely proportional to the order parameter, in agreement with previous numerical simulations. The upper bound in the scaling law is obtained by explicit constructions. The lower bound is obtained using rigidity arguments, and as an important intermediate step we show that the Friesecke–James–Müller geometric rigidity estimate holds with a uniform constant for uniformly Lipschitz domains.

• Higher Sobolev Regularity of Convex Integration Solutions in Elasticity

  2016

  arXiv (Cornell University) · 3 Zitationen · DOI

  In this article we discuss quantitative properties of convex integration solutions arising in problems modeling shape-memory materials. For a two-dimensional, geometrically linearized model case, the hexagonal-to-rhombic phase transformation, we prove the existence of convex integration solutions $u$ with higher Sobolev regularity, i.e. there exists $θ_0&gt;0$ such that $\nabla u \in W^{s,p}_{loc}(\mathbb{R}^2)\cap L^{\infty}(\mathbb{R}^2)$ for $s\in(0,1)$, $p\in(1,\infty)$ with $0

• Energy scaling laws for microstructures: from helimagnets to martensites

  2023

  Calculus of Variations and Partial Differential Equations · 2 Zitationen · DOI

  Abstract We consider scalar-valued variational models for pattern formation in helimagnetic compounds and in shape memory alloys. Precisely, we consider a non-convex multi-well bulk energy term on the unit square, which favors four gradients $$({\pm }\,\alpha ,{\pm }\,\beta )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo>±</mml:mo> <mml:mspace/> <mml:mi>α</mml:mi> <mml:mo>,</mml:mo> <mml:mo>±</mml:mo> <mml:mspace/> <mml:mi>β</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , regularized by a singular perturbation in terms of the total variation of the second derivative. We derive scaling laws for the minimal energy in the case of incompatible affine boundary conditions in terms of the singular perturbation parameter as well as the ratio $$\alpha /\beta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>/</mml:mo> <mml:mi>β</mml:mi> </mml:mrow> </mml:math> and the incompatibility of the boundary condition. We discuss how well-studied models for martensitic microstructures in shape-memory alloys arise as a limiting case, and relations between the different models in terms of scaling laws. In particular, we show that scaling regimes arise in which an interpolation between the rather different branching-type constructions in the spirit of Kohn and Müller (Commun Pure Appl Math 47(4):405–435, 1994) and Ginster and Zwicknagl (J Nonlinear Sci 33:20, 2023), respectively, occurs. A particular technical difficulty in the lower bounds arises from the fact that the energy scalings involve various logarithmic terms that we capture in matching upper and lower scaling bounds.

• Baltische Föderale Immanuel-Kant-Universität

  From Modeling and Analysis to Approximation

  university

• Kharkiv National Universität

  Dynamic Phenomena in Elasticity Problems

  university

• National Academy of Sciences of Ukraine

  Dynamic Phenomena in Elasticity Problems

  university

• Nationale Wassyl-Stus-Universität Donezk

  Qualitative properties of PDEs with non-standard growth conditions

  university

• Russian Academy of Sciences

  From Modeling and Analysis to Approximation

  other

• Staatliche Universität Sankt Petersburg

  From Modeling and Analysis to Approximation

  university

• Technische Universität Berlin

  Q13 - Variational models for pattern formation in biomembranes

  university

• Technische Universität Braunschweig

  From Modeling and Analysis to Approximation

  university

• Technische Universität München

  SFB/TRR 109/3: Geometrische Rigidität in Spinsystemen (TP B11)

  university

• Universität zu Lübeck

  From Modeling and Analysis to Approximation

  university

Name

Prof. Dr. Barbara Zwicknagl

Titel

Prof. Dr.

Fakultät

Mathematisch-Naturwissenschaftliche Fakultät

Institut

Institut für Mathematik

Arbeitsgruppe

Angewandte Analysis

Telefon

+49 30 2093-45338

HU-FIS-Profil

Quelle ↗

Zuletzt gescrapt

26.4.2026, 01:14:37