GRK 1294:  Analysis, Simulation and Design of Nanotechnological Processes

• Subject Area: Mathematics
• Term: from 2006 to 2015
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 12406856

• Final Report Year: 2015

Final Report Abstract

Das Graduiertenkolleg wird von Wissenschaftlerinnen und Wissenschaftlern des Forschungsschwerpunkts Partielle Differentialgleichungen der Fakultät für Mathematik am Karlsruher Institut für Technologie (KIT) organisiert. Die Promovierenden und Postdocs sollen die mathematischen Aspekte konkreter aus der Nanotechnologie stammender Probleme bearbeiten. Die Anwendungsprobleme stammen hauptsächlich aus dem Bereich photonische Kristalle und der nichtlinearen Optik. Zum Beispiel besitzen photonische Kristalle Bandlücken bei der Leitung von Licht, wie sie auch von der Stromleitung bei Halbleitern her bekannt sind. Ein anderes Problem besteht darin, die Existenz und Stabilität räumlich lokalisierter Lichtpulse in periodischen und periodisch gestörten Materialien zu zeigen. Analog dazu sollen stehende Wellen in nichtlinearen Materialien untersucht werden. In vielen Anwendungen sind die optischen Eigenschaften des untersuchten Materials nicht bekannt und können nur indirekt durch Messungen des Streufeldes bestimmt werden. Bei der Analysis und der Simulation der diese Strukturen beschreibenden nichtlinearen partiellen Differentialgleichungen ergeben sich zahlreiche mathematische Schwierigkeiten.Diese reichen von der Behandlung der Nichtlinearitäten, dem gleichzeitigen Auftreten von kontinuierlichem und diskretem Spektrum, dem Auftreten unterschiedlicher Skalen bis hin zur Schlechtgestelltheit der Probleme. Bei den hier verwendeten Modellen treten die Maxwell-Gleichungen und die Nichtlineare Schrödinger-Gleichung in verschiedenen Geometrien auf. Die erfolgreiche Bearbeitung dieser Probleme erfordert eine Vielzahl von Methoden, die von mathematischer Modellierung über Analysis und Numerik partieller Differentialgleichungen bis zur Theorie inverser Probleme und Optimierung reichen. Zu nennen sind hier die Multiskalenanalysis, variationelle Methoden und Verzweigungstheorie, Fourier- und Blochwellenanalysis, adaptive Finite Elemente, unstetige Galerkin-Verfahren, implizite Runge-Kutta-Verfahren, exponentielle Integratoren und Splitting-Verfahren, Wavelet-Kollokation und direkte und inverse Streutheorie. Das Graduiertenkolleg bietet exzellente Möglichkeiten zur interdisziplinären Zusammenarbeit und Qualifikation. Durch das Graduiertenkolleg wurde die interdisziplinäre Kooperation der Arbeitsgruppen in der Analysis und Numerik signifikant verbessert. Zudem trugen die strukturellen Maßnahmen zur Verbesserung der Qualifizierungs- und Betreuungssituation von Promotionen in der Fakultät für Mathematik bei. Die Nachhaltigkeit der 9-jährigen Förderung spiegelt sich in der erfolgreichen Beantragung von SFB 1173 ”Wellenphänomene: Analysis und Numerik” wider, an dem alle PIs der zweiten Förderperiode beteiligt sind.

Publications

• A regularization technique for the factorization method. Inverse Probl., 22:1605–1625, 2006
  A. Lechleiter
  
• Multistable solitons in the cubic-quintic discrete nonlinear Schrödinger equation. Physica D, 216:77–89, 2006
  R. Carretero-Gonzáles, J. D. Talley, C. Chong, and B. A. Malomed
  
• Newton regularizations for impedance tomography: A numerical study. Inverse Probl., 22:1967–1987, 2006
  A. Lechleiter and A. Rieder
  
• On the interaction of NLS-described modulating pulses with different carrier waves. Math. Meth. Appl. Sci., 30:1965–1978, 2007
  M. Chirilus-Bruckner, G. Schneider, and H. Uecker
  
• Optimality of the fully discrete filtered backprojection algorithm for tomographic inversion. Numer. Math., 108:151–175, 2007
  A. Rieder and A. Schneck
  
• Regularization of the factorization method and an application in impedance tomography. In Inverse Problems in Wave Scattering, volume 13, pages 14–16. Oberwolfach Reports, 2007
  A. Lechleiter, N. Hyvönen, and H. Hakula
  
• A remark about the justification of the nonlinear Schrödinger equation in quadratic spatially periodic media. ZAMP, 59:554–557, 2008
  C. Blank, M. Chirilus-Bruckner, C. Chong, V. Lescarret, G. Schneider, and H. Uecker
  
• Factorization methods for photonics and rough surfaces. PhD thesis, Karlsruhe Institute of Technology, 2008
  A. Lechleiter
  
• Newton regularizations for impedance tomography: Convergence by local injectivity. Inverse Probl., 24:065009, 2008
  A. Lechleiter and A. Rieder
  
• Seperation of internal and interaction dynamics for NLS-described wave packets with different carrier waves. J. Math. Anal. Appl., 347:304–314, 2008
  M. Chirilus-Bruckner, C. Chong, G. Schneider, and H. Uecker
  
• The factorization method applied to the complete electrode model of impedance tomography. SIAM J. Appl. Math., 68:1097–1121, 2008
  A. Lechleiter, N. Hyvönen, and H. Hakula
  
• The factorization method for inverse problems. Oxford Lecture Series in Mathematics and its Applications 36, Oxford University Press, 2008
  A. Kirsch, N. Grinberg
  
• Variational formulations for scattering in a threedimensional acoustic waveguide. Math. Meth. Appl. Sci., 31:821–847, 2008
  T. Arens, D. Gintides, and A. Lechleiter
  
• A computer-assisted proof for photonic band gaps. Z. Angew. Math. Phys., 60:1035–1052, 2009
  V. Hoang, M. Plum, and C. Wieners
  
• Bounds for optimization of the reflection coefficient by constrained optimization in Hardy spaces. PhD thesis, Karlsruhe Institute of Technology, 2009
  A. Schneck
  
• Computations of lossy Bloch waves in twodimensional photonic crystals. J. Comput. Theor. Nanosci., 6:775–783, 2009
  C. Engström, C. Hafner, and K. Schmidt
  
• Constrained Hardy space approximation II: Numerics, 2009
  A. Schneck
  (See online at https://doi.org/10.1016/j.jat.2010.03.006)
• Coupled mode equations and gap solitons for the 2D Gross-Pitaevskii equation with a non-separable periodic potential. Physica D, 238:860–879, 2009
  T. Dohnal and H. Uecker
  
• Coupled-mode equations and gap solitons for a two-dimensional nonlinear elliptic problem with a separable periodic potential. J. Nonlin. Sci., 19:95–131, 2009
  T. Dohnal, D. Pelinovsky, and G. Schneider
  
• Diverging probability density functions for flat-top solitary waves. Phys. Rev. E, 80:026602, 2009
  A. Peleg, Y. Chung, T. Dohnal, and Q. M. Nguyen
  
• HOPT – a MATLAB package for constrained optimization in Hardy spaces, 2009
  A. Schneck
  
• Labeling of n-dimensional images with choosable adjacency of the pixels. Image Anal. Stereol., 28:45–61, 2009
  K. Sandfort and J. Ohser
  
• Localized modes of the linear periodic Schrodinger operator with a nonlocal perturbation. SIAM J. Math. Anal., 41:1967–1993, 2009
  T. Dohnal, M. Plum, and W. Reichel
  
• Modeling optical technologies with continuous and discrete Nonlinear Schrödinger equations. PhD thesis, Karlsruhe Institute of Technology, 2009
  C. Chong
  
• Multistable solitons in higher-dimensional cubic-quintic nonlinear Schrödinger lattices. Physica D, 238:126–136, 2009
  C. Chong, R. Carretero-Gonzaléz, B. A. Malomed, and P. G. Kevrekidis
  
• MUSIC for extended scatterers as an instance of the factorization method. SIAM J. Appl. Math., 70:1283–1304, 2009
  T. Arens, A. Lechleiter, and D. R. Luke
  
• Nonlinear Interaction of Pulses. PhD thesis, Karlsruhe Institute of Technology, 2009
  M. Chirilus-Bruckner
  
• On spectral bounds for photonic crystal waveguides. In C. Bandle, A. Gilanyi, L. Losonczi, Z. Pales, and M. Plum, editors, Inequalities and Applications, volume 157 of International Series of Numerical Mathematics, pages 23–30, 2009
  M. Brown, V. Hoang, M. Plum, and I. Wood
  
• On the spectrum of an operator pencil with applications to wave propagation in periodic and frequency dependent materials. SIAM J. Appl. Math., 70:231–247, 2009
  C. Engström and M. Richter
  
• Perfectly matched layers for coupled nonlinear Schrodinger equations with mixed derivatives. J. Comput. Phys., 228:8752–8765, 2009
  T. Dohnal
  
• Standing generalized modulating pulse solutions for a nonlinear wave equation in periodic media. Nonlinearity, 22:1869–1898, 2009
  V. Lescarret, C. Blank, M. Chirilus-Bruckner, C. Chong, and G. Schneider
  
• The factorization method is independent of transmission eigenvalues. Inv. Prob. Imag., 3:123–138, 2009
  A. Lechleiter
  
• The linear sampling method revisited. J. Int. Eq. Appl., 21:179– 202, 2009
  T. Arens and A. Lechleiter
  
• The operator equations of Lippmann-Schwinger type for acoustic and electromagnetic scattering problems in L2 . Appl. Anal., 88:807–830, 2009
  A. Kirsch and A. Lechleiter
  
• Constrained Hardy space approximation. J. Approx. Theory, 162:1466– 1483, 2010
  A. Schneck
  
• Floquet Theory for a Class of Periodic Evolution Equations in an Lp-Setting. PhD thesis, Karlsruhe Institute of Technology, 2010
  T. Gauss
  
• Imaging of periodic dielectrics. BIT Numer. Math., 50:59–83, 2010
  A. Lechleiter
  
• Intermediate model for spatial evolution in nonlinear optics. Math. Models Methods Appl. Sci., 20:1209–1249, 2010
  V. Lescarret
  
• On the Existence of Breathers Solutions in non-linear Klein-Gordon equations with periodic coefficients. PhD thesis, Karlsruhe Institute of Technology, 2010
  C. Blank
  
• Optimization of photonic band structures. PhD thesis, Karlsruhe Institute of Technology, 2010
  M. Richter
  
• Parallel multigrid methods for the band structure computation of 3D photonic crystals with higher order finite elements. PhD thesis, Karlsruhe Institute of Technology, 2010
  A. Bulovyatov
  
• The factorization method for inverse scattering from periodic inhomogeneous media. PhD thesis, Karlsruhe Institute of Technology, 2010
  K. Sandfort
  
• The Principles of Limit Absorption and Limit Amplitude for Periodic Operators. PhD thesis, Karlsruhe Institute of Technology, 2010
  M. Radosz
  
• Adaptivity in Bandstructure Calculations of Photonic Crystals. PhD thesis, Karlsruhe Institute of Technology, 2011
  A. Krämer
  
• An hp-efficient residual-based a posteriori error estimator for Maxwell’s equations. Proc. Appl. Math. Mech., 11:869–870, 2011
  M. Bürg
  
• Analysis of light propagation in slotted resonator based systems via coupled-mode theory. Opt. Express, 19:8641–8655, 2011
  K. R. Hiremath, J. Niegeman, and K. Busch
  
• Boundary Element Approximation for Maxwell’s Eigenvalue Problem. PhD thesis, Karlsruhe Institute of Technology, 2011
  J. Xin
  
• Breather solutions in periodic media. Comm. Math. Phys., 302:815–841, 2011
  C. Blank, M. Chirilus-Bruckner, V. Lescarret, and G. Schneider
  
• Convergence of an adaptive hp finite element strategy in higher space-dimensions. Appl. Numer. Math., 61:1132–1146, 2011
  M. Bürg and W. Dörfler
  
• Families of surface gap solitons and their stability via the numerical Evans function method. SIAM J. Appl. Dyn. Syst., 10:667–706, 2011
  E. Blank and T. Dohnal
  
• Nichtlineare photonische Kristalle: Computerunterstützter Beweis einer spektralen Bandlücke. PhD thesis, Karlsruhe Institute of Technology, 2011
  M. Reimers
  
• Numerical methods of localization of Wannier Functions in the modeling of Photonic Crystals. PhD thesis, Karlsruhe Institute of Technology, 2011
  T. Bulovyatova
  
• Numerical Simulation of a Micro-ring Resonator with Adaptive Wavelet Collocation Method. PhD thesis, Karlsruhe Institute of Technology, 2011
  H. Li
  
• Photonic crystals: Mathematical Analysis and Numerical Implementation, volume 42 of Oberwolfach Seminars. Birkhäuser, Basel, 2011
  W. Dörfler, A. Lechleiter, M. Plum, G. Schneider, and C. Wieners
  
• Surface gap soliton ground states for the nonlinear Schrödinger equation. Comm. Math. Phys., 308:511–542, 2011
  T. Dohnal, M. Plum, and W. Reichel
  
• The limiting absorption principle in a semi-infinite periodic waveguide. SIAM J. Appl. Math., 71:791–810, 2011
  V. Hoang
  
• Variational approximations of bifurcations of asymmetric solitons in cubic-quintic nonlinear Schrödinger lattices. Discrete and Continuous Dynamical Systems, Series S, 4:1019–1031, 2011
  C. Chong and D. E. Pelinovsky
  
• A Fully Automatic hp-Adaptive Refinement Strategy. PhD thesis, Karlsruhe Institute of Technology, 2012
  M. Bürg
  
• hp-FEM for Two-component Flows with Applications in Optofluidics. PhD thesis, Karlsruhe Institute of Technology, 2012
  S. Ronnas
  
• Vortex families near a spectral edge in the Gross-Pitaevskii equation with a two-dimensional periodic potential. Phys. Rev. E, 85:026605, 2012
  T. Dohnal and D. Pelinovsky
  (See online at https://doi.org/10.1103/PhysRevE.85.026605)
• Analysing Ewald’s method for the evaluation of Green’s functions for periodic media. IMA J. Appl. Math., 78:405–431, 2013
  T. Arens, K. Sandfort, and S. Schmitt
  
• Boundary element approximation for Maxwell’s eigenvalue problem. Math. Methods Appl. Sci., 36:2524–2539, 2013
  C. Wieners and J. Xin
  (See online at https://doi.org/10.1002/mma.2772)
• Convergence of an automatic hp-adaptive finite element strategy for Maxwell’s equations. Appl. Numer. Math., 72:188–206, 2013
  
  
• Coupled mode equations modeling for out-of-plane gap solitons in 2D photonic crystals. SIAM Multiscale Model. Simul., 11:162–191, 2013
  T. Dohnal and W. Dörfler
  (See online at https://doi.org/10.1137/120865914)
• Die unstetige Galerkinmethode für die Maxwell-Gleichungen: Anwendung auf Rotationskörper und Kerr-Nichtlinearitäten. PhD thesis, Karlsruhe Institute of Technology, 2013
  E. Blank
  
• Distributional solutions of the stationary nonlinear Schrodinger equation: singularities, regularity and exponential decay. ZAA, 32:55–82, 2013
  R. Mandel and W. Reichel
  
• Error analysis of implicit and exponential time integration of linear Maxwell’s equations. PhD thesis, Karlsruhe Institute of Technology, 2013
  T. Pažur
  
• Grenzflächenprobleme bei der nichtlinearen Schrödingergleichung. PhD thesis, Karlsruhe Institute of Technology, 2013
  H. J. Freisinger
  
• Grundzustände, Verzweigungen und singulare Lösungen nichtlinearer Schrödingersysteme. PhD thesis, Karlsruhe Institute of Technology, 2013
  R. Mandel
  
• Interfaces supporting surface gap soliton ¨ ground states in the 1D nonlinear Schrödinger equation. J. Math. Anal. Appl., 407:425– 435, 2013
  T. Dohnal, K. Nagatou, M. Plum, and W. Reichel
  
• Interpolatory Weighted–H2 Model Reduction. Automatica, 49(5):1275–1280, 2013
  Branimir Anić, Christopher A. Beattie, Serkan Gugercin, Athanasios C. Antoulas
  (See online at https://doi.org/10.1016/j.automatica.2013.01.040)
• Local wellposedness of a quasilinear wave equation. Applicable Analysis
  W. Dörfler, H. Gerner, and R. Schnaubelt
  (See online at https://doi.org/10.1080/00036811.2015.1089236)
• On the Spectral Properties of Dispersive Photonic Crystals. PhD thesis, Karlsruhe Institute of Technology, 2013
  P. Schmalkoke
  
• Stetige Galerkinverfahren für zeitabhängige Maxwellgleichungen mit Kerr-Nichtlinearität. PhD thesis, Karlsruhe Institute of Technology, 2013
  H. Gerner
  
• The Bloch Transform on Lp-Spaces. PhD thesis, Karlsruhe Institute of Technology, 2013
  B. Barth
  
• The Fourier-Galerkin Method for Band Structure Computations of 2D and 3D Photonic Crystals. PhD thesis, Karlsruhe Institute of Technology, 2013
  B. Anić
  
• The optimal shape of a pipe. Zeitschrift für Angewandte Mathematik und Physik, 64(4):1177–1185, 2013
  A. Schulz
  
• Absence of bound states for waveguides in 2D periodic structures. J. Math. Phys., 55:033506, 2014
  V. Hoang and M. Radosz
  (See online at https://doi.org/10.1063/1.4868480)
• An exponential integrator for non-autonomous parabolic problems. ETNA, 41:497–511, 2014
  D. Hipp, M. Hochbruck, and A. Ostermann
  
• Conditional space-time stability of collocation Runge-Kutta for parabolic evolution equations. Electron. Trans. Numer. Anal., 41:62–80, 2014
  R. Andreev and J. Schweitzer
  
• Eine flexible Klasse von local time stepping Verfahren. PhD thesis, Karlsruhe Institute of Technology, 2014
  A. Demirel
  
• Mesh-independent a priori bounds for nonlinear elliptic finite difference boundary value problems. J. Math. Anal. Appl., 419(1):496– 524, 2014
  P. J. McKenna, W. Reichel, and A. Verbitsky
  (See online at https://doi.org/10.1016/j.jmaa.2014.04.044)
• Minimal energy solutions for repulsive nonlinear Schrödinger systems. J. Differential Equations, 257:450–468, 2014
  R. Mandel
  (See online at https://doi.org/10.1016/j.jde.2014.04.006)
• Opening up and control of spectral gaps of the Laplacian in periodic domains. J. Math. Phys., 55(12):121502, 2014
  A. Khrabustovskyi
  (See online at https://doi.org/10.1063/1.4902935)
• Positive Solutions for the Discrete Nonlinear Schrödinger Equation: A Priori Estimates and Convergence. PhD thesis, Karlsruhe Institute of Technology, 2014
  A. Verbitsky
  
• Spectral properties of elliptic operator with doublecontrast coefficients near a hyperplane
  A. Khrabustovskyi and M. Plum
  
• Spectrum created by line defects in periodic structures. Math. Nachr., 287:1972-1985, 2014
  M. Brown, V. Hoang, M. Plum, and I. Wood
  (See online at https://doi.org/10.1002/mana.201300165)
• Uniqueness results for semilinear elliptic systems on Rn . Mathematische Nachrichten, 287:1828–1836, 2014
  R. Mandel
  (See online at https://doi.org/10.1002/mana.201300130)
• Well-posedness for a general class of quasilinear evolution equations with applications to Maxwell’s equations. PhD thesis, Karlsruhe Institute of Technology, 2014
  D. Müller
  
• Domain optimization for an acoustic waveguide scattering problem. Applied Mathematics & Optimization, 72(1):101-146, 2015
  J. Ott
  (See online at https://doi.org/10.1007/s00245-014-9273-1)
• Efficient multiple time-stepping algorithms of higher order. J. Comp. Phys., 285:133–148, 2015
  A. Demirel, J. Niegemann, K. Busch, and M. Hochbruck
  (See online at https://doi.org/10.1016/j.jcp.2015.01.018)
• Efficient time integration for discontinuous Galerkin approximations of linear wave equations. ZAMM, 95(3):237– 259, 2015
  M. Hochbruck, T. Pažur, A. Schulz, E. Thawinan, and C. Wieners
  (See online at https://doi.org/10.1002/zamm.201300306)
• Error analysis of a second order locally implicit method for linear Maxwell’s equations. CRC-Preprint 2015/1, Karlsruhe Institute of Technology, CRC 1173, 2015
  M. Hochbruck and A. Sturm
  (See online at https://doi.org/10.1137/15M1038037)
• Error analysis of implicit Euler methods for quasilinear hyperbolic evolution equations. Technical report, Karlsruhe Institute of Technology, 2015
  M. Hochbruck and T. Pažur
  (See online at https://doi.org/10.1007/s00211-016-0810-5)
• Gaps in the spectrum of the Neumann Laplacian generated by a system of periodically distributed trap. Math. Meth. Appl. Sci., 38(1):11– 26, 2015
  A. Khrabustovskyi and E. Khruslov
  (See online at https://doi.org/10.1002/mma.3046)
• High order numerical methods for highly oscillatory problems, ESAIM:M2AN, 49(3):695–711, 2015
  D. Cohen and J. Schweitzer
  (See online at https://doi.org/10.1051/m2an/2014056)
• Implicit Runge–Kutta methods and discontinuous Galerkin, discretizations for linear Maxwell’s equations. SIAM J. Numer. Anal., 53(1):485–507, 2015
  M. Hochbruck and T. Pažur
  (See online at https://doi.org/10.1137/130944114)
• Mathematical Theory of Time-harmonic Maxwell’s Equations. Springer Series in Applied Mathematical Siences 190, Springer International Publishing, 2015
  A. Kirsch, F. Hettlich
  
• Minimal energy solutions for cooperative nonlinear Schrödinger systems. No- DEA, 22:239–262, 2015
  R. Mandel
  (See online at https://dx.doi.org/10.1007%2Fs00030-014-0281-2)
• Neumann spectral problem in a domain with very corrugated boundary. Journal of Differential Equations, 259(6):2333–2367, 2015
  G. Cardone and A. Khrabustovskyi
  (See online at https://doi.org/10.1016/j.jde.2015.03.031)
• Numerical Analysis of the Electro-Magnetic Perfectly Matched Layer in a Discontinuous Galerkin Discretization. PhD thesis, Karlsruhe Institute of Technology, 2015
  A. Schulz
  
• Numerical optimization of a waveguide transition using finite element beam propagation. International Journal of Numerical Modelling: Electronic Networks, Devices and Fields, 28(2):201–212, 2015
  S. Findeisen and W. Dörfler
  (See online at https://doi.org/10.1002/jnm.1997)
• On the spectrum of narrow Neumann waveguide with periodically distributed δ traps. Journal of Physics A: Mathematical and Theoretical, 48 (31):315301 (2015)
  A. Khrabustovskyi and P. Exner
  (See online at https://doi.org/10.1088/1751-8113/48/31/315301)
• Spectrally Localized Strichartz Estimates and Nonlinear Schrödinger Equations. PhD thesis, Karlsruhe Institute of Technology, 2015
  A. Bolleyer