A prewavelet-based algorithm for the solution of second-order elliptic differential equations with variable coefficients on sparse grids - Hartmann, Rainer; Pflaum, Christoph (Numerical Algorithms, 2017)
We present a Ritz-Galerkin discretization on sparse grids using prewavelets, which allows us to solve elliptic differential equations with variable coefficients for dimensions d ≥ 2. The method applies multilinear finite elements. We introduce an efficient algorithm for matrix vector multiplication using a Ritz-Galerkin discretization and semi-orthogonality. This algorithm is based on standard 1-dimensional restrictions and prolongations, a simple prewavelet stencil, and the classical operator-dependent stencil for multilinear finite elements. Numerical simulation results are presented for a three-dimensional problem on a curvilinear bounded domain and for a six-dimensional problem with variable coefficients. Simulation results show a convergence of the discretization according to the approximation properties of the finite element space. The condition number of the stiffness matrix can be bounded below 10 using a standard diagonal preconditione
UnivIS-Import:2017-11-06:Pub.2017.tech.IMMD.c3sim.aprewa
Keywords: Sparse grid; Prewavelets; Semi-orthogonality; Variable coefficients; Conjugate gradient method; Finite element method

A Sparse Grid Discretization of the Helmholtz Equation with Variable Coefficients in High Dimensions - Pflaum, Christoph; Hartmann, Rainer (SIAM Journal on Numerical Analysis, 2016)
The computational effort for solving elliptic differential equations can significantly be reduced by using sparse grids. We present a new Ritz–Galerkin discretization of the Helmholtz equation with variable coefficients on sparse grids. This discretization uses prewavelets and a semi-orthogonality property on sparse grids. A detailed convergence analysis is given for the arbitrary dimension d. The linear equation system of the discretization can efficiently be solved by a multigrid Q-cycle. The computational effort for solving elliptic differential equations can significantly be reduced by using sparse grids. We present a new Ritz–Galerkin discretization of the Helmholtz equation with variable coefficients on sparse grids. This discretization uses prewavelets and a semi-orthogonality property on sparse grids. A detailed convergence analysis is given for the arbitrary dimension d. The linear equation system of the discretization can efficiently be solved by a multigrid Q-cycle. Read More: http://epubs.siam.org/doi/abs/10.1137/15M101508X The computational effort for solving elliptic differential equations can significantly be reduced by using sparse grids. We present a new Ritz–Galerkin discretization of the Helmholtz equation with variable coefficients on sparse grids. This discretization uses prewavelets and a semi-orthogonality property on sparse grids. A detailed convergence analysis is given for the arbitrary dimension d. The linear equation system of the discretization can efficiently be solved by a multigrid Q-cycle. Read More: http://epubs.siam.org/doi/abs/10.1137/15M101508X The computational effort for solving elliptic differential equations can significantly be reduced by using sparse grids. We present a new Ritz–Galerkin discretization of the Helmholtz equation with variable coefficients on sparse grids. This discretization uses prewavelets and a semi-orthogonality property on sparse grids. A detailed convergence analysis is given for the arbitrary dimension d. The linear equation system of the discretization can efficiently be solved by a multigrid Q-cycle. Read More: http://epubs.siam.org/doi/abs/10.1137/15M101508X
UnivIS-Import:2017-03-24:Pub.2016.tech.IMMD.c3sim.aspars
Keywords: sparse grids; prewavelets; discretization error; variable coefficients; high dimensions; finite elements; multigrid methods

Calculations of Eigenpolarization in Nd:YAG Laser Rods Due to Thermally Induced Birefringence - Hartmann, Rainer; Pflaum, Christoph; Graupeter, Thomas (IEEE Journal of Quantum Electronics, 2014)
Laser crystals like Nd:YAG are widely used in laser resonators. The stress-induced birefringence of such laser crystals has a strong influence on beam quality, power, and polarization of the output laser beam. Detailed simulations using 3-D finite-element analysis and a 2-D Jones matrix analysis were performed to analyze these effects. The finite-element analysis is used to calculate stress and birefringence depending on the cut direction of the crystal and its rotation. Eigenvalues and eigenvectors of the electromagnetic field inside the resonators are calculated by Jones matrix analysis. The analysis includes resonators with Brewster plates. Output power and beam quality are calculated by dynamic multimode analysis. Simulation results are presented for [100]-cut, [110]-cut, and [111]-cut Nd:YAG crystals.
UnivIS-Import:2015-04-14:Pub.2014.tech.IMMD.c3sim.calcul
Keywords: Birefringence; Photoelasticity; Rod lasers; Thermal and stress effects; YAG lasers

Efficient Ritz-Galerkin Discretization of PDEs with Variable Coefficients in arbitrary Dimensions using Sparse Grids - Hartmann, Rainer; Pflaum, Christoph (PAMM, 2018)
UnivIS-Import:2018-09-06:Pub.2018.tech.IMMD.c3sim.fficie
Keywords: sparse grid; variable coefficients; Ritz-Galerkin; Finite Elements

Effiziente Verfahren zur Lösung partieller Differentialgleichungen mit variablen Koeffizienten auf dünnen Gittern - Hartmann, Rainer (Friedrich-Alexander-Universität Erlangen-Nürnberg, 2018)

Modeling and simulation of ultra-short pulse amplification - Pflaum, Christoph; Hartmann, Rainer; Rahimi, Zhabiz (Solid State Lasers XXV: Technology and Devices, 2016)
Ultra-short pulses with high average power are required for a variety of technical and medical applications. Single, multi-pass, and regenerative amplifiers are used, in order to increase the power of ultra-short lasers. Typical laser crystals for such amplifiers include Ti:Sapphire or Yb:YAG laser crystals. Difficulties in the amplification of ultra-short pulses include gain narrowing effects and dispersion effects in the laser crystal. In particular, these complications arise, when a pulse stretcher is needed before amplification of the laser beam. We present a technique to model and simulate the amplification of ultra-short pulses. This technique allows to model both gain narrowing effects and decrease of beam quality caused by amplification of the laser beam. This requires a detailed 3-dimensional simulation of population inversion. Gain narrowing effects are taken into account by analyzing the gain of the spectrum of the laser beam. It is important to distinguish amplifiers with one or only two passes and a regenerative amplifier. These two different kind of amplifiers are modeled by different approaches. A regenerative amplifier is modeled by a set of time dependent rate equations. However, a single pass amplifier is modeled by a set of spatial dependent rate equations. In both cases, a system of rate equations arises from spectral discretization of the laser beam. Detailed simulation results are presente
UnivIS-Import:2017-07-10:Pub.2016.tech.IMMD.c3sim.modeli
Keywords: laser simulation

Structural characterization of intracellular C-terminal domains of group III metabotropic glutamate receptors. - Seebahn, Angela; Dinkel, Holger; Mohrlüder, J.; Hartmann, Rainer; Vogel, N.; Becker, Cord-Michael; Sticht, Heinrich; Enz, Ralf (Febs Letters, 2011)
Metabotropic glutamate receptors (mGluRs) are regulated by interacting proteins that mostly bind to their intracellular C-termini. Here, we investigated if mGluR6, mGluR7a and mGluR8a C-termini form predefined binding surfaces or if they were rather unstructured. Limited tryptic digest of purified peptides argued against the formation of stable globular folds. Circular dichroism, 1H NMR and 1H 15N HSQC spectra indicated the absence of rigid secondary structure elements. Furthermore, we localized short linear binding motifs in the unstructured receptor domains. Our data provide evidence that protein interactions of the analyzed mGluR C-termini are mediated rather by short linear motifs than by preformed folds. © 2011 Federation of European Biochemical Societies. Published by Elsevier B.V. All rights reserved.
UnivIS-Import:2015-04-14:Pub.2011.med.IB.profes_1.struct
Keywords: G-protein coupled receptor; Metabotropic glutamate receptor; Neurotransmitter receptor; Short linear motif