We discuss realization, properties and performance of the adaptive finite element approach to the design of nano-photonic components. Central issues are the construction of vectorial finite elements and the embedding of bounded components into the unbounded and possibly heterogeneous exterior. We apply the finite element method to the optimization of the design of a hollow core photonic crystal fiber. Thereby we look at the convergence of the method and discuss automatic and adaptive grid refinement and the performance of higher order elements.
We study time-delayed feedback control of noise-induced oscillations analytically and numerically for the paradigmatic model of the Van der Pol oscillator under the influence of white noise. We focus on the regime below the Hopf bifurcation where the deterministic system has a stable fixed point and does not exhibit oscillations. Analytical expressions for the power spectral density and the coherence properties of the stochastic delay differential equation in dependence upon noise intensity, delay time, and feedback strength are derived on the basis of a mean field approximation, and are in good agreement with our numerical simulations of the full nonlinear model. Our analytical results elucidate how the correlation time of the controlled stochastic oscillations can be maximized as a function of delay time and feedback strength. Introduction.-Time delayed feedback as a control mechanism is often used in systems with deterministic chaos [1], where an unstable periodic orbit embedded in a chaotic attractor can be stabilized [2]. The control scheme uses the difference of a system variable s(t) at time t and the same variable at a delayed time, s(t − τ), to generate a control force which is coupled back to the system. Variants of this scheme have also been studied, e.g. [3-5]. In contrast to control of deterministic chaos, the control of noise-induced phenomena is still an open problem. Recently, a number of methods were suggested for the control of stochastic resonance [6, 7] and of periodically forced [8], multistable [9,10], or self-oscillating [11] systems in the presence of noise. In contrast to those investigations, a passive self-adaptive method for the control of oscillations induced merely by noise was proposed only recently [12-14]. It uses time-delayed feedback control to change the coherence of the oscillations, and tune their timescale. In the present work we study a generic model for noise-induced oscillations, the Van der Pol (VdP) oscillator, below the Hopf bifurcation, i.e. in the regime where the deterministic system does not oscillate autonomously. It is applicable to a diversity of nonlinear systems. We have been able to obtain analytical results for our problem that go beyond the usual linearization and take into account the nonlinearity by a self-consistent mean field approach. This allows us to predict analytically the dependence of the coherence and the spectral properties upon the noise intensity, the control feedback strength, and the delay time τ , which complements the numerical studies in [12, 13].
High-Q optical resonances in photonic microcavities are investigated numerically using a time-harmonic finiteelement method.
We propose a new method for fast estimation of error bounds for outputs of interest in the reduced basis context, efficiently applicable to real world 3D problems. Geometric parameterizations of complicated 2D, or even simple 3D, structures easily leads to affine expansions consisting of a high number of terms (∝ 100 − 1000). Application of state-of-the-art techniques for computation of error bounds becomes practically impossible. As a way out we propose a new error estimator, inspired by the subdomain residuum method, which leads to substantial savings (orders of magnitude) regarding online and offline computational times and memory consumption. We apply certified reduced basis techniques with the newly developed error estimator to 3D electromagnetic scattering problems on unbounded domains. A numerical example from computational lithography demonstrates the good performance and effectivity of the proposed estimator. Introduction.Numerical design and optimization, as well as inverse reconstruction and parameter estimation, usually requires the multiple solution of a parameterized model, described by a partial differential equation (PDE). In these many-query or real-time contexts, short online computational times become indispensable.The reduced basis method allows us to split up the solution process of a parameterized problem into an expensive offline and a cheap online phase [1]. In the offline phase the problem is solved rigorously several times. These solutions build the reduced basis. The full problem is projected onto the reduced basis space, which results in a significant reduction of the problem size. In the online phase only the reduced system is solved. Usually the reduced basis method is applied to input-output relationships, where the inputs are parameters to a PDE and the outputs are functionals of the PDE solution. Error estimators assure the reliability of the reduced basis solution and are, therefore, of great importance.For online-offline decomposition in the reduced basis setup, an affine decomposition of the underlying PDE is essential. Geometries with complicated parameter dependencies in two dimensions and already simple geometries in three dimensions can, however, lead to affine decompositions with a very high number of terms (∝ 100− 1000). This results in poor online performance of the reduced model, when state-ofthe-art techniques are applied; see, for example, a discussion in [1]. Also, offline construction of the error estimator becomes orders of magnitude more expensive than construction of the reduced basis system itself. In the following we introduce a novel and much cheaper error estimator for the reduced basis method, inspired by the
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