A mixture density network approach to modelling and exploiting uncertainty in nonlinear control problems

In nonlinear and stochastic control problems, learning an efficient feed-forward controller is not amenable to conventional neurocontrol methods. For these approaches, estimating and then incorporating uncertainty in the controller and feed-forward models can produce more robust control results. Here, we introduce a novel inversion-based neurocontroller for solving control problems involving uncertain nonlinear systems which could also compensate for multi-valued systems. The approach uses recent developments in neural networks, especially in the context of modelling statistical distributions, which are applied to forward and inverse plant models. Provided that certain conditions are met, an estimate of the intrinsic uncertainty for the outputs of neural networks can be obtained using the statistical properties of networks. More generally, multicomponent distributions can be modelled by the mixture density network. Based on importance sampling from these distributions a novel robust inverse control approach is obtained. This importance sampling provides a structured and principled approach to constrain the complexity of the search space for the ideal control law. The developed methodology circumvents the dynamic programming problem by using the predicted neural network uncertainty to localise the possible control solutions to consider. A nonlinear multi-variable system with different delays between the input-output pairs is used to demonstrate the successful application of the developed control algorithm. The proposed method is suitable for redundant control systems and allows us to model strongly non-Gaussian distributions of control signal as well as processes with hysteresis. © 2004 Elsevier Ltd. All rights reserved.

Nonlinear Time-Fractional Differential Equations in Combustion Science

MSC 2010: 34A08 (main), 34G20, 80A25

Increasing the Calculation Speed of an Acceleration Scheme for an Age-Structured Diffusion Model

Дойчин Бояджиев, Галена Пеловска - В статията се предлага оптимизиран алгоритъм, който е по-бърз в сравнение с по- рано описаната ускорена (модифицирана STS) диференчна схема за възрастово структуриран популационен модел с дифузия. Запазвайки апроксимацията на модифицирания STS алгоритъм, изчислителното времето се намаля почти два пъти. Това прави оптимизирания метод по-предпочитан за задачи с нелинейност или с по-висока размерност.

Nonlinear Normalization in Limit Theorems for Extremes

2000 Mathematics Subject Classification: 60G70, 60F05.

Strong Consistency of the Conditional Least Squares Estimator for a Nonstationary Process. Example of the Garch Model

2000 Mathematics Subject Classification: Primary: 62M10, 62J02, 62F12, 62M05, 62P05, 62P10; secondary: 60G46, 60F15.

Validation of a multi-component digital dissolution model for irregular particles

A new mesoscale simulation model for solids dissolution based on an computationally efficient and versatile digital modelling approach (DigiDiss) is considered and validated against analytical solutions and published experimental data for simple geometries. As the digital model is specifically designed to handle irregular shapes and complex multi-component structures, use of the model is explored for single crystals (sugars) and clusters. Single crystals and the cluster were first scanned using X-ray microtomography to obtain a digital version of their structures. The digitised particles and clusters were used as a structural input to digital simulation. The same particles were then dissolved in water and the dissolution process was recorded by a video camera and analysed yielding: the overall dissolution times and images of particle size and shape during the dissolution. The results demonstrate the coherence of simulation method to reproduce experimental behaviour, based on known chemical and diffusion properties of constituent phase. The paper discusses how further sophistications to the modelling approach will need to include other important effects such as complex disintegration effects (particle ejection, uncertainties in chemical properties). The nature of the digital modelling approach is well suited to for future implementation with high speed computation using hybrid conventional (CPU) and graphical processor (GPU) systems.

A general Approach to Methods with a Sparse Jacobian for Solving Nonlinear Systems of Equations

2000 Mathematics Subject Classification: 65H10.

Quantization of the Inhomogeneous Bianchi I model:quasi-Heisenberg picture

The quantization scheme is suggested for a spatially inhomogeneous 1+1 Bianchi I model. The scheme consists in quantization of the equations of motion and gives the operator (so called quasi-Heisenberg) equations describing explicit evolution of a system. Some particular gauge suitable for quantization is proposed. The Wheeler-DeWitt equation is considered in the vicinity of zero scale factor and it is used to construct a space where the quasi-Heisenberg operators act. Spatial discretization as a UV regularization procedure is suggested for the equations of motion.

Modeling of CW Yb-doped fiber lasers with highly nonlinear cavity dynamics

We develop a theoretical framework for modeling of continuous wave Yb-doped fiber lasers with highly nonlinear cavity dynamics. The developed approach has shown good agreement between theoretical predictions and experimental results for particular scheme of Yb-doped laser with large spectral broadening during single round trip. The model is capable to accurately describe main features of the experimentally measured laser outputs such as power efficiency slope, power leakage through fibre Bragg gratings, spectral broadening and spectral shape of generated radiation. © 2011 Optical Society of America.

Filter-based dispersion-managed versatile ultrafast fibre laser

We present the operation of an ultrafast passively mode-locked fibre laser, in which flexible control of the pulse formation mechanism is readily realised by an in-cavity programmable filter the dispersion and bandwidth of which can be software configured. We show that conventional soliton, dispersion- managed (DM) soliton (stretched-pulse) and dissipative soliton mode-locking regimes can be reliably targeted by changing the filter’s dispersion and bandwidth only, while no changes are made to the physical layout of the laser cavity. Numerical simulations are presented which confirm the different nonlinear pulse evolutions inside the laser cavity. The proposed technique holds great potential for achieving a high degree of control over the dynamics and output of ultrafast fibre lasers, in contrast to the traditional method to control the pulse formation mechanism in a DM fibre laser, which involves manual optimisation of the relative length of fibres with opposite-sign dispersion in the cavity. Our versatile ultrafast fibre laser will be attractive for applications requiring different pulse profiles such as in optical signal processing and optical communications.

Perception of global image contrast is predicted by the same spatial integration model of gain control as detection and discrimination

How are the image statistics of global image contrast computed? We answered this by using a contrast-matching task for checkerboard configurations of ‘battenberg’ micro-patterns where the contrasts and spatial spreads of interdigitated pairs of micro-patterns were adjusted independently. Test stimuli were 20 × 20 arrays with various sized cluster widths, matched to standard patterns of uniform contrast. When one of the test patterns contained a pattern with much higher contrast than the other, that determined global pattern contrast, as in a max() operation. Crucially, however, the full matching functions had a curious intermediate region where low contrast additions for one pattern to intermediate contrasts of the other caused a paradoxical reduction in perceived global contrast. None of the following models predicted this: RMS, energy, linear sum, max, Legge and Foley. However, a gain control model incorporating wide-field integration and suppression of nonlinear contrast responses predicted the results with no free parameters. This model was derived from experiments on summation of contrast at threshold, and masking and summation effects in dipper functions. Those experiments were also inconsistent with the failed models above. Thus, we conclude that our contrast gain control model (Meese & Summers, 2007) describes a fundamental operation in human contrast vision.

Comparison of DSP-based nonlinear equalizers for intra-channel nonlinearity compensation in coherent optical OFDM

A novel versatile digital signal processing (DSP)-based equalizer using support vector machine regression (SVR) is proposed for 16-quadrature amplitude modulated (16-QAM) coherent optical orthogonal frequency-division multiplexing (CO-OFDM) and experimentally compared to traditional DSP-based deterministic fiber-induced nonlinearity equalizers (NLEs), namely the full-field digital back-propagation (DBP) and the inverse Volterra series transfer function-based NLE (V-NLE). For a 40 Gb/s 16-QAM CO-OFDM at 2000 km, SVR-NLE extends the optimum launched optical power (LOP) by 4 dB compared to V-NLE by means of reduction of fiber nonlinearity. In comparison to full-field DBP at a LOP of 6 dBm, SVR-NLE outperforms by ∼1 dB in Q-factor. In addition, SVR-NLE is the most computational efficient DSP-NLE.

Strongly localized moving discrete dissipative breather-solitons in Kerr nonlinear media supported by intrinsic gain

We investigate the mobility of nonlinear localized modes in a generalized discrete Ginzburg-Landau-type model, describing a one-dimensional waveguide array in an active Kerr medium with intrinsic, saturable gain and damping. It is shown that exponentially localized, traveling discrete dissipative breather-solitons may exist as stable attractors supported only by intrinsic properties of the medium, i.e., in the absence of any external field or symmetry-breaking perturbations. Through an interplay by the gain and damping effects, the moving soliton may overcome the Peierls-Nabarro barrier, present in the corresponding conservative system, by self-induced time-periodic oscillations of its power (norm) and energy (Hamiltonian), yielding exponential decays to zero with different rates in the forward and backward directions. In certain parameter windows, bistability appears between fast modes with small oscillations and slower, large-oscillation modes. The velocities and the oscillation periods are typically related by lattice commensurability and exhibit period-doubling bifurcations to chaotically "walking" modes under parameter variations. If the model is augmented by intersite Kerr nonlinearity, thereby reducing the Peierls-Nabarro barrier of the conservative system, the existence regime for moving solitons increases considerably, and a richer scenario appears including Hopf bifurcations to incommensurately moving solutions and phase-locking intervals. Stable moving breathers also survive in the presence of weak disorder. © 2014 American Physical Society.

Kelvin-wave cascade in the vortex filament model

The small-scale energy-transfer mechanism in zero-temperature superfluid turbulence of helium-4 is still a widely debated topic. Currently, the main hypothesis is that weakly nonlinear interacting Kelvin waves (KWs) transfer energy to sufficiently small scales such that energy is dissipated as heat via phonon excitations. Theoretically, there are at least two proposed theories for Kelvin-wave interactions. We perform the most comprehensive numerical simulation of weakly nonlinear interacting KWs to date and show, using a specially designed numerical algorithm incorporating the full Biot-Savart equation, that our results are consistent with the nonlocal six-wave KW interactions as proposed by L'vov and Nazarenko.

A Harrod modell strukturális stabilitása (Structural stability of the Harrod model)

In this study it is shown that the nontrivial hyperbolic fixed point of a nonlinear dynamical system, which is formulated by means of the adaptive expectations, corresponds to the unstable equilibrium of Harrod. We prove that this nonlinear dynamical (in the sense of Harrod) model is structurally stable under suitable economic conditions. In the case of structural stability, small changes of the functions (C1-perturbations of the vector field) describing the expected and the true time variation of the capital coefficients do not influence the qualitative properties of the endogenous variables, that is, although the trajectories may slightly change, their structure is the same as that of the unperturbed one, and therefore these models are suitable for long-time predictions. In this situation the critique of Lucas or Engel is not valid. There is no topological conjugacy between the perturbed and unperturbed models; the change of the growth rate between two levels may require different times for the perturbed and unperturbed models.