Direct Numerical Optimization Research Articles

K-Surfaces: Bézier-Splines Interpolating at Gaussian Curvature Extrema

K-surfaces are an interactive modeling technique for Bézier-spline surfaces. Inspired by k -curves by [Yan et al. 2017], each patch provides a single control point that is being interpolated at a local extremum of Gaussian curvature. The challenge is to solve the inverse problem of finding the center control point of a Bézier patch given the boundary control points and the handle. Unlike the situation in 2D, bi-quadratic Bézier patches may exhibit none, one, or several extrema, and finding them is non-trivial. We solve the difficult inverse problem, including the possible selection among several extrema, by learning the desired function from samples, generated by computing Gaussian curvature of random patches. This approximation provides a stable solution to the ill-defined inverse problem and is much more efficient than direct numerical optimization, facilitating the interactive modeling framework. The local solution is used in an iterative optimization incorporating continuity constraints across patches. We demonstrate that the surface varies smoothly with the handle location and that the resulting modeling system provides local and generally intuitive control. The idea of learning the inverse mapping from handles to patches may be applicable to other parametric surfaces.

Polarization Sensitive Imaging with Qubits

We compare reconstructed quantum state images of a birefringent sample using direct quantum state tomography and inverse numerical optimization technique. Qubits are used to characterize birefringence in a flat transparent plastic sample by means of polarization sensitive measurement using density matrices of two-level quantum entangled photons. Pairs of entangled photons are generated in a type-II nonlinear crystal. About half of the generated photons interact with a birefringent sample, and coincidence counts are recorded. Coincidence rates of entangled photons are measured for a set of sixteen polarization states. Tomographic and inverse numerical techniques are used to reconstruct the density matrix, the degree of entanglement, and concurrence for each pixel of the investigated sample. An inverse numerical optimization technique is used to obtain a density matrix from measured coincidence counts with the maximum probability. Presented results highlight the experimental noise reduction, greater density matrix estimation, and overall image enhancement. The outcome of the entanglement distillation through projective measurements is a superposition of Bell states with different amplitudes. These changes are used to characterize the birefringence of a 3M tape. Well-defined concurrence and entanglement images of the birefringence are presented. Our results show that inverse numerical techniques improve overall image quality and detail resolution. The technique described in this work has many potential applications.

Impacts of Model Fidelity on Trajectory Optimization for Autonomous Vehicles in Extreme Maneuvers

In this paper, we quantify the impacts of model fidelity on the effectiveness of trajectory optimization for autonomous vehicles when driving at the limits of friction through experiments with a full-size vehicle. Models ranging from a double-track model with lateral and longitudinal load transfer dynamics to a simple point-mass model are used in combination with direct numerical optimization to generate optimal trajectories subject to the limits imposed by each model. The effectiveness of each model for trajectory planning is evaluated by testing the trajectories on an automated vehicle across friction conditions ranging from ice to dry asphalt. Comparisons between the outright performance of the car and the car's ability to track the optimal trajectory are made across the various models. The tests reveal that the advantage of more complex models is less that they better predict the vehicle's behavior, but that they provide a more nuanced view of the vehicle's limits and guidance on the proper coordination of the various actuators on the vehicle in order to make most efficient use of the available tire friction.

Analytical Solution to the Wave-Drag Minimization Problem for an Axisymmetric Fore-Body Using Local Linearization

The problem of constructing an axisymmetric forebody with minimal wave drag with given constraints on volume and dimensions has been solved within the framework of the local linearization of the relationship between the geometric parameters and the gas-dynamic functions. The optimal fore-body has been determined by varying the shape of the cone with equivalent elongation, for which the approximation for the objective function (wave drag related with volume) has been determined on the basis of the known exact values of the flow parameters. Characteristic features of the optimal shape are bluntness of the nose across the front face and a smooth conjugation with closing cylinder for bodies with sufficiently large volume, the latter exceeding the value depending on the elongation and the Mach number. Comparison with the results of direct numerical optimization in the framework of Euler’s model has shown that the proposed analytical solution provides a near-minimum wave drag.

A procedure for the extraction of a nonlinear microwave GaN FET model

SummaryIn this paper, we describe an extraction procedure of nonlinear models for microwave field‐effect transistor (FET). We use a nonlinear model available in commercial CAD tools, and we extract the parameters by combining direct extraction and numerical optimization. We determine a first estimate of the model parameters by few DC and S‐parameter measurements. Next, we refine the parameters by optimization against low‐frequency and high‐frequency vector‐calibrated large‐signal measurements gathered with a Large‐Signal‐Network Analyzer (LSNA). As case study we consider a 0.25 × 200 µm2 GaN FET on SiC for power amplifier applications. Ultimately, we want to show that a good accuracy level can be achieved while minimizing the extraction effort and that an accurate model can be built and suitably tailored depending on the final application. Copyright © 2016 John Wiley & Sons, Ltd.

Thermal analysis of HVAC and solar panels using genetic optimization algorithm

The present study investigates the use of HVAC and solar heating systems inside residential zones using a combination of direct numerical modeling and genetic optimization method in different climates of Iran. The direct numerical modeling is capable of calculating temperature field inside buildings. Then, direct modelling algorithm is used to determine energy consumption rate of the building in the different climates of Iran. The results of direct numerical modelling are checked against those of carrier and energy-plus software. The results show good accordance with those of numerical simulations presented in the open literature.

How to discretize a quantum bath for real-time evolution

Many numerical techniques for the description of quantum systems that are coupled to a continuous bath require the discretization of the latter. To this end, a wealth of methods has been developed in the literature, which we classify as (i) direct discretization, (ii) orthogonal polynomial, and (iii) numerical optimization strategies. We recapitulate strategies (i) and (ii) to clarify their relation. For quadratic Hamiltonians, we show that (ii) is the best strategy in the sense that it gives the numerically exact time evolution up to a maximum time ${t}_{\text{max}}$, for which we give a simple expression. For nonquadratic Hamiltonians, we show that no such best strategy exists. We present numerical examples relevant to open quantum systems and strongly correlated systems, as treated by dynamical mean-field theory (DMFT).

An adaptive wavelet collocation method for solving optimal control problem

A sequential solution approach based on an adaptive wavelet collocation method is proposed for solving optimal control problems. By expanding the state and control variables with wavelet multi-resolution decomposition, an original optimal control problem can be transcribed into a nonlinear programming problem that can be solved by general methods. In the proposed framework, an iterative algorithm starts from an initial coarse grid with few collocations. Subsequent solutions can thus be obtained on a dynamically refined grid by an adaptive wavelet collocation method, while the previous coarser level solution can be taken as the initial guess for the next iteration. Consequently, the computational grid for direct numerical optimization method is able to automatically adapt to any irregularities or discontinuities in the solution. The efficiency and accuracy of the proposed algorithm are verified by two typical examples.

Identification of metabolic network models from incomplete high-throughput datasets

Motivation: High-throughput measurement techniques for metabolism and gene expression provide a wealth of information for the identification of metabolic network models. Yet, missing observations scattered over the dataset restrict the number of effectively available datapoints and make classical regression techniques inaccurate or inapplicable. Thorough exploitation of the data by identification techniques that explicitly cope with missing observations is therefore of major importance.Results: We develop a maximum-likelihood approach for the estimation of unknown parameters of metabolic network models that relies on the integration of statistical priors to compensate for the missing data. In the context of the linlog metabolic modeling framework, we implement the identification method by an Expectation-Maximization (EM) algorithm and by a simpler direct numerical optimization method. We evaluate performance of our methods by comparison to existing approaches, and show that our EM method provides the best results over a variety of simulated scenarios. We then apply the EM algorithm to a real problem, the identification of a model for the Escherichia coli central carbon metabolism, based on challenging experimental data from the literature. This leads to promising results and allows us to highlight critical identification issues.Contact: sara.berthoumieux@inria.fr; eugenio.cinquemani@inria.frSupplementary information: Supplementary data are available at Bioinformatics online.

Spatial models generated by nested stochastic partial differential equations, with an application to global ozone mapping

A new class of stochastic field models is constructed using nested stochastic partial differential equations (SPDEs). The model class is computationally efficient, applicable to data on general smooth manifolds, and includes both the Gaussian Mat\'{e}rn fields and a wide family of fields with oscillating covariance functions. Nonstationary covariance models are obtained by spatially varying the parameters in the SPDEs, and the model parameters are estimated using direct numerical optimization, which is more efficient than standard Markov Chain Monte Carlo procedures. The model class is used to estimate daily ozone maps using a large data set of spatially irregular global total column ozone data.

Development and validation of an efficient direct numerical optimisation approach for aerofoil shape design

AbstractIntelligent shape optimisation architecture is developed, validated and applied in the design of high-altitude long endurance aerofoil (HALE). The direct numeric optimisation (DNO) approach integrating a geometrical shape parameterisation model coupled to a validated flow solver and a population based search algorithm are applied in the design process. The merit of the DNO methodology is measured by computational time efficiency and feasibility of the optimal solution. Gradient based optimisers are not suitable for multi-modal solution topologies. Thus, a novel particle swarm optimiser with adaptive mutation (AM-PSO) is developed. The effect of applying the PARSEC and a modified variant of the original function, as a shape parameterisation model on the global optimal is verified. Optimisation efficiency is addressed by mapping the solution topology for HALE aerofoil designs and by computing the sensitivity of aerofoil shape variables on the objective function. Variables with minimal influence are identified and eliminated from shape optimisation simulations. Variable elimination has a negligible effect on the aerodynamics of the global optima, with a significant reduction in design iterations to convergence. A novel data-mining technique is further applied to verify the accuracy of the AM-PSO solutions. The post-processing analysis, to swarm optimisation solutions, indicates a hybrid optimisation methodology with the integration of global and local gradient based search methods, yields a true optima. The findings are consistent for single and multi-point designs.

Comparative Study of the Model and Optimum Cascades for Multicomponent Isotope Separation

The possibility of estimating the minimum total flow in a cascade with concentrations of a target component given in the product and waste flows by means of a model match abundance ratio cascade (MARC) is studied. The parameters required to describe MARC characteristics are the total number of separation stages, the feed flow location, and the M* parameter, which is equal to a half-sum of mass numbers of the target and the supporting components. Specific research carried out independently in two scientific labs in China and Russia has demonstrated that the integral parameters of the MARC, optimized by the M* parameter, are very close to that of the optimum by the minimum total flow cascade found by means of numerical optimization. The calculation is performed for separation of krypton isotopes when the end component 78Kr and the intermediate component 83Kr are considered to be the targets. It paves the way to use the optimized MARC parameters for two purposes: first, for fast and easy evaluation of the real cascade parameters and second, as an initial guess in its further direct numerical optimization, thereby allowing significant savings in computation time.

Tuning of PID controllers based on simplified single parameter optimisation

A simple method to design PID controllers in the frequency domain based on a simplified constrained optimisation is proposed. The method is based on the use of a single tuning parameter, defined as the quotient between the final crossover frequency and the zero of the controller. The tuning procedure is based on the maximisation of the controller gain subject to an equality constraint in the phase margin and an inequality constraint in the gain margin. The main advantage of the proposed method is that, even though the maximisation of the controller gain is straightforward, since there is only one parameter to be tuned, the solution is close to the optimal tuning obtained with direct numerical optimisation methods. Moreover the method is applicable to any linear model structure, including dead time and non-minimum phase systems.

DepmixS4: AnRPackage for Hidden Markov Models

<b>depmixS4</b> implements a general framework for defining and estimating dependent mixture models in the <b>R</b> programming language. This includes standard Markov models, latent/hidden Markov models, and latent class and finite mixture distribution models. The models can be fitted on mixed multivariate data with distributions from the <b>glm</b> family, the (logistic) multinomial, or the multivariate normal distribution. Other distributions can be added easily, and an example is provided with the <i>exgaus</i> distribution. Parameters are estimated by the expectation-maximization (EM) algorithm or, when (linear) constraints are imposed on the parameters, by direct numerical optimization with the <b>Rsolnp</b> or <b>Rdonlp2</b> routines.

Optimization of Rotorcraft Simultaneous Noninterfering Noise Abatement Approach Procedures

T HE concept of simultaneous noninterfering (SNI) approaches for rotorcraft is currently being explored in both Europe and the United States. The SNI concept was initially conceived in the United States to enable the integration of both fixed wing and rotorcraft into the terminal airspace system to reduce delays and increase capacity [1]. It comprises primarily specific instrument flight rules (IFR) departure and approach procedures which enable rotorcraft to operate independently from fixed-wing traffic IFR streams. Because of the much lower approach speeds of helicopters, sequencing helicopter operations into the IFR approach stream tends to result in congestion and undesirable delays. At present, helicopter operators often prefer to operate under visual flight rules to avoid complex IFR procedures, but clearly this does not provide a reliable air transport service in all weather conditions. Also, in Europe, research pertaining to SNI operations is well underway, notably in the sixth framework project OPTIMAL (Optimized Procedures and Techniques for Improvement of Approach and Landing) [2]. The main aim of the OPTIMAL project is to increase airport capacity and efficiency, while reducing the noise footprint. A downside of the SNI concept is that new SNI routes will be located in previously unused airspace and consequently may overfly noise-sensitive communities that were not affected by traffic movements before. This, coupled with the fact that helicopter approach operations tend to be relatively noisy, indicates a clear need for optimization of the SNI routes and flight paths such as to reduce the noise impact in the affected residential communities. Optimization of SNI approaches with respect to community noise impact was first explored by Atkins and Xue [3,4]. In [3,4], they present an optimization technique for segmented three-dimensional SNI trajectory design based on an incremental search strategy that combines a k-ary tree with Dijkstra’s algorithm. The objective function is based on a validated rotorcraft noise model as well as terminal area population density data. Fixed-wing airspace corridors were treated as impenetrable obstacles. In this approach, a trajectory is described as a sequence of trimmed flight segments that connect initial approach and landing sites. The number of segments considered in this study remains limited (typicallyfive or less) primarily due to the fact that the computational load increases rapidly with the number of segments. To make the approach numerically tractable, a number of simplifying assumptions were introduced. In particular, it was assumed that transitions between trim states do not appreciably affect solution cost. The optimization of noise abatement procedures for fixed-wing aircraft has received considerablymore attention. In [5], an optimization approach is presented that, similar to [3,4], relies on segmented routes. The segmented routes, optimized using a genetic algorithm, enable a specification of simple procedures that are amenable to fast online computer solutions that can be readily implemented in a guidance system. In [6–8], a noise-optimized approach and departure trajectories are computed based on a direct numerical optimization techniques that enable one to generate (piecewise) continuous optimal trajectories for a point-mass modeled fixed-wing aircraft. The numerical tool that has been used in these particular studies is called NOISHHH. The NOISHHH tool essentially combines a noise model, a dose-response relationship, an emission inventory model, a geographic information system, and a dynamic trajectory optimization algorithm. NOISHHH generates routings and flight paths for both arrivals and departures that minimize the environmental impact in the residential communities surrounding the airport, while satisfying all imposed operational and safety constraints. The present study is aimed at extending the NOISHHH tool to permit the computation of noise-optimized SNI trajectories for rotorcraft. The extension does not only entail a modification of the dynamic vehiclemodel and the operational context, but also involves an adaptation of the implemented noisemodel. The environmentallyoptimized SNI trajectories calculated with the modified version of NOISHHH are illustrated in an example scenario involving an SNI instrument approach of a Robinson R22 helicopter [9] to a helispot on runway 22 of Schiphol airport (Amsterdam) in the Netherlands.

Design of multimode axisymmetric hypersonic nozzles with the use of optimization methods

On the basis of the solution of direct problems with the use of various medium models and numerical meth- ods of integration of gas flow equations, methods for designing super- and hypersonic nozzles have been de- veloped. Multimode nozzles in the ranges of Mach numbers 8-14 and 14-20 satisfying given conditions have been designed. optimization. Introduction. It is known that good quality of flow in a wind tunnel is provided by the application of thor- oughly shaped nozzles accelerating the working gas to a given velocity. Such nozzles have a spatial structure, are made with a high precision, and are rather expensive. Practically in all existing wind tunnels single-mode nozzles de- signed for definite operating conditions are used. Because of the complex structure of single-mode nozzles and the high cost of their production, the number of modes of operation of a wind tunnel is always limited. There exist two approaches to the design of multimode shaped nozzles in which the main outlet section has a fixed geometry and the small separable part adjoining the minimal section makes it possible to vary the Mach number at the outlet from the nozzle. The first approach is based on constructing the nozzle contour with the use of the method of characteristics (1-4), and the second approach is based on methods of direct numerical optimization. For example, in (5) the func- tional equal to the sum of standard deviations of the Mach number from its average values in a given region of the flow was minimized. Both approaches did not take into account the presence of viscosity and, accordingly, of the boundary layer that radically changes the characteristics of hypersonic nozzles (6). Therefore, to solve the design prob- lem stated, we used a complex approach combining the method of characteristics that permits calculating fairly easily supersonic nozzles with a uniform outlet characteristic for a perfect nonviscous non-heat-conducting gas, direct meth- ods based on the calculation of viscous flows, and methods of numerical optimization. Construction of the Initial Approximation. The first stage of designing a nozzle consists of constructing by the method of characteristics the nozzle contour for the maximum value of the Mach number at the outlet and deter- mining the size of the acceleration part of the nozzle for the minimum Mach number. This makes it possible to divide the contour into a fixed equailizing part and a variable accelerating part (much smaller in size) whose shape is deter- mined for the chosen value of the Mach number at the outlet from the nozzle (7). This stage is an important element of the chosen strategy since the solution of optimization problems for viscous flows required large computational re- sources and is successful, as a rule, only given a good initial approximation. Design of Multimode Nozzles. Since the viscous flow in a hypersonic nozzle of a given geometry was sup- posed to be calculated by means of the Fluent package, it was necessary to test the accuracy of the solution of the Navier-Stokes equation with the k-ω-model of turbulence. The contour obtained by the method of (5) was used in creating a real axisymmetric nozzle with a nozzle outlet diameter of 0.40 m meant for obtaining Mach numbers 8, 10, 12, and 14 at the outlet. The comparison between the experimental and numerical distributions of the Mach number

A new real-time guidance strategy for aerodynamic ascent flight

Reusable launch vehicles are conceived to constitute the future space transportation system. If these vehicles use air-breathing propulsion and lift taking-off horizontally, the optimal steering for these vehicles exhibits completely different behavior from that in conventional rockets flight. In this paper, the new guidance strategy is proposed. This method derives from the optimality condition as for steering and an analysis concludes that the steering function takes the form comprised of Linear and Logarithmic terms, which include only four parameters. The parameter optimization of this method shows the acquired terminal horizontal velocity is almost same with that obtained by the direct numerical optimization. This supports the parameterized Liner Logarithmic steering law. And here is shown that there exists a simple linear relation between the terminal states and the parameters to be corrected. The relation easily makes the parameters determined to satisfy the terminal boundary conditions in real-time. The paper presents the guidance results for the practical application cases. The results show the guidance is well performed and satisfies the terminal boundary conditions specified. The strategy built and presented here does guarantee the robust solution in real-time excluding any optimization process, and it is found quite practical.

Feedback Approximation of the Stochastic Growth Model by Genetic Neural Networks

A direct numerical optimization method is developed to approximate the one-sector stochastic growth model. The feedback investment policy is parameterized as a neural network and trained by a genetic algorithm to maximize the utility functional over the space of time-invariant investment policies. To eliminate the dependence of training on the initial conditions, at any generation, the same stationary investment policy (the same network) is used to repeatedly solve the problem from differing initial conditions. The fitness of a given policy rule is then computed as the sum of payoffs over all initial conditions. The algorithm performs quite well under a wide set of parameters. Given the general purpose nature of the method, the flexibility of neural network parametrization and the global nature of the genetic algorithm search, it can be easily extended to tackle problems with higher dimensional nonlinearities, state spaces and/or discontinuities.

Optimal strictly positive real controllers using direct optimization

In this paper, we consider the H 2 -optimal control problem subject to the constraint that the resulting controller be strictly positive real. A direct numerical optimization approach is adopted in conjunction with a controller parametrization that is linear in the unknown parameters. The SPR constraint is easily expressed at each frequency in the form of a linear inequality. The method is applied to a numerical example from the literature and good results are achieved. In particular, the proposed method is particularly adept at determining low order controllers.

Analysis of optical waveguides with multilayer dielectric coatings using plane wave expansion

A numerical analysis of optical waveguides with multilayer dielectric coatings is presented. The simulation is based on the plane wave expansion method, taking into account the exact guided mode profile of the waveguide. The method is free from any assumption about the homogeneity of the waveguide. The power reflectivity is calculated for both antireflection coatings and high-reflection coatings. The accuracy of the model is verified by comparing with the exact results and other simulation methods. The speed and accuracy of the plane wave expansion method make it suitable for direct numerical optimization. The method can also be used as an analytic tool to analyze the behavior of waveguides with multidielectric coatings.