Trust Region Algorithm For Optimization Research Articles

Development of soil water characteristic equations for various soil texture classes

The investigation of unsaturated soil through numerical modeling requires the prior information on soil properties including Soil Water Characteristic (SWC) equation. Acquiring these data for different soil texture classes is laborious and time-consuming. Consequently, there is a need for predefined dataset numerical modeling of unsaturated zone in different soil texture classes. In this study, 32 soil samples were collected from different locations, and soil physical properties like particle size distribution, particle density, bulk density, porosity, and saturated hydraulic conductivity were determined. Six different soil texture classes have been found in this study and average properties were computed for each soil texture class. SWC curves were developed for all soil samples using pressure plate apparatus. Based on average SWC curves, optimal parameters of four different SWC equations (Brooks and Corey, Campbell, Van Genuchten, and Fredlund and Xing models) were determined using non-linear regression with the Trust Region Optimization algorithm with R 2 and RMSE values ranging from 0.9650 to 0.9983 and from 0.0018 to 0.0219 cm3/cm3, respectively. All the four SWC models were compared using Taylor diagrams, which show that Fredlund and Xing models are the best models among the four. Further, the performance of developed SWC equations was compared with the moisture content of UNSODA datasets and results were satisfactory as RMSE ranges from 0.0381 to 0.1265 cm3/cm3. Thus, this study provides a reliable SWC equation for different soils in Bihar.

Optimization of ultrathin polarization insensitive metamaterial absorbers using trust region algorithm based on Co-kriging model

In this paper, a trust region optimization algorithm integrated with a co-kriging surrogate-model is newly formulated to optimize compact ultrathin wideband metamaterial absorbers with high absorption. The reported algorithm is linked with the full vectorial finite element method to simulate and optimize the studied absorbers. In order to show the power of the suggested algorithm, the geometrical parameters of a previous metamaterial absorber (MA) with L-shaped resonators are optimized to achieve high absorption (over 90 %) over a frequency band from 13.8 GHz to 24.5 GHz covering most of KU and K bands. Then, a novel design of L-shaped MA with triangular resonators is optimized, analyzed and fabricated with high absorption (above 90 %) through a large frequency band from 11.9 GHz to 22 GHz covering the entire KU-band and most of the K-band. The reported design is also robust against the variations of the incident angle (above 80 % up to 50°) for both transverse electric (TE) and transverse magnetic (TM) polarized modes. Therefore, the proposed algorithm could be used efficiently for the design and optimization of metamaterial absorbers.

Reconstruction of incidence reporting rate for SARS-CoV-2 Delta variant of COVID-19 pandemic in the US

In recent years, advanced regularization techniques have emerged as a powerful tool aimed at stable estimation of infectious disease parameters that are crucial for future projections, prevention, and control. Unlike other system parameters, i.e., incubation and recovery rates, the case reporting rate, Ψ, and the time-dependent effective reproduction number, Re(t), are directly influenced by a large number of factors making it impossible to pre-estimate these parameters in any meaningful way. In this study, we propose a novel iteratively-regularized trust-region optimization algorithm, combined with SuSvIuIvRD compartmental model, for stable reconstruction of Ψ and Re(t) from reported epidemic data on vaccination percentages, incidence cases, and daily deaths. The innovative regularization procedure exploits (and takes full advantage of) a unique structure of the Jacobian and Hessian approximation for the nonlinear observation operator. The proposed inversion method is thoroughly tested with synthetic and real SARS-CoV-2 Delta variant data for different regions in the United States of America from July 9, 2021, to November 25, 2021. Our study shows that case reporting rate during the Delta wave of COVID-19 pandemic in the US is between 12% and 37%, with most states being in the range from 15% to 25%. This confirms earlier accounts on considerable under-reporting of COVID-19 cases due to the impact of ”silent spreaders” and the limitations of testing.

Gradient-based optimization for multi-scale geographically weighted regression

Multi-scale geographically weighted regression (MGWR) is among the most popular methods to analyze non-stationary spatial relationships. However, the current model calibration algorithm is computationally intensive: its runtime has a cubic growth with the sample size, while its memory use grows quadratically. We propose calibrating MGWR with gradient-based optimization. This is obtained by analytically deriving the gradient vector and the Hessian matrix of the corrected Akaike information criterion (AICc) and wrapping them with a trust-region optimization algorithm. We evaluate the model quality empirically. Our method converges to the same coefficients and produces the same inference as the current method but it has a substantial computational gain when the sample size is large. It reduces the runtime to quadratic convergence and makes the memory use linear with respect to sample size. Our new algorithm outperforms the existing alternatives and makes MGWR feasible for large spatial datasets.

A tandem trust-region optimization approach for ill-posed falling weight deflectometer backcalculation

Falling weight deflectometer backcalculation is a structural health monitoring approach for estimating the dynamic modulus of flexible pavements. It consists of two key aspects: a computational pavement model and an optimization routine. When using gradient-based methods, the optimization problem is commonly ill-posed, whereby a unique solution does not necessarily exist. In this paper, a new tandem trust-region optimization algorithm is proposed for ill-posed falling weight deflectometer backcalculation. The algorithm’s performance is tested against existing optimization methods in the context of dynamic modulus estimation for flexible pavements, and the performance tests are simulated computationally using practical values for material properties and geometry. The tandem trust-region algorithm combines the relative strengths of the subspace trust-region interior reflective method with those of the Levenberg–Marquardt algorithm. The increased computational expense of executing these two methods in parallel is negligible compared to the expense of other essential steps in backcalculation. For ill-posed problems, the performance tests indicate the tandem trust region algorithm has an overall reliability that is 33.9% higher than using only the subspace trust-region interior reflective method, and 56.9% higher than using only the Levenberg–Marquardt algorithm. Further, the new optimizer is 13.5% more reliable than a robust commercial option.

Trust-Region Approximation of Extreme Trajectories in Power System Dynamics

In this work we present a novel technique, based on a trust-region optimization algorithm and second-order trajectory sensitivities, to compute the extreme trajectories of power system dynamic simulations given a bounded set that represents parametric uncertainty. We show how this method, while remaining computationally efficient compared with sampling-based techniques, overcomes the limitations of previous sensitivity-based techniques to approximate the bounds of the trajectories, when the local approximation loses validity because of the nonlinearity. In addition, we show how this method can be adapted to account for those cases in which the initial conditions depend on the uncertain parameter. To conclude, we present several numerical experiments that showcase the accuracy and scalability of the technique, including a demonstration on the IEEE New England test system.

Quartic Riemannian Trust Region Algorithm for Cognitive Radar Ambiguity Function Shaping

Waveform adaptation grants cognitive radar (CR) the ability to adapt to its environment, which requires an effective framework to synthesize waveforms sharing a desired ambiguity function (AF). In this letter, we propose a novel method for shaping the slow-time AF in order to adaptively suppress the interference power. The problem is formulated as minimizing the average value of the slow-time AF over some range Doppler bins spanned by the interference, which can be identified exploiting a plurality of knowledge sources. From a technical point of view, this is tantamount to optimizing a complex quartic-order polynomial with a constant modulus (CM) constraint on each optimization variable. To solve this problem, we proposed a quartic Riemannian trust region algorithm. This algorithm first transforms the optimization into an unconstrained one in a complex circle Riemannian manifold, then devises a new Riemannian trust region optimization algorithm that invokes Riemannian gradient and Hessian matrix to obtain an iterative solution with super-linear convergence rate and ability to escape potential saddle points. Simulation results demonstrated that our proposed algorithm outperformed state-of-the art approaches for AF shaping while being computationally less expensive.

Identification of Laminar Composition in Cerebral Cortex Using Low-Resolution Magnetic Resonance Images and Trust Region Optimization Algorithm.

Pathological changes in the cortical lamina can cause several mental disorders. Visualization of these changes in vivo would enhance their diagnostics. Recently a framework for visualizing cortical structures by magnetic resonance imaging (MRI) has emerged. This is based on mathematical modeling of multi-component T1 relaxation at the sub-voxel level. This work proposes a new approach for their estimation. The approach is validated using simulated data. Sixteen MRI experiments were carried out on healthy volunteers. A modified echo-planar imaging (EPI) sequence was used to acquire 105 individual volumes. Data simulating the images were created, serving as the ground truth. The model was fitted to the data using a modified Trust Region algorithm. In single voxel experiments, the estimation accuracy of the T1 relaxation times depended on the number of optimization starting points and the level of noise. A single starting point resulted in a mean percentage error (MPE) of 6.1%, while 100 starting points resulted in a perfect fit. The MPE was <5% for the signal-to-noise ratio (SNR) ≥ 38 dB. Concerning multiple voxel experiments, the MPE was <5% for all components. Estimation of T1 relaxation times can be achieved using the modified algorithm with MPE < 5%.

A novel fast Fourier transform accelerated off-grid exhaustive search method for cryo-electron microscopy fitting

This paper presents a novel fast Fourier transform (FFT)-based exhaustive search method extended to off-grid translational and rotational degrees of freedom. The method combines the advantages of the FFT-based exhaustive search, which samples all the conformations of a system under study on a grid, with a local optimization technique that guarantees to find the nearest optimal off-grid conformation. The method is demonstrated on a fitting problem and can be readily applied to a docking problem. The algorithm first samples a scoring function on a six-dimensional grid of sizeN6using the FFT. This operation has an asymptotic complexity ofO(N6logN). Then, the method performs the off-grid search using a local quadratic approximation of the cost function and the trust-region optimization algorithm. The computation of the quadratic approximation is also accelerated by FFT at the same additional asymptotic cost ofO(N6logN). The method is demonstrated by fitting atomic protein models into several simulated and experimental maps from cryo-electron microscopy. The method is available at https://team.inria.fr/nano-d/software/offgridfit.

Modified Trust Region Algorithm for Dispersion Optimization of Photonic Crystal Fibers

In this paper, a modified derivative-free surrogate-based trust region optimization algorithm is proposed for optimizing the dispersion properties of photonic crystal fibers (PCFs) for the first time to the best of our knowledge. The modal analysis of the PCF is made using the full vectorial finite difference method with perfect matched layer boundary condition. The numerical results are compared with another trust region algorithm and other metaheuristic techniques to show the strength of the reported technique. Further, the modified algorithm is also used to achieve a nearly ultra-flattened zero dispersion over a wide range of wavelengths from 1.45 $\mu {\rm{m}}$ to 1.6 $\mu {\rm{m}}$ using an index guiding soft glass PCF selectively infiltrated with a nematic liquid crystal. In order to show the strength of the reported algorithm, a highly negative flat dispersion compensation PCF is also designed over wavelength range from 1.4 $\mu {\rm{m}}$ to 1.6 $\mu {\rm{m}}$ . Such a design has a negative dispersion of −155 ± 0.5 ps/Km·nm over the studied wavelength range. The trust region algorithms show a strong potential as an efficient tool for the design and optimization of different photonic devices. Further, these algorithms can be used as powerful techniques for solving the inverse problems.

A Nonmonotone Trust Region Method for Nonlinear Programming with Simple Bound Constraints

In this paper we propose a nonmonotone trust region algorithm for optimization with simple bound constraints. Under mild conditions, we prove the global convergence of the algorithm. For the monotone case it is also proved that the correct active set can be identified in a finite number of iterations if the strict complementarity slackness condition holds, and so the proposed algorithm reduces finally to an unconstrained minimization method in a finite number of iterations, allowing a fast asymptotic rate of convergence. Numerical experiments show that the method is efficient.

A globally convergent trust region algorithm for optimization with general constraints and simple bounds

In this paper, we introduce a concept of substationary points and present a new trust region-based method for the optimization problems with general nonlinear equality constraints and simple bounds. Without the linear independent assumption on the gradients of the equalitiy constraints, we prove the global convergence results for the main algorithm and indicate that they extend the results on SQP and those on trust region methods for equality constrained optimization and for optimization with simple bounds. Moreover, since any nonlinear programming problem can be converted into the standard nonlinear programming by introducing slack variables, the trust region method presented in this paper can be used for solving general nonlinear programming problems.

OPTIMIZATION METHODS FOR SOLVING NON-LINEAR EQUATIONS IN VISCOELASTIC FLOW PROBLEMS

Numerical simulation of complex flows of viscoelastic fluids requires the use of efficient algorithms for solving the non-linear governing equations. In this paper, optimization algorithms, associated to a non-linear least squares methods are considered. The governing equations are written in the context of stream-tube analysis. This method is particularly adapted to handle memory-integral constitutive equations and permits calculation of flows by considering elementary sub-domains in a mapped computational domain where the transformed streamlines are parallel and straight. The procedure used to solve the equations is the Trust Region optimization algorithm, which satisfies a global convergence property. Using an integral codeformational equation, various tests involving the influence of parameters of the algorithms are applied to different flow geometries and underline the robustness and efficiency of the Trust Region algorithm.

A PDE Sensitivity Equation Method for Optimal Aerodynamic Design

The use of gradient-based optimization algorithms in inverse design is well established as a practical approach to aerodynamic design. A typical procedure uses a simulation scheme to evaluate the objective function (from the approximate states) and its gradient, then passes this information to an optimization algorithm. Once the simulation scheme (CFD flow solver) has been selected and used to provide approximate function evaluations, there are several possible approaches to the problem of computing gradients. One popular method is to differentiate the simulation scheme and compute design sensitivities that are then used to obtain gradients. Although this black-box approach has many advantages in shape optimization problems, one must compute mesh sensitivities in order to compute the design sensitivity. In this paper, we present an alternative approach using the PDE sensitivity equation to develop algorithms for computing gradients. This approach has the advantage that mesh sensitivities need not be computed. Moreover, when it is possible to use the CFD scheme for both the forward problem and the sensitivity equation, then there are computational advantages. An apparent disadvantage of this approach is that it does not always produce consistent derivatives. However, for a proper combination of discretization schemes, one can showasymptotic consistencyunder mesh refinement, which is often sufficient to guarantee convergence of the optimal design algorithm. In particular, we show that when asymptotically consistent schemes are combined with a trust-region optimization algorithm, the resulting optimal design method converges. We denote this approach as thesensitivity equation method.The sensitivity equation method is presented, convergence results are given, and the approach is illustrated on two optimal design problems involving shocks.

Global Convergence of a Class of Trust Region Algorithms for Optimization Using Inexact Projections on Convex Constraints

A class of trust region-based algorithms is presented for the solution of nonlinear optimization problems with a convex feasible set. At variance with previously published analyses of this type, the theory presented allows for the use of general norms. Furthermore, the proposed algorithms do not require the explicit computation of the projected gradient, and can therefore be adapted to cases where the projection onto the feasible domain may be expensive to calculate. Strong global convergence results are derived for the class. It is also shown that the set of linear and nonlinear constraints that are binding at the solution are identified by the algorithms of the class in a finite number of iterations.

Convergence of Trust Region Algorithms for Optimization with Bounds when Strict Complementarity Does Not Hold

Conn, Gould, and Toint [SIAM J. Numer. Anal., 25 (1988), pp. 433–460; 26 (1989), p. 764] have proposed a class of trust region algorithms for minimizing nonlinear functions whose variables are subjected to simple bound constraints. In their convergence analysis, they show that if the strict complementarily condition holds, the considered algorithms reduce to an unconstrained calculation after finitely many iterations, allowing fast asymptotic rates of convergence. This paper analyses the behaviour of these iterative processes in the case where the strict complementarily condition is violated. It is proved that inexact Newton methods lead to superlinear or quadratic rates of convergence, even if the set of active bounds at the solution is not entirely detected. Practical criteria for stopping the inner iterations of the algorithms are deduced, ensuring these rates of convergence.

Correction to the Paper on Global Convergence of a Class of Trust Region Algorithms for Optimization with Simple Bounds

A correction is made to the paper entitled “Global convergence of a class of trust region algorithms for optimization with simple bounds” [Corm, Gould, and Toint, SIAM J. Numer. Anal., 25 (1988), pp. 433–460]. First, an error is pointed out in the proof of the fact that the correct set of active bounds is determined after a finite number of iterations. A new proof of the relevant theorem is given.

Global Convergence of a Class of Trust Region Algorithms for Optimization with Simple Bounds

This paper extends the known excellent global convergence properties of trust region algorithms for unconstrained optimization to the case where bounds on the variables are present. Weak conditions on the accuracy of the Hessian approximations are considered. It is also shown that, when the strict complementarily condition holds, the proposed algorithms reduce to an unconstrained calculation after finitely many iterations, allowing a fast asymptotic rate of convergence.