convergence rates
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2022 ◽  
Vol 41 (1) ◽  
pp. 1-10
Jonas Zehnder ◽  
Stelian Coros ◽  
Bernhard Thomaszewski

We present a sparse Gauss-Newton solver for accelerated sensitivity analysis with applications to a wide range of equilibrium-constrained optimization problems. Dense Gauss-Newton solvers have shown promising convergence rates for inverse problems, but the cost of assembling and factorizing the associated matrices has so far been a major stumbling block. In this work, we show how the dense Gauss-Newton Hessian can be transformed into an equivalent sparse matrix that can be assembled and factorized much more efficiently. This leads to drastically reduced computation times for many inverse problems, which we demonstrate on a diverse set of examples. We furthermore show links between sensitivity analysis and nonlinear programming approaches based on Lagrange multipliers and prove equivalence under specific assumptions that apply for our problem setting.

2022 ◽  
Vol 13 (1) ◽  
pp. 0-0

In this paper, we propose a hybrid algorithm combining two different metaheuristic methods, “Genetic Algorithms (GA)” and “Sperm Swarm Optimization (SSO)”, for the global optimization of multimodal benchmarks functions. The proposed Hybrid Genetic Algorithm and Sperm Swarm Optimization (HGASSO) operates based on incorporates concepts from GA and SSO in which generates individuals in a new iteration not only by crossover and mutation operations as proposed in GA, but also by techniques of local search of SSO. The main idea behind this hybridization is to reduce the probability of trapping in local optimum of multi modal problem. Our algorithm is compared against GA, and SSO metaheuristic optimization algorithms. The experimental results using a suite of multimodal benchmarks functions taken from the literature have evinced the superiority of the proposed HGASSO approach over the other approaches in terms of quality of results and convergence rates in which obtained good results in solving the multimodal benchmarks functions that include cosine, sine, and exponent in their formulation.

Nonlinearity ◽  
2021 ◽  
Vol 35 (2) ◽  
pp. 1061-1092
Theresa Lange

Abstract We provide a rigorous derivation of the ensemble Kalman–Bucy filter as well as the ensemble transform Kalman–Bucy filter in case of nonlinear, unbounded model and observation operators. We identify them as the continuous time limit of the discrete-time ensemble Kalman filter and the ensemble square root filters, respectively, together with concrete convergence rates in terms of the discretisation step size. Simultaneously, we establish well-posedness as well as accuracy of both the continuous-time and the discrete-time filtering algorithms.

Water ◽  
2021 ◽  
Vol 14 (1) ◽  
pp. 63
Su Li ◽  
Zhihong Yan ◽  
Jinxia Sha ◽  
Jing Gao ◽  
Bingqing Han ◽  

The reasonable allocation of water resources using different optimization technologies has received extensive attention. However, not all optimization algorithms are suitable for solving this problem because of its complexity. In this study, we applied an ameliorative multi-objective gray wolf optimizer (AMOGWO) to the problem. For AMOGWO, which is based on the multi-objective gray wolf optimizer, we improved the distance control parameter calculation method, added crowding degree for the archive, and optimized the selection mechanism for leader wolves. Subsequently, AMOGWO was used to solve the multi-objective optimal allocation of water resources in Handan, China, for 2035, with the maximum economic benefit and minimum social water shortage used as objective functions. The optimal results obtained indicate a total water demand in Handan of 2740.43 × 106 m3, total water distribution of 2442.23 × 106 m3, and water shortage of 298.20 × 106 m3, which is consistent with the principles of water resource utilization in Handan. Furthermore, comparison results indicate that AMOGWO has substantially enhanced convergence rates and precision compared to the non-dominated sorting genetic algorithm II and the multi-objective particle swarm optimization algorithm, demonstrating relatively high reliability and applicability. This study thus provides a new method for solving the multi-objective optimal allocation of water resources.

2021 ◽  
Vol 32 (1) ◽  
Umberto Amato ◽  
Anestis Antoniadis ◽  
Italia De Feis ◽  
Irène Gijbels

AbstractThis article studies M-type estimators for fitting robust additive models in the presence of anomalous data. The components in the additive model are allowed to have different degrees of smoothness. We introduce a new class of wavelet-based robust M-type estimators for performing simultaneous additive component estimation and variable selection in such inhomogeneous additive models. Each additive component is approximated by a truncated series expansion of wavelet bases, making it feasible to apply the method to nonequispaced data and sample sizes that are not necessarily a power of 2. Sparsity of the additive components together with sparsity of the wavelet coefficients within each component (group), results into a bi-level group variable selection problem. In this framework, we discuss robust estimation and variable selection. A two-stage computational algorithm, consisting of a fast accelerated proximal gradient algorithm of coordinate descend type, and thresholding, is proposed. When using nonconvex redescending loss functions, and appropriate nonconvex penalty functions at the group level, we establish optimal convergence rates of the estimates. We prove variable selection consistency under a weak compatibility condition for sparse additive models. The theoretical results are complemented with some simulations and real data analysis, as well as a comparison to other existing methods.

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