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1. Local and Global Optimum Design in Aero-Structural Optimization of Long-Span Bridges Considering Flutter

   https://doi.org/10.2139/ssrn.5038974  

   Cid_Montoya, Miguel; Hernandez, Santiago; Jurado, José Ángel (December 2024, SSRN Electronic Journal)

   The aero-structural design of bridges is mainly controlled by the deck cross-section design. Design modifications on bridge decks impact the deck aerodynamics and the deck mechanical contribution, which also affect the bridge aeroelastic responses. The nonlinear inherent nature of bluff body aerodynamics combined with the nonlinearities of multimodal aeroelastic analyses result in complex relationships between the full bridge aeroelastic responses and deck shape design variables. This fact impacts the design process as it may lead to the development of complex feasible design regions in the chosen design domain, including disjoint feasible regions that may cause local minima. Given the limitations of metaheuristic optimization methods to deal with optimization problems with large sets of design variables, as required in holistic bridge design problems, gradient-based optimization algorithms can be recast to address global optimization problems. In this study, we propose the use of tunneling optimization methods to address this challenge. 

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2. Leader Stochastic Gradient Descent for Distributed Training of Deep Learning Models

   Teng, Y.; Gao, W.; Chalus, F.; Choromanska, A. E.; Goldfarb, D.; Weller, A. (January 2019, Advances in neural information processing systems)

   We consider distributed optimization under communication constraints for training deep learning models. We propose a new algorithm, whose parameter updates rely on two forces: a regular gradient step, and a corrective direction dictated by the currently best-performing worker (leader). Our method differs from the parameter-averaging scheme EASGD in a number of ways:(i) our objective formulation does not change the location of stationary points compared to the original optimization problem;(ii) we avoid convergence decelerations caused by pulling local workers descending to different local minima to each other (ie to the average of their parameters);(iii) our update by design breaks the curse of symmetry (the phenomenon of being trapped in poorly generalizing sub-optimal solutions in symmetric non-convex landscapes); and (iv) our approach is more communication efficient since it broadcasts only parameters of the leader rather than all workers. We provide theoretical analysis of the batch version of the proposed algorithm, which we call Leader Gradient Descent (LGD), and its stochastic variant (LSGD). Finally, we implement an asynchronous version of our algorithm and extend it to the multi-leader setting, where we form groups of workers, each represented by its own local leader (the best performer in a group), and update each worker with a corrective direction comprised of two attractive forces: one to the local, and one to the global leader (the best performer among all workers). The multi-leader setting is well-aligned with current hardware architecture, where local workers forming a group lie within a single computational node and … 

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3. Breaking Reversibility Accelerates Langevin Dynamics for Non-Convex Optimization

   Xuefeng Gao, Mert Gurbuzbalaban (January 2020, Proceedings of Machine Learning Research)

   null (Ed.)

   Langevin dynamics (LD) has been proven to be a powerful technique for optimizing a non-convex objective as an efficient algorithm to find local minima while eventually visiting a global minimum on longer time-scales. LD is based on the first-order Langevin diffusion which is reversible in time. We study two variants that are based on non-reversible Langevin diffusions: the underdamped Langevin dynamics (ULD) and the Langevin dynamics with a non-symmetric drift (NLD). Adopting the techniques of Tzen et al. (2018) for LD to non-reversible diffusions, we show that for a given local minimum that is within an arbitrary distance from the initialization, with high probability, either the ULD trajectory ends up somewhere outside a small neighborhood of this local minimum within a recurrence time which depends on the smallest eigenvalue of the Hessian at the local minimum or they enter this neighborhood by the recurrence time and stay there for a potentially exponentially long escape time. The ULD algorithm improves upon the recurrence time obtained for LD in Tzen et al. (2018) with respect to the dependency on the smallest eigenvalue of the Hessian at the local minimum. Similar results and improvements are obtained for the NLD algorithm. We also show that non-reversible variants can exit the basin of attraction of a local minimum faster in discrete time when the objective has two local minima separated by a saddle point and quantify the amount of improvement. Our analysis suggests that non-reversible Langevin algorithms are more efficient to locate a local minimum as well as exploring the state space. 

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4. Repulsive Curves

   Yu, Chris; Schumacher, Henrik; Crane, Keenan (January 2021, ACM transactions on graphics)

   null (Ed.)

   Curves play a fundamental role across computer graphics, physical simulation, and mathematical visualization, yet most tools for curve design do nothing to prevent crossings or self-intersections. This paper develops efficient algorithms for (self-)repulsion of plane and space curves that are well-suited to problems in computational design. Our starting point is the so-called tangent-point energy, which provides an infinite barrier to self-intersection. In contrast to local collision detection strategies used in, e.g., physical simulation, this energy considers interactions between all pairs of points, and is hence useful for global shape optimization: local minima tend to be aesthetically pleasing, physically valid, and nicely distributed in space. A reformulation of gradient descent, based on a Sobolev-Slobodeckij inner product enables us to make rapid progress toward local minima---independent of curve resolution. We also develop a hierarchical multigrid scheme that significantly reduces the per-step cost of optimization. The energy is easily integrated with a variety of constraints and penalties (e.g., inextensibility, or obstacle avoidance), which we use for applications including curve packing, knot untangling, graph embedding, non-crossing spline interpolation, flow visualization, and robotic path planning. 

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5. Geometrically non-linear topology optimization via geometry projection

   https://doi.org/10.1016/j.cma.2024.117636  

   Hu, Jingyu; Wallin, Mathias; Ristinmaa, Matti; Norato, JA; Liu, Shutian (February 2025, Computer Methods in Applied Mechanics and Engineering)

   Geometry projection-based topology optimization has attracted a great deal of attention because it enables the design of structures consisting of a combination of geometric primitives and simplifies the integration with computer-aided design (CAD) systems. While the approach has undergone substantial development under the assumption of linear theory, it remains to be developed for non-linear hyperelastic problems. In this study, a geometrically non-linear explicit topology optimization approach is proposed in the framework of the geometry projection method. The energy transition strategy is adopted to mitigate excessive distortion in low-stiffness regions that might cause the equilibrium iterations to diverge. A neo-Hookean hyperelastic strain energy potential is used to model the material behavior. Design sensitivities of the functions passed to the gradient-based optimizer are detailed and verified. The proposed method is used to solve benchmark problems for which the output displacement in a compliant mechanism is maximized and the structural compliance is minimized. 

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