On the Existence of a Progressive Variational Vademecum based on the Proper Generalized Decomposition for a Class of Elliptic Parameterized Problems

In this study, we present the mathematical analysis needed to explain the convergence of a progressive variational vademecum based on the proper generalized decomposition (PGD). The PGD is a novel technique that was developed recently for solving problems with high dimensions, and it also provides new approaches for obtaining the solutions of elliptic and parabolic problems via the abstract separation of variables method. This new scenario requires a mathematical framework in order to justify its application to the solution of numerical problems and the PGD can help in the change to this paradigm. The main aim of this study is to provide a mathematical environment for defining the notion of progressive variational vademecum. We prove the convergence of this iterative procedure and we also provide the first order optimality conditions in order to construct the numerical approximations of the parameterized solutions. In particular, we illustrate this methodology based on a robot path planning problem. This is one of the common tasks when designing the trajectory or path of a mobile robot. The construction of a progressive variational vademecum provides a novel methodology for computing all the possible paths from any start and goal positions derived from a harmonic potential field in a predefined map

A Path Planning Algorithm for a Dynamic Environment Based on Proper Generalized Decomposition

[EN] A necessity in the design of a path planning algorithm is to account for the environment. If the movement of the mobile robot is through a dynamic environment, the algorithm needs to include the main constraint: real-time collision avoidance. This kind of problem has been studied by different researchers suggesting different techniques to solve the problem of how to design a trajectory of a mobile robot avoiding collisions with dynamic obstacles. One of these algorithms is the artificial potential field (APF), proposed by O. Khatib in 1986, where a set of an artificial potential field is generated to attract the mobile robot to the goal and to repel the obstacles. This is one of the best options to obtain the trajectory of a mobile robot in real-time (RT). However, the main disadvantage is the presence of deadlocks. The mobile robot can be trapped in one of the local minima. In 1988, J.F. Canny suggested an alternative solution using harmonic functions satisfying the Laplace partial differential equation. When this article appeared, it was nearly impossible to apply this algorithm to RT applications. Years later a novel technique called proper generalized decomposition (PGD) appeared to solve partial differential equations, including parameters, the main appeal being that the solution is obtained once in life, including all the possible parameters. Our previous work, published in 2018, was the first approach to study the possibility of applying the PGD to designing a path planning alternative to the algorithms that nowadays exist. The target of this work is to improve our first approach while including dynamic obstacles as extra parameters.This research was funded by the GVA/2019/124 grant from Generalitat Valenciana and by the RTI2018-093521-B-C32 grant from the Ministerio de Ciencia, Innovacion y Universidades.Falcó, A.; Hilario, L.; Montés, N.; Mora, MC.; Nadal, E. (2020). A Path Planning Algorithm for a Dynamic Environment Based on Proper Generalized Decomposition. Mathematics. 8(12):1-11. https://doi.org/10.3390/math8122245S111812Gonzalez, D., Perez, J., Milanes, V., & Nashashibi, F. (2016). A Review of Motion Planning Techniques for Automated Vehicles. IEEE Transactions on Intelligent Transportation Systems, 17(4), 1135-1145. doi:10.1109/tits.2015.2498841Rimon, E., & Koditschek, D. E. (1992). Exact robot navigation using artificial potential functions. IEEE Transactions on Robotics and Automation, 8(5), 501-518. doi:10.1109/70.163777Khatib, O. (1986). Real-Time Obstacle Avoidance for Manipulators and Mobile Robots. The International Journal of Robotics Research, 5(1), 90-98. doi:10.1177/027836498600500106Kim, J.-O., & Khosla, P. K. (1992). Real-time obstacle avoidance using harmonic potential functions. IEEE Transactions on Robotics and Automation, 8(3), 338-349. doi:10.1109/70.143352Connolly, C. I., & Grupen, R. A. (1993). The applications of harmonic functions to robotics. Journal of Robotic Systems, 10(7), 931-946. doi:10.1002/rob.4620100704Garrido, S., Moreno, L., Blanco, D., & Martín Monar, F. (2009). Robotic Motion Using Harmonic Functions and Finite Elements. Journal of Intelligent and Robotic Systems, 59(1), 57-73. doi:10.1007/s10846-009-9381-3Bai, X., Yan, W., Cao, M., & Xue, D. (2019). Distributed multi‐vehicle task assignment in a time‐invariant drift field with obstacles. IET Control Theory & Applications, 13(17), 2886-2893. doi:10.1049/iet-cta.2018.6125Bai, X., Yan, W., Ge, S. S., & Cao, M. (2018). An integrated multi-population genetic algorithm for multi-vehicle task assignment in a drift field. Information Sciences, 453, 227-238. doi:10.1016/j.ins.2018.04.044Falcó, A., & Nouy, A. (2011). Proper generalized decomposition for nonlinear convex problems in tensor Banach spaces. Numerische Mathematik, 121(3), 503-530. doi:10.1007/s00211-011-0437-5Chinesta, F., Leygue, A., Bordeu, F., Aguado, J. V., Cueto, E., Gonzalez, D., … Huerta, A. (2013). PGD-Based Computational Vademecum for Efficient Design, Optimization and Control. Archives of Computational Methods in Engineering, 20(1), 31-59. doi:10.1007/s11831-013-9080-xFalcó, A., Montés, N., Chinesta, F., Hilario, L., & Mora, M. C. (2018). On the Existence of a Progressive Variational Vademecum based on the Proper Generalized Decomposition for a Class of Elliptic Parameterized Problems. Journal of Computational and Applied Mathematics, 330, 1093-1107. doi:10.1016/j.cam.2017.08.007Domenech, L., Falcó, A., García, V., & Sánchez, F. (2016). Towards a 2.5D geometric model in mold filling simulation. Journal of Computational and Applied Mathematics, 291, 183-196. doi:10.1016/j.cam.2015.02.043Falcó, A., & Nouy, A. (2011). A Proper Generalized Decomposition for the solution of elliptic problems in abstract form by using a functional Eckart–Young approach. Journal of Mathematical Analysis and Applications, 376(2), 469-480. doi:10.1016/j.jmaa.2010.12.003Falcó, A., & Hackbusch, W. (2012). On Minimal Subspaces in Tensor Representations. Foundations of Computational Mathematics, 12(6), 765-803. doi:10.1007/s10208-012-9136-6Canuto, C., & Urban, K. (2005). Adaptive Optimization of Convex Functionals in Banach Spaces. SIAM Journal on Numerical Analysis, 42(5), 2043-2075. doi:10.1137/s0036142903429730Ammar, A., Chinesta, F., & Falcó, A. (2010). On the Convergence of a Greedy Rank-One Update Algorithm for a Class of Linear Systems. Archives of Computational Methods in Engineering, 17(4), 473-486. doi:10.1007/s11831-010-9048-

Snapshot-Based Methods and Algorithms

An increasing complexity of models used to predict real-world systems leads to the need for algorithms to replace complex models with far simpler ones, while preserving the accuracy of the predictions. This two-volume handbook covers methods as well as applications. This second volume focuses on applications in engineering, biomedical engineering, computational physics and computer science

Model Order Reduction

An increasing complexity of models used to predict real-world systems leads to the need for algorithms to replace complex models with far simpler ones, while preserving the accuracy of the predictions. This two-volume handbook covers methods as well as applications. This second volume focuses on applications in engineering, biomedical engineering, computational physics and computer science