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

Falcó, Antonio; Montés, Nicolás; Chinesta, Francisco; Hilario, Lucía; Mora, Marta C.

doi:10.1016/j.cam.2017.08.007

Cited by 10 publications

(10 citation statements)

References 13 publications

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“…The standard approach is the progressive PGD strategy [59], which is used in this paper as well. For a prove of the convergence of the progressive PGD for elliptic problems, the authors refer to Falco et al [60]. The problem is solved iteratively…”

Section: Proper Generalized Decomposition For Structural Reliabilitymentioning

confidence: 99%

Efficient structural reliability analysis by using a PGD model in an adaptive importance sampling schema

Robens-Radermacher

Unger

2020

Adv. Model. and Simul. in Eng. Sci.

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One of the most important goals in civil engineering is to guarantee the safety of the construction. Standards prescribe a required failure probability in the order of 10 −4 to 10 −6. Generally, it is not possible to compute the failure probability analytically. Therefore, many approximation methods have been developed to estimate the failure probability. Nevertheless, these methods still require a large number of evaluations of the investigated structure, usually finite element (FE) simulations, making full probabilistic design studies not feasible for relevant applications. The aim of this paper is to increase the efficiency of structural reliability analysis by means of reduced order models. The developed method paves the way for using full probabilistic approaches in industrial applications. In the proposed PGD reliability analysis, the solution of the structural computation is directly obtained from evaluating the PGD solution for a specific parameter set without computing a full FE simulation. Additionally, an adaptive importance sampling scheme is used to minimize the total number of required samples. The accuracy of the failure probability depends on the accuracy of the PGD model (mainly influenced on mesh discretization and mode truncation) as well as the number of samples in the sampling algorithm. Therefore, a general iterative PGD reliability procedure is developed to automatically verify the accuracy of the computed failure probability. It is based on a goal-oriented refinement of the PGD model around the adaptively approximated design point. The methodology is applied and evaluated for 1D and 2D examples. The computational savings compared to the method based on a FE model is shown and the influence of the accuracy of the PGD model on the failure probability is studied.

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Section: Proper Generalized Decomposition For Structural Reliabilitymentioning

confidence: 99%

Efficient structural reliability analysis by using a PGD model in an adaptive importance sampling schema

Robens-Radermacher

Unger

2020

Adv. Model. and Simul. in Eng. Sci.

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“…This definition and the next theorem were introduced in [20]. The important factor in the previous procedure is the minimization problem ( * ) because for each m J(u m−1 + •) is considered a map…”

Section: Definition 1 ([20] Progressive Variational Vademecum) Sincmentioning

confidence: 99%

“…Our previous work, references [20,21] developed a PGD-based computational vademecum (PGD-vademecum) or abacus to solve the Laplace equation, allowing the use of the potential flow theory in RT applications when the robot is guided in a static environment [22]. In this paper, the formulation of the PGD-vademecum for dynamic environments with dynamic obstacles is derived, the progressive PGD-vademecum, where the obstacles are considered in the representation as extra parameters.…”

Section: Introductionmentioning

confidence: 99%

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

et al. 2020

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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.

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“…Many problems in science and engineering are hard to compute due their numerical complexity. Moreover, in the analysis of complex systems under a real time constraint, the evaluation of all possible scenarios appears as a necessity [1]. Despite the improvements in techniques used in high dimensional problems, some challenging questions remain unresolved due to the efficiency of our computers.…”

Section: Introductionmentioning

confidence: 99%

“…The strategy is to include in an equivalent non-parametrized problem all possible parameter values as extra-coordinates. The name of this particular PGD based approach is Progressive Variational Vademecum [1], and it can be implemented offline. As a consequence, the PGD based approach opens the possibility of solving problems in industry with a different strategy not envisioned until now.…”

Section: Introductionmentioning

confidence: 99%

Towards a Vector Field Based Approach to the Proper Generalized Decomposition (PGD)

et al. 2020

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A novel algorithm called the Proper Generalized Decomposition (PGD) is widely used by the engineering community to compute the solution of high dimensional problems. However, it is well-known that the bottleneck of its practical implementation focuses on the computation of the so-called best rank-one approximation. Motivated by this fact, we are going to discuss some of the geometrical aspects of the best rank-one approximation procedure. More precisely, our main result is to construct explicitly a vector field over a low-dimensional vector space and to prove that we can identify its stationary points with the critical points of the best rank-one optimization problem. To obtain this result, we endow the set of tensors with fixed rank-one with an explicit geometric structure.

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

Cited by 10 publications

References 13 publications

Efficient structural reliability analysis by using a PGD model in an adaptive importance sampling schema

Efficient structural reliability analysis by using a PGD model in an adaptive importance sampling schema

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

Towards a Vector Field Based Approach to the Proper Generalized Decomposition (PGD)

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