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Probabilistic Learning on Manifolds (with Applications)

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USMW01 - Introduction to Uncertainty Quantification in Mechanics of Materials

Lecture: Friday 14 July 2023, 9:00 – 10:00Title: Probabilistic learning on manifolds (PLoM)Presenter: Christian SOIZE 1. Setting the problem of the probabilistic learning on manifolds (PLoM) Statistical surrogate model of a parameterized solution of a stochastic computational model. Training dataset and random vector X. Role played by the learned dataset generated by probabilistic learning in the construction of a statistical surrogate model. Proposed probabilistic learning on manifolds.- Illustration of a scattering of learned realizations. 2. Methodology and algorithm of the probabilistic learning on manifolds- Methodology, steps of the algorithm.- A few mathematical results. 3. Illustrations- Illustration 1: manifold as a helical in 3D Euclidean space. Illustration 2: analysis of a petro-physics experimental database. Illustration 3: optimization under uncertainties using a limited number of function evaluations. 4. Probabilistic learning on manifolds under constraints- Example: probabilistic learning inference for stochastic boundary value problem. Formulation of PLoM under constraints using the Kullback-Leibler divergence minimization principle. Methodology for solving the Kullback minimization problem.- Types of constraints and their consideration in PLoM algorithm. 5. Probabilistic learning inference for 3D stochastic homogenization of heterogeneous material- Framework: non-separation case of mesoscale with macroscale. Stochastic elliptic boundary value problem. Prior random elasticity field and learning inference.- Numerical results and validation. A few papers related to Probabilistic Learning on Manifolds (PLoM)

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