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Probabilistic Modeling, Multiscale and Validation Roger Ghanem University of Southern California Los Angeles, California

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Outline Introduction and Objectives Representation of Information Model Validation Efficiency Issues

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Introduction Objectives: –Determine certifiable confidence in model-based predictions: Certifiable = amenable to analysis Accept the possibility that certain statements, given available resources, cannot be certified. –Compute actions to increase confidence in model predictions: change the information available to the prediction. More experimental/field data, more detailed physics, more resolution for numerics… Stochastic models package information in a manner suitable for analysis: –Adapt this packaging to the needs of our decision-maker Craft a mathematical model that is parameterized with respect to the relevant uncertainties.

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Two meaningful questions Nothing new here. What is new is: sensor technology computing technology Can/must adapt our “packaging” of information and knowledge accordingly.

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Theoretical basis: Cameron-Martin Theorem The polynomial chaos decomposition of any square-integrable functional of the Brownian motion converges in mean-square as N goes to infinity. For a finite-dimensional representation, the coefficients are functions of the missing dimensions: they are random variables (Slud,1972).

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The random quantities are resolved as surfaces in a normalized space: These could be, for example: Parameters in a PDE Boundaries in a PDE (e.g. Geometry) Field Variable in a PDE Multidimensional Orthogonal Polynomials Independent random variables Dimension of vector reflects complexity of Representation of Uncertainty

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Uncertainty due to small experimental database or anything else. Uncertainty in model parameters Dimension of vector reflects complexity of Representation of Uncertainty

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Galerkin Projections Maximum Likelihood Maximum Entropy Bayes Theorem Ensemble Kalman Filter Characterization of Uncertainty

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Characterization of Uncertainty Maximum Entropy Estimation / Spatio-Temporal Processes Temperature is measured as function of time along cables. Temperature fluctuations affect sound speed in the ocean.

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Temperature time histories,, at various depths. Characterization of Uncertainty Maximum Entropy Estimation / Spatio-Temporal Processes

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Reduced order model of KL expansion A typical plot of marginal pdf for a Karhunen- Loeve variable. Characterization of Uncertainty Maximum Entropy Estimation with Histogram Constraints Spearman Rank Correlation Coefficient is also matched:

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Uncertainty Propagation: Stochastic Projection

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Example Application: W76 Foam study Built-up structure with shell, foam and devices. Foam domain. 1. Modeled as non-stationary random field. 2. Accounting for random and structured variations 3. Limited observations are assumed: selected 30 locations on the foam. Limited statistical observations: Correlation estimator from small sample size: interval bounds on correlation matrix. System has 10320 HEX elements. Stochastic block has 2832 elements.

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Example Application: W76 Foam study Polynomial Chaos representation of epistemic information Constrained polynomial chaos construction Radial Basis function consistent spatial interpolation Cubature integration in high-dimensions

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Foam study Statistics of maximum acceleration Histogram of average of maximum acceleration

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Foam study Statistics of maximum acceleration Plots of density functions of the maximum acceleration

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Effect of missing information

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Estimate %95 probability box Remarks: Confidence intervals are due to finite sample size. CDF of calibrated stochastic parameters (3 out of 9 shown)

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Sample Mean ofSample Variance of 0.08350.000830 Remark: Based on only 25 samples. Validation Challenge Problem Treated as a random variables: Criterion for certifying a design (we would like to assess it without full- scale experiments:

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Solution with 3 rd order chaos Solution with 3 rd order chaos and enrichment Exact solution Efficiency Issues: Basis Enrichment

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NANO-RESONATOR WITH RANDOM GEOMETRY

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semiconductor conductor OBJETCIVE: 1.Determine requirements on manufacturing tolerance. 2.Determine relationship between manufacturing tolerance and performance reliability.

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APPROACH Define the problem on some underlying deterministic geometry. Define a random mapping from the deterministic geometry to the random geometry. Approximate this mapping using a polynomial chaos decomposition. Solve the governing equations using coupled FEM-BEM. Compare various implementations.

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TREATMENT OF RANDOM GEOMETRY Ref: Tartakovska & Xiu, 2006.

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GOVERNING EQUATIONS Interior Electrostatic BVP : Elastic BVP for Semiconductor: Exterior Electrostatic BVP :

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A couple of realizations of solution (deformed shape and charge distribution)

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More Significant Probabilistic Results PDF of Vertical Displacement at tip. PDF of Maximum Principal Stress at a point.

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Typical Challenge

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Comparison of Monte Carlo, Quadrature and Exact Evaluations of the Element Integrations

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Only one deterministic solve required. Minimal change to existing codes. Need iterative solutions with multiple right hand sides. Integrated into ABAQUS (not commercially). Using Components of Existing Analysis Software

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Implementation: Sundance

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Non-intrusive implementation

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Implementation: Dakota

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Example Application (non-intrusive)

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joints

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Example Application

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Conclusions Personal Experience: Every time I have come close to concluding on PCE, new horizons have unfolded in Applications Models Algorithms Software

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