Calculation of Generalized Polynomial-Chaos Basis Functions and Gauss Quadrature Rules in Hierarchical Uncertainty Quantification

Stochastic spectral methods are efficient techniques for uncertainty quantification. Recently they have shown excellent performance in the statistical analysis of integrated circuits. In stochastic spectral methods, one needs to determine a set of orthonormal polynomials and a proper numerical quadrature rule. The former are used as the basis functions in a generalized polynomial chaos expansion. The latter is used to compute the integrals involved in stochastic spectral methods. Obtaining such information requires knowing the density function of the random input {\it a-priori}. However, individual system components are often described by surrogate models rather than density functions. In order to apply stochastic spectral methods in hierarchical uncertainty quantification, we first propose to construct physically consistent closed-form density functions by two monotone interpolation schemes. Then, by exploiting the special forms of the obtained density functions, we determine the generalized polynomial-chaos basis functions and the Gauss quadrature rules that are required by a stochastic spectral simulator. The effectiveness of our proposed algorithm is verified by both synthetic and practical circuit examples.Comment: Published by IEEE Trans CAD in May 201

Stochastic Testing Method for Transistor-Level Uncertainty Quantification Based on Generalized Polynomial Chaos

Uncertainties have become a major concern in integrated circuit design. In order to avoid the huge number of repeated simulations in conventional Monte Carlo flows, this paper presents an intrusive spectral simulator for statistical circuit analysis. Our simulator employs the recently developed generalized polynomial chaos expansion to perform uncertainty quantification of nonlinear transistor circuits with both Gaussian and non-Gaussian random parameters. We modify the nonintrusive stochastic collocation (SC) method and develop an intrusive variant called stochastic testing (ST) method. Compared with the popular intrusive stochastic Galerkin (SG) method, the coupled deterministic equations resulting from our proposed ST method can be solved in a decoupled manner at each time point. At the same time, ST requires fewer samples and allows more flexible time step size controls than directly using a nonintrusive SC solver. These two properties make ST more efficient than SG and than existing SC methods, and more suitable for time-domain circuit simulation. Simulation results of several digital, analog and RF circuits are reported. Since our algorithm is based on generic mathematical models, the proposed ST algorithm can be applied to many other engineering problems

Mathematical Theory and Modelling in Atmosphere-Ocean-Science

Participants from around the world gathered to review application and development of mathematics in relation to problems in the atmospheric, oceanic and climate sciences

Metamodel-based inverse uncertainty quantification of nuclear reactor simulators under the Bayesian framework

Mathematical modeling and computer simulations have long been the central technical topics in practically all branches of science and technology. Tremendous progress has been achieved in revealing quantitative connections between numerical predictions and real-world observations. However, because computer models are reduced representations of the real phenomena, there are always discrepancies between ideal in silico designed systems and real-world manufactured ones. As a consequence, uncertainties must be quantified along with the simulation outputs to facilitate optimal design and decision making, ensure robustness, performance or safety margins. Forward uncertainty propagation requires knowledge in the statistical information for computer model random inputs, for example, the mean, variance, Probability Density Functions (PDFs), upper and lower bounds, etc. Historically, ``expert judgment'' or ``user self-evaluation'' have been used to specify the uncertainty information associated with random input parameters. Such ad hoc characterization is unscientific and lacks mathematical rigor. In this thesis, we attempt to solve such ``lack of uncertainty information'' issue with inverse Uncertainty Quantification (UQ). Inverse UQ is the process to seek statistical descriptions of the random input parameters that are consistent with available high-quality experimental data. We formulate the inverse UQ process under the Bayesian framework using the ``model updating equation''. Markov Chain Monte Carlo (MCMC) sampling is applied to explore the posterior distributions and generate samples from which we can extract statistical information for the uncertain input parameters. To greatly alleviate the computational burden during MCMC sampling, we used systematically and rigorously developed metamodels based on stochastic spectral techniques and Gaussian Processes (also known as Kriging) emulators. We demonstrated the developed methodology based on three problems with different levels of sophistication: (1) Point Reactor Kinetics Equation (PRKE) coupled with lumped parameter thermal-hydraulics feedback model based on synthetic experimental data; (2) best-estimate system thermal-hydraulics code TRACE physical model parameters based on OECD/NRC BWR Full-size Fine-Mesh Bundle Tests (BFBT) benchmark steady-state void fraction data; (3) fuel performance code BISON Fission Gas Release (FGR) model based on Risø-AN3 on-line time-dependent FGR measurement data. Metamodels constructed with generalized Polynomial Chaos Expansion (PCE), Sparse Gird Stochastic Collocation (SGSC) and GP were applied respectively for these three problems to replace the full models during MCMC sampling. We proposed an improved modular Bayesian approach that can avoid extrapolating the model discrepancy that is learnt from the inverse UQ domain to the validation/prediction domain. The improved approach is organized in a structure such that the posteriors achieved with data in inverse UQ domain is informed by data in the validation domain. Therefore, over-fitting can be avoided while extrapolation is not required. A sequential approach was also developed for test source allocation (TSA) for inverse UQ and validation. This sequential TSA methodology first select tests for validation that has a full coverage of the test domain to avoid extrapolation of model discrepancy term when evaluated at input setting of tests for inverse UQ. Then it select tests that tend to reside in the unfilled zones of the test domain for inverse UQ, so that inverse UQ can extract the most information for posteriors of calibration parameters using only a relatively small number of tests. The inverse UQ process successfully quantified the uncertainties associated with input parameters that are consistent with the experimental observations. The quantified uncertainties are necessary for future uncertainty and sensitivity study of nuclear reactor simulators in system design and safety analysis. We applied and extended several advanced metamodeling approaches to nuclear engineering practice to greatly reduce the computational cost. The current research bridges the gap between models and data by solving ``lack of uncertainty information'' issue, as well as providing guidance for improving nuclear reactor simulators through the validation process