Gradient-Informed Basis Adaptation for Legendre Chaos Expansions

Gaussian processes with built-in dimensionality reduction: Applications to high-dimensional uncertainty propagation

Compressive sampling of polynomial chaos expansions: Convergence analysis and sampling strategies

Multi-scale stochastic dynamic response analysis of offshore risers with lognormal uncertainties

The development of high-performance MEMS devices necessitates efficient design optimization and yield analysis in the presence of unavoidable process variations. The existing algorithms are not able to integrate the different requirements of optimization and uncertainty quantification (UQ), highlighting the necessity for a comprehensive algorithmic framework. To this end, a novel algorithm framework is proposed for the optimization and UQ of MEMS devices. The deep neural network (DNN) and polynomial chaos expansion (PCE) are respectively leveraged to build the global surrogate model for the nondominated sorting genetic algorithm II (NSGA-II) and the local surrogate model for the UQ analysis. The DNN is refined by the bi-fidelity data training scheme to reduce the training data demanded, and the active subspace dimension reduction (AS) is employed to fully utilize the information within the global surrogate model, thereby minimizing the computational resources needed for the local surrogate model construction. This algorithm is applied to a kind of MEMS thermal flow sensor. The proposed algorithmic simultaneously increases sensor sensitivity by 14.77% while reducing computational expenditure by 44.4% during the optimization phase. Besides, the yield analysis is also derived accurately, with computational cost reductions of 73.0%. These results indicate that this novel algorithm supplies an effective, accurate, and efficient technique for the optimization and yield analysis for MEMS devices. [2025-0231]

The purpose of history matching is to achieve geological realizations calibrated to the historical performance of the reservoir. For complex geological structures it is usually intractable to run tens of thousands of full reservoir simulation to trace the most probable geological model. Hence the inadequacy of the history-matching results frequently leads to poor estimation of the true model and high uncertainty in production forecasting. Reduced-order modeling procedures, which have been applied in many application areas including reservoir simulation, represent a promising means for constructing efficient surrogate models. Nonlinear dimensionality reduction techniques allow for encapsulating the high-resolution complex geological description of reservoir into a low-dimensional subspace, which significantly reduces number of unknowns and provides an efficient way to construct a proxy model based on the the reduced-dimension parameters. Polynomial Chaos Expansions (PCE) is a powerful tool to quantify uncertainty in dynamical system when there is probabilistic uncertainty in the system parameters. In reservoir simulation it has been shown to be more accurate and efficient compared to traditional experimental design (ED). PCEs have a significant advantage over other response surfaces as the convergence to the true probability distribution is proved when the order of the PCE is increased. Accordingly PCE proxy can be used as the pseudo-simulator to represent the surface responses of the uncertain variables. When the objective and constraints of a reservoir model is described by multivariate polynomial functions, there are very efficient algorithms to compute the global solutions. We have developed a workflow at which incorporates PCE to find the global minimum of the misfit surface and assess the uncertainty associated with. The accuracy of the PCE proxy increases with the additional trial runs of the reservoir simulator. We conduct a two dimensional synthetic case study of a fluvial channel as well as a real field example to demonstrate the effectiveness of this approach. Kernel Principal Component Analysis (KPCA) is used to parameterize the complex geological structure. The study has revealed useful reservoir information and delivered more reliable production forecasts. PCE-based history match enhances the quality and efficiency of the estimation of the most probable geological model and improve the confidence interval of production forecasts.

Polynomial chaos (PC) expansions are used for the propagation of uncertainty through dynamical systems as an alternative to Monte Carlo methods. Model parameters in a given dynamical system are assumed to have known expansions, which correspond to simple standard distributions, and one is usually interested in the polynomial expansion of the system solution. We are concerned with the problem of estimating the PC expansion of a parameter vector when only realizations from its distribution are given. To this end we apply ideas from optimal transportation theory and network optimization.

In this paper a novel improvement to the polynomial chaos (PC) approach for the uncertainty analysis of high speed circuits is presented. The key feature of this work is the development of an alternative hyperbolic truncation scheme to replace the conventional linear truncation scheme used in generation of PC expansions. This hyperbolic truncation scheme results in a sparse PC expansion for marginal loss of accuracy. The computational effort required to evaluate the coefficients of the resultant sparse expansion is only a small fraction of that required for full-blown PC expansions. A greedy adaptive methodology to determine the number of basis terms and evaluate the corresponding coefficients for the sparse expansion is also presented. This approach is validated on a nonlinear radio-frequency (RF) circuit against conventional full-blown PC methods.

Performing uncertainty quantification (UQ) and sensitivity analysis (SA) is vital when developing a patient-specific physiological model because it can quantify model output uncertainty and estimate the effect of each of the model's input parameters on the mathematical model. By providing this information, UQ and SA act as diagnostic tools to evaluate model fidelity and compare model characteristics with expert knowledge and real world observation. Computational efficiency is an important part of UQ and SA methods and thus optimization is an active area of research. In this work, we investigate a new efficient sampling method for least-squares polynomial approximation, weighted approximate Fekete points (WAFP). We analyze the performance of this method by demonstrating its utility in stochastic analysis of a cardiovascular model that estimates changes in oxyhemoglobin saturation response. Polynomial chaos (PC) expansion using WAFP produced results similar to the more standard Monte Carlo in quantifying uncertainty and identifying the most influential model inputs (including input interactions) when modeling oxyhemoglobin saturation, PC expansion using WAFP was far more efficient. These findings show the usefulness of using WAFP based PC expansion to quantify uncertainty and analyze sensitivity of a oxyhemoglobin dissociation response model. Applying these techniques could help analyze the fidelity of other relevant models in preparation for clinical application.

Parametric uncertainty assessment in hydrological modeling using the generalized polynomial chaos expansion

Here, we examine the suitability of truncated Polynomial Chaos Expansions (PCE) and truncated Gram-Charlier Expansions (GrChE) as possible methods for uncertainty quantification (UQ) in nonlinear systems with intermittency and positive Lyapunov exponents.These two methods rely on truncated Galerkin projections of either the system variables in a fixed polynomial basis spanning the 'uncertain' subspace (PCE) or a suitable eigenfunction expansion of the joint probability distribution associated with the uncertain evolution of the system (GrChE).Based on a simple, statistically exactly solvable non-linear and non-Gaussian test model, we show in detail that methods exploiting truncated spectral expansions, be it PCE or GrChE, have significant limitations for uncertainty quantification in systems with intermittent instabilities or parametric uncertainties in the damping.Intermittency and fat-tailed probability densities are hallmark features of the inertial and dissipation ranges of turbulence and we show that in such important dynamical regimes PCE performs, at best, similarly to the vastly simpler Gaussian moment closure technique utilized earlier by the authors in a different context for UQ within a framework of Empirical Information Theory.Moreover, we show that the non-realizability of the GrChE approximations is linked to the onset of intermittency in the dynamics and it is frequently accompanied by an erroneous blow-up of the second-order statistics at short times.These limitations of the two types of truncated spectral expansions arise from the following: (i) Non-uniform convergence in time of PCE and GrChE resulting in a rapidly increasing number of terms necessary for a good approximation of the random process as time evolves, (ii) Fundamental problems with capturing the constant flux of randomness due to white Gaussian noise forcing via finite truncations of the spectral representation of the associated Wiener process, (iii) Slow decay of PCE and GrChE coefficients in the presence of intermittency, hampering implementation of sparse truncation methods which have been widely used in nearly elliptic problems or in low Reynolds number flows.Rigorous justification of these limitations is richly illustrated by straightforward tests exploiting a simple nonlinear and non-Gaussian but statistically exactly solvable test model which is proposed here as a challenging benchmark for algorithms for UQ in systems with intermittency.

Gradient free active subspace construction using Morris screening elementary effects

We present a distributed active subspace method for training surrogate models of complex physical processes with high-dimensional inputs and function valued outputs. Specifically, we represent the model output with a truncated Karhunen–Loeve (KL) expansion, screen the structure of the input space with respect to each KL mode via the active subspace method, and finally form an overall surrogate model of the output by combining surrogates of individual output KL modes. To ensure scalable computation of the gradients of the output KL modes, needed in active subspace discovery, we rely on adjoint-based gradient computation. The proposed method combines benefits of active subspace methods for input dimension reduction and KL expansions used for spectral representation of the output field. We provide a mathematical framework for the proposed method and conduct an error analysis of the mixed KL active subspace approach. Specifically, we provide an error estimate that quantifies errors due to active subspace projection and truncated KL expansion of the output. We demonstrate the numerical performance of the surrogate modeling approach with an application example from biotransport.

We present a distributed active subspace method for training surrogate models of complex physical processes with high-dimensional inputs and function valued outputs. Specifically, we represent the model output with a truncated Karhunen-Lo\`eve (KL) expansion, screen the structure of the input space with respect to each KL mode via the active subspace method, and finally form an overall surrogate model of the output by combining surrogates of individual output KL modes. To ensure scalable computation of the gradients of the output KL modes, needed in active subspace discovery, we rely on adjoint-based gradient computation. The proposed method combines benefits of active subspace methods for input dimension reduction and KL expansions used for spectral representation of the output field. We provide a mathematical framework for the proposed method and conduct an error analysis of the mixed KL active subspace approach. Specifically, we provide an error estimate that quantifies errors due to active subspace projection and truncated KL expansion of the output. We demonstrate the numerical performance of the surrogate modeling approach with an application example from biotransport.

Neumann enriched polynomial chaos approach for stochastic finite element problems

Reliability analysis and design of a single diode solar cell model using polynomial chaos and active subspace