Haiyan Cheng

Efficient uncertainty quantification with the polynomial chaos method for stiff systems

The polynomial chaos (PC) method has been widely adopted as a computationally feasible approach for uncertainty quantification (UQ). Most studies to date have focused on non-stiff systems. When stiff systems are considered, implicit numerical integration requires the solution of a non-linear system of equations at every time step. Using the Galerkin approach the size of the system state increases from n to Sxn, where S is the number of PC basis functions. Solving such systems with full linear algebra causes the computational cost to increase from O(n^3) to O(S^3n^3). The S^3-fold increase can make the computation prohibitive. This paper explores computationally efficient UQ techniques for stiff systems using the PC Galerkin, collocation, and collocation least-squares (LS) formulations. In the Galerkin approach, we propose a modification in the implicit time stepping process using an approximation of the Jacobian matrix to reduce the computational cost. The numerical results show a run time reduction with no negative impact on accuracy. In the stochastic collocation formulation, we propose a least-squares approach based on collocation at a low-discrepancy set of points. Numerical experiments illustrate that the collocation least-squares approach for UQ has similar accuracy with the Galerkin approach, is more efficient, and does not require any modification of the original code.

Collocation least-squares polynomial chaos method

The polynomial chaos (PC) method has been used in many engineering applications to replace the traditional Monte Carlo (MC) approach for uncertainty quantification (UQ) due to its better convergence properties. Many researchers seek to further improve the efficiency of PC, especially in higher dimensional space with more uncertainties. The intrusive PC Galerkin approach requires the modification of the deterministic system, which leads to a stochastic system with a much bigger size. The non-intrusive collocation approach imposes the system to be satisfied at a set of collocation points to form and solve the linear system equations. Compared with the intrusive approach, the collocation method is easy to implement, however, choosing an optimal set of the collocation points is still an open problem. In this paper, we first propose using the low-discrepancy Hammersley/Halton dataset and Smolyak datasets as the collocation points, then propose a least-squares (LS) collocation approach to use more collocation points than the required minimum to solve for the system coefficients. We prove that the PC coefficients computed with the collocation LS approach converges to the optimal coefficients. The numerical tests on a simple 2-dimensional problem show that PC collocation LS results using the Hammersley/Halton points approach to optimal result.

Numerical study of uncertainty quantification techniques for implicit stiff systems

Galerkin polynomial chaos and collocation methods have been widely adopted for uncertainty quantification purpose. However, when the stiff system is involved, the computational cost can be prohibitive, since stiff numerical integration requires the solution of a nonlinear system of equations at every time step. Applying the Galerkin polynomial chaos to stiff system will cause a computational cost increase from O(n3) to O(S3n3). This paper explores uncertainty quantification techniques for stiff chemical systems using Galerkin polynomial chaos, collocation and collocation least-square approaches. We propose a modification in the implicit time stepping process. The numerical test results show that with the modified approach, the run time of the Galerkin polynomial chaos is reduced. We also explore different methods of choosing collocation points in collocation implementations and propose a collocation least-square approach. We conclude that the collocation least-square for uncertainty quantification is at least as accurate as the Galerkin approach, and is more efficient with a well-chosen set of collocation points.

Uncertainty apportionment for air quality forecast models

Effective environmental protection policy making depends on comprehensive and accurate Air Quality Model (AQM) prediction results. The confidence level associated with the model prediction, as well as the uncertainty sources that contribute to the prediction uncertainty are important information that should not be neglected when interpreting simulation results. In this work, we explore the capability of the polynomial chaos (PC) method for uncertainty quantification (UQ) and propose a uncertainty apportionment (UA) approach that can be easily applied to any forecast models. The numerical tests on the STEM (Sulfur Transport Eulerian Model) for the northeast region of the United States provide a categorization for the major uncertainty sources that contribute to the uncertainty in the ozone concentration prediction. This information can be used to guide the optimal investment decisions as to which input measurement accuracy should be improved to make the maximum impact on reducing the uncertainty in the prediction result.

On the Stopping Criteria for Conjugate Gradient Solutions of First-Kind Integral Equations in Two Variables

In this chapter, we investigate the conjugate gradient method for solving first-kind integral equations in two variables. A typical application problem would be the 2-D image reconstruction problem. For example, if we take a picture from far above the atmosphere, owing to the precision of the physical equipment, there will be some deviation between the picture and the original object, not to mention the effects posed by air turbulence, clouds, and perhaps pollution.

On the Use of the Conjugate Gradient Method for the Numerical Solution of First-Kind Integral Equations in Two Variables

In this chapter we study the use of the conjugate gradient method for solving Fredholm integral equations of the first kind of the form

Uncertainty quantification and apportionment in air quality models using the polynomial chaos method

\int_{0}^{1} {\int_{0}^{1} {K(x - s)K(y - t)f(s,t)dsdt = g(x,y),} }

A Robust Non-Gaussian Data Assimilation Method for Highly Non-Linear Models

(8.1) where the limits of integration are taken to be 0 and 1 without loss of generality. Such equations occur often in imaging problems, and when the kernels are nondegenerate, are marked by being ill-posed [1]. These problems yield discretizations that are linear systems and must be handled with great care because the coefficient matrices are quite ill-conditioned. To cope with this ill conditioning, we choose the method of conjugate gradients (see [2] and [3]), which has several advantages. The dependence upon the condition number is milder and in addition we may use the number of iterations as a regularizing parameter.