Residual Minimizing Model Interpolation for Parameterized Nonlinear Dynamical Systems

We present a method for approximating the solution of a parameterized, nonlinear dynamical system using an affine combination of solutions computed at other points in the input parameter space. The coefficients of the affine combination are computed with a nonlinear least squares procedure that minimizes the residual of the governing equations. The approximation properties of this residual minimizing scheme are comparable to existing reduced basis and POD-Galerkin model reduction methods, but its implementation requires only independent evaluations of the nonlinear forcing function. It is particularly appropriate when one wishes to approximate the states at a few points in time without time marching from the initial conditions. We prove some interesting characteristics of the scheme including an interpolatory property, and we present heuristics for mitigating the effects of the ill-conditioning and reducing the overall cost of the method. We apply the method to representative numerical examples from kinetics - a three state system with one parameter controlling the stiffness - and conductive heat transfer - a nonlinear parabolic PDE with a random field model for the thermal conductivity.Comment: 28 pages, 8 figures, 2 table

Bayesian Restricted Likelihood Methods: Conditioning on Insufficient Statistics in Bayesian Regression

Bayesian methods have proven themselves to be successful across a wide range of scientific problems and have many well-documented advantages over competing methods. However, these methods run into difficulties for two major and prevalent classes of problems: handling data sets with outliers and dealing with model misspecification. We outline the drawbacks of previous solutions to both of these problems and propose a new method as an alternative. When working with the new method, the data is summarized through a set of insufficient statistics, targeting inferential quantities of interest, and the prior distribution is updated with the summary statistics rather than the complete data. By careful choice of conditioning statistics, we retain the main benefits of Bayesian methods while reducing the sensitivity of the analysis to features of the data not captured by the conditioning statistics. For reducing sensitivity to outliers, classical robust estimators (e.g., M-estimators) are natural choices for conditioning statistics. A major contribution of this work is the development of a data augmented Markov chain Monte Carlo (MCMC) algorithm for the linear model and a large class of summary statistics. We demonstrate the method on simulated and real data sets containing outliers and subject to model misspecification. Success is manifested in better predictive performance for data points of interest as compared to competing methods

Scheduling of Multiple Chillers in Trigeneration Plants

The scheduling of both absorption cycle and vapour compression cycle chillers in trigeneration plants is investigated in this work. Many trigeneration plants use absorption cycle chillers only but there are potential performance advantages to be gained by using a combination of absorption and compression chillers especially in situations where the building electrical demand to be met by the combined heat and power (CHP) plant is variable. Simulation models of both types of chillers are developed together with a simple model of a variable-capacity CHP engine developed by curve-fitting to supplier’s data. The models are linked to form an optimisation problem in which the contribution of both chiller types is determined at a maximum value of operating cost (or carbon emission) saving. Results show that an optimum operating condition arises at moderately high air conditioning demands and moderately low power demand when the air conditioning demand is shared between both chillers, all recovered heat is utilised, and the contribution arising from the compression chiller results in an increase in CHP power generation and, hence, engine efficiency

Fault Detection and Diagnosis in Air Conditioners and Refrigerators

A fault detection and diagnosis (FDD) method was used to detect and diagnose faults on both a refrigerator and an air conditioner during normal cycling operation. The objective of the method is to identify a set of sensors that can detect faults reliably before they severely hinder system performance. Unlike other methods, this one depends on the accuracy of a number of small, on-line linear models, each of which is valid over a limited range of operating conditions. To detect N faults, N sensors are needed. Using M>N sensors can further reduce the risk of false positives. For both the refrigerator and air conditioner systems, about 1000 combinations of candidate sensor locations were examined. Through inspection of matrix condition numbers and each sensor's contribution to fault detection calculation, the highest quality sets of sensors were identified. The issue of detecting simultaneous multiple faults was also addressed, with varying success. Fault detection was verified using both model simulations and experimental data. The results were similar, although in practice only one of the two would probably be used. Both load-type faults (such as door gasket leaks) and system faults were simulated on the refrigerator. It was found that system faults were generally more easily detectable than load faults. Refrigerator experiments were performed on a typical household refrigerator because it was readily available in a laboratory, but the results of this project may be more immediately useful on larger commercial, industrial or transport refrigeration systems. Air conditioner experiments were performed on a 3-ton split system. Again, the economic benefits of this type of fault detection scheme may also be more feasible for larger field-assembled systems.Air Conditioning and Refrigeration Project 8

Sensitivity analysis and parameter estimation for distributed hydrological modeling: potential of variational methods

Variational methods are widely used for the analysis and control of computationally intensive spatially distributed systems. In particular, the adjoint state method enables a very efficient calculation of the derivatives of an objective function (response function to be analysed or cost function to be optimised) with respect to model inputs. In this contribution, it is shown that the potential of variational methods for distributed catchment scale hydrology should be considered. A distributed flash flood model, coupling kinematic wave overland flow and Green Ampt infiltration, is applied to a small catchment of the Thoré basin and used as a relatively simple (synthetic observations) but didactic application case. It is shown that forward and adjoint sensitivity analysis provide a local but extensive insight on the relation between the assigned model parameters and the simulated hydrological response. Spatially distributed parameter sensitivities can be obtained for a very modest calculation effort (~6 times the computing time of a single model run) and the singular value decomposition (SVD) of the Jacobian matrix provides an interesting perspective for the analysis of the rainfall-runoff relation. For the estimation of model parameters, adjoint-based derivatives were found exceedingly efficient in driving a bound-constrained quasi-Newton algorithm. The reference parameter set is retrieved independently from the optimization initial condition when the very common dimension reduction strategy (i.e. scalar multipliers) is adopted. Furthermore, the sensitivity analysis results suggest that most of the variability in this high-dimensional parameter space can be captured with a few orthogonal directions. A parametrization based on the SVD leading singular vectors was found very promising but should be combined with another regularization strategy in order to prevent overfitting