Efficient Adaptive Stochastic Collocation Strategies for Advection-Diffusion Problems with Uncertain Inputs

Physical models with uncertain inputs are commonly represented as parametric partial differential equations (PDEs). That is, PDEs with inputs that are expressed as functions of parameters with an associated probability distribution. Developing efficient and accurate solution strategies that account for errors on the space, time and parameter domains simultaneously is highly challenging. Indeed, it is well known that standard polynomial-based approximations on the parameter domain can incur errors that grow in time. In this work, we focus on advection-diffusion problems with parameter-dependent wind fields. A novel adaptive solution strategy is proposed that allows users to combine stochastic collocation on the parameter domain with off-the-shelf adaptive timestepping algorithms with local error control. This is a non-intrusive strategy that builds a polynomial-based surrogate that is adapted sequentially in time. The algorithm is driven by a so-called hierarchical estimator for the parametric error and balances this against an estimate for the global timestepping error which is derived from a scaling argument.Comment: 29 pages, 14 figure

Evaluation of state of the art numerical integration schema

Tese de mestrado integrado. Engenharia Química. Faculdade de Engenharia. Universidade do Porto. 201

Kalman Filtering and its Application to On-Line State Estimation of a Once-Through Boiler

This thesis contributes to non-linear continuous-discrete Kalman filtering of multiplex systems through the development of two main ideas, namely, integration of the unscented transforms with linearly implicit methods and incorporation of simulation errors in the state estimation problem. The newly developed techniques are then applied to the technically relevant problem of state estimation on the main components of a utility boiler. State estimators in industrial systems are used as soft-sensors in monitoring and control applications as the most cost effective and practical alternative to telemetering all variables of interest. One such example is in utility boilers where reliable and real-time data characterising its behaviour is used to detect faults and optimise performance. With respect to the state-of-the-art, state estimators display limitations in real-time applications to large-scale systems. This motivates theoretical developments in state estimation as a first part in this thesis. These developments are aimed at producing more practical and efficient algorithms in non-linear continuous discrete Kalman filtering for stiff large-scale industrial systems. This is achieved using two novel ideas. The first is to exploit the similarities between the extended and unscented Kalman filter in order to estimate the Jacobian required for linearly implicit schemes, thereby tightly coupling state propagation and continuous-time simulation. The second is to account for numerical integration error by appending a stochastic local error model to the system's stochastic differential equation. This allows for coarser integration time steps in systems that are otherwise only suited to relatively small step sizes, making the filter more computationally efficient without lowering its potential to construct accurate estimates. The second part of this thesis uses these algorithms to demonstrate the feasibility of on-line state estimation on the main components of a once-through utility power boiler that require in excess of a hundred state variables to capture its behaviour with adequate fidelity. Two separate models of the boiler are developed, a MATLAB® and a Flownex® model, comprising the economiser, evaporators, reheaters, superheaters and furnace. The mathematical MATLAB® model is better suited to real-time execution and is used in the filter. The more sophisticated model is based on a commercial thermal-hydraulic simulation environment, Flownex® , and is used to validate the mathematical modelling philosophies and construct filter observation data. After validating the performance of the filter against ground-truth data provided by the Flownex® model, the filter is demonstrated on historical plant data to illustrate its utility