Imperial College London

DrFangxinFang

Faculty of EngineeringDepartment of Earth Science & Engineering

Senior Research Fellow
 
 
 
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Contact

 

+44 (0)20 7594 1912f.fang CV

 
 
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Location

 

4.90Royal School of MinesSouth Kensington Campus

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Summary

 

Publications

Publication Type
Year
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87 results found

Cheng M, Fang F, Navon IM, Pain CCet al., 2021, A real-time flow forecasting with deep convolutional generative adversarial network: Application to flooding event in Denmark, PHYSICS OF FLUIDS, Vol: 33, ISSN: 1070-6631

Journal article

Zheng J, Wu X, Fang F, Li J, Wang Z, Xiao H, Zhu J, Pain C, Linden P, Xiang Bet al., 2021, Numerical study of COVID-19 spatial-temporal spreading in London, PHYSICS OF FLUIDS, Vol: 33, ISSN: 1070-6631

Journal article

Wang Y, Ding X, Hu K, Fang F, Navon IM, Lin Get al., 2021, Feasibility of DEIM for retrieving the initial field via dimensionality reduction, JOURNAL OF COMPUTATIONAL PHYSICS, Vol: 429, ISSN: 0021-9991

Journal article

Cheng M, Fang F, Pain CC, Navon IMet al., 2020, An advanced hybrid deep adversarial autoencoder for parameterized nonlinear fluid flow modelling, Computer Methods in Applied Mechanics and Engineering, Vol: 372, Pages: 1-19, ISSN: 0045-7825

Considering the high computation cost required in conventional computation fluid dynamic simulations, machine learning methods have been introduced to flow dynamic simulations in years, aiming on reducing CPU time. In this work, we propose a hybrid deep adversarial autoencoder (VAE-GAN) to integrate generative adversarial network (GAN) and variational autoencoder (VAE) for predicting parameterized nonlinear fluid flows in spatial and temporal dimensions. High-dimensional inputs are compressed into the low-dimensional representations by nonlinear functions in a convolutional encoder. In this way, the predictive fluid flows reconstructed in a convolutional decoder contain the dynamic fluid flow physics of high nonlinearity and chaotic nature. In addition, the low-dimensional representations are applied to the adversarial network for model training and parameter optimization, which enables fast computation process. The capability of the hybrid VAE-GAN is illustrated by varying inputs on a flow past a cylinder test case as well as a second case of water column collapse. Numerical results show that this hybrid VAE-GAN has successfully captured the spatio-temporal flow features with CPU speed-up of three orders of magnitude. These promising results suggest that the hybrid VAE-GAN can play a critical role in efficiently and accurately predicting complex flows in future research efforts.

Journal article

Cheng M, Fang F, Kinouchi T, Navon IM, Pain CCet al., 2020, Long lead-time daily and monthly streamflow forecasting using machine learning methods, Journal of Hydrology, Vol: 590, Pages: 1-13, ISSN: 0022-1694

Long lead-time streamflow forecasting is of great significance for water resources planning and management in both the short and long terms. Despite of some studies using machine learning methods in streamflow forecasting, only few studies have been conducted to explore long lead-time forecasting capabilities of these methods, and gain an insight into systematic comparison of model forecasting performance in both the short and long terms. In this work, an artificial neural network (ANN) and a long short term memory (LSTM), a powerful tool for learning long-term temporal dependencies and capturing nonlinear relationship, have been adopted to forecast streamflow at daily and monthly scales for a long lead-time period. For long lead-time streamflow forecasting, a recursive forecasting procedure, which takes the last one-step-ahead forecast as a new input for the next-step-ahead forecast, is used in the ANN and LSTM forecasting systems. Two models are trained and validated for streamflow forecasting using the rainfall and runoff datasets collected from the Nan River Basin and Ping River Basin, Thailand, covering the period 1974 to 2014. To further explore the impact of parameter settings on model performance, two parameters, i.e. the length of time lag and the number of maximum epochs, are examined in the ANN and LSTM models. The main findings are highlighted here. First, with an optimal setting up of model parameters, both the ANN and LSTM model can provide accurate daily forecasting (up to 20 days ahead). Second, in comparison to the ANN model, the LSTM model exhibits better model performance in long lead-time daily forecasting, but less satisfactory in multi-monthly forecasting due to lack of large monthly training dataset. Third, the selection of the length of the time lag and number of maximum epochs used in both ANN and LSTM modelling are the key for long lead-time streamflow forecasting at daily and monthly scales. These findings suggest that the LSTM could be ad

Journal article

Cheng M, Fang F, Pain CC, Navon IMet al., 2020, Data -driven modelling of nonlinear spatio-temporal fluid flows using a deep convolutional generative adversarial network, COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING, Vol: 365, ISSN: 0045-7825

Journal article

Zheng J, Fang F, Wang Z, Zhu J, Li J, Xiao H, Pain CCet al., 2020, A new anisotropic adaptive mesh photochemical model for ozone formation in power plant plumes, ATMOSPHERIC ENVIRONMENT, Vol: 229, ISSN: 1352-2310

Journal article

Steppeler J, Li J, Navon IM, Fang F, Xiao Zet al., 2020, Medium range forecasts using cut-cells: a sensitivity study, METEOROLOGY AND ATMOSPHERIC PHYSICS, Vol: 132, Pages: 171-179, ISSN: 0177-7971

Journal article

Arcucci R, Casas CQ, Xiao D, Mottet L, Fang F, Wu P, Pain C, Guo Y-Ket al., 2020, A Domain Decomposition Reduced Order Model with Data Assimilation (DD-RODA), Conference on Parallel Computing - Technology Trends (ParCo), Publisher: IOS PRESS, Pages: 189-198, ISSN: 0927-5452

Conference paper

Li J, Steppeler J, Fang F, Pain CC, Zhu J, Peng X, Dong L, Li Y, Tao L, Leng W, Wang Y, Zheng Jet al., 2019, Potential numerical techniques and challenges for atmospheric modeling, Bulletin of the American Meteorological Society, Vol: 125, Pages: ES239-ES242, ISSN: 0003-0007

Journal article

Xiao D, Fang F, Heaney CE, Navon IM, Pain CCet al., 2019, A domain decomposition method for the non-intrusive reduced order modelling of fluid flow, Computer Methods in Applied Mechanics and Engineering, Vol: 354, Pages: 307-330, ISSN: 0045-7825

In this paper we present a new domain decomposition non-intrusive reduced order model (DDNIROM) for the Navier–Stokes equations. The computational domain is partitioned into subdomains and a set of local basis functions is constructed in each subdomain using Proper Orthogonal Decomposition (POD). A radial basis function (RBF) method is then used to generate a set of hypersurfaces for each subdomain. Each local hypersurface represents, not only the fluid dynamics over the subdomain to which it belongs, but also the interactions with the surrounding subdomains. This implicit coupling between the subdomains provides the global coupling necessary to enforce incompressibility and is a means of providing boundary conditions for each subdomain.The performance of this DDNIROM is illustrated numerically by three examples: flow past a cylinder, and air flow over 2D and 3D street canyons. The results show that the DDNIROM exhibits good agreement with the high-fidelity full model while the computational cost is reduced by several orders of magnitude. The domain decomposition (DD) method provides the flexibility to choose different numbers of local basis functions for each subdomain depending on the complexity of the flow therein. The fact that the RBF surface representation takes input only from its current subdomain and the surrounding subdomains, means that, crucially, there is a reduction in the dimensionality of the hypersurface when compared with a more traditional, global NIROM. This comes at the cost of having a larger number of hypersurfaces.

Journal article

Hu R, Fang F, Pain CC, Navon IMet al., 2019, Rapid spatio-temporal flood prediction and uncertainty quantification using a deep learning method, Journal of Hydrology, Vol: 575, Pages: 911-920, ISSN: 0022-1694

Recently accrued attention has been given to machine learning approaches for flooding prediction. However, most of these studies focused mainly on time-series flooding prediction at specified sensors, rarely on spatio-temporal prediction of inundations. In this work, an integrated long short-term memory (LSTM) and reduced order model (ROM) framework has been developed. This integrated LSTM-ROM has the capability of representing the spatio-temporal distribution of floods since it takes advantage of both ROM and LSTM. To reduce the dimensional size of large spatial datasets in LSTM, the proper orthogonal decomposition (POD) and singular value decomposition (SVD) approaches are introduced. The LSTM training and prediction processes are carried out over the reduced space. This leads to an improvement of computational efficiency while maintaining the accuracy. The performance of the LSTM-ROM developed here has been evaluated using Okushiri tsunami as test cases. The results obtained from the LSTM-ROM have been compared with those from the full model (Fluidity). In predictive analytics, it is shown that the results from both the full model and LSTM-ROM are in a good agreement whilst the CPU cost using the LSTM-ROM is decreased by three orders of magnitude compared to full model simulations. Additionally, prescriptive analytics has been undertaken to estimate the uncertainty in flood induced conditions. Given the time series of the free surface height at a specified detector, the corresponding induced wave conditions along the coastline have then been provided using the LSTM network. Promising results indicate that the use of LSTM-ROM can provide the flood prediction in seconds, enabling us to provide real-time predictions and inform the public in a timely manner, reducing injuries and fatalities.

Journal article

Steppeler J, Li J, Fang F, Zhu J, Ullrich PAet al., 2019, o3o3: A variant of spectral elements with a regular collocation grid, Monthly Weather Review, Vol: 147, Pages: 2067-2082, ISSN: 0027-0644

In this study, an alternative local Galerkin method (LGM), the o3o3 scheme, is proposed. o3o3 is a variant or generalization of the third-order spectral element method (SEM3). It uses third-order piecewise polynomials for the representation of a field and piecewise third-degree polynomials for fluxes. For the discretization, SEM3 uses the irregular Legendre–Gauss–Lobatto grid while o3o3 uses a regular collocation grid. o3o3 can be regarded as an inhomogeneous finite-difference scheme on a uniform grid, which means that the finite-difference equations are different for each group with three points. A particular version of o3o3 is set as an example of many possibilities to construct LGM schemes on piecewise polynomial spaces in which the basis functions used are continuous at corner points and function spaces having continuous derivatives are shortly discussed. We propose a standard o3o3 scheme and a spectral o3o3 scheme as alternatives to the standard method of using the quadrature approximation. These two particular schemes selected were chosen for ease of implementation rather than optimal performance. In one dimension, compared to standard SEM3, o3o3 has a larger CFL condition benefiting from the use of a regular collocation grid. While SEM3 uses the irregular Legendre–Gauss–Lobatto collocation grid, o3o3 uses a regular grid. This is considered an advantage for physical parameterizations. The shortest resolved wave is marginally smaller than that with SEM3. In two dimensions, o3o3 is implemented on a sparse grid where only a part of the points on the underlying regular grid are used for forecasting.

Journal article

Steppeler J, Li J, Fang F, Zhu Jet al., 2019, Test of a cubic spline interface for physical processes with a 1-D third-order spectral element model, Tellus Series A: Dynamic Meteorology and Oceanography, Vol: 71, Pages: 1-6, ISSN: 0280-6495

A common way to introduce physical processes into numerical models of the atmosphere is to call the parameterization at every grid point. This can lead to considerable errors. A simple 1-D example is proposed to illustrate that when a physical process occurs at one grid point only, a considerable sampling error may occur, with the result that only a fraction of the true impact of this process is seen. The interface to the physical parameterization in numerical weather prediction model using a third-order 1-D spectral element method (SEM3) model is investigated by homogeneous advection. In SEM3, the grid points, called principal nodes, are at boundaries of computational intervals and two more collocation points in the interior of each cell. This article argues that it is sufficient to do the physical parameterization for principal nodes only that creating the interior grid-point values of physics schemes by linear interpolation. This is called the spline interface method. A simple condensation model of water is taken as an example. Compared to the standard paramaterization, which computes the physical processes at every grid point, the spline interface method is more accurate and has a potential to save computer time. It turns out that the standard method creates a noisy wave which can easily be filtered by hyperviscosity. In the spline interface to the condensation physics, the condensation is done at every third grid point only. Third-order spline methods are used to represent the condensation at other points. The method using a smaller grid to compute condensation represented the condensation process more accurately and produced less of the computational noise. This version could be run without hyperviscosity, as no significant computational noise mode was generated by condensation. By doing physical processes only at every third grid point computer time may be saved.

Journal article

Xiao D, Heaney CE, Fang F, Mottet L, Hu R, Bistrian DA, Aristodemou E, Navon IM, Pain CCet al., 2019, A domain decomposition non-intrusive reduced order model for turbulent flows, Computers and Fluids, Vol: 182, Pages: 15-27, ISSN: 0045-7930

In this paper, a new Domain Decomposition Non-Intrusive Reduced Order Model (DDNIROM) is developed for turbulent flows. The method works by partitioning the computational domain into a number of subdomains in such a way that the summation of weights associated with the finite element nodes within each subdomain is approximately equal, and the communication between subdomains is minimised. With suitably chosen weights, it is expected that there will be approximately equal accuracy associated with each subdomain. This accuracy is maximised by allowing the partitioning to occur through areas of the domain that have relatively little flow activity, which, in this case, is characterised by the pointwise maximum Reynolds stresses.A Gaussian Process Regression (GPR) machine learning method is used to construct a set of local approximation functions (hypersurfaces) for each subdomain. Each local hypersurface represents not only the fluid dynamics over the subdomain it belongs to, but also the interactions of the flow dynamics with the surrounding subdomains. Thus, in this way, the surrounding subdomains may be viewed as providing boundary conditions for the current subdomain.We consider a specific example of turbulent air flow within an urban neighbourhood at a test site in London and demonstrate the effectiveness of the proposed DDNIROM.

Journal article

Yang P, Xiang J, Fang F, Pain CCet al., 2019, A fidelity fluid-structure interaction model for vertical axis tidal turbines in turbulence flows, APPLIED ENERGY, Vol: 236, Pages: 465-477, ISSN: 0306-2619

Journal article

Xiao D, Fang F, Pain C, Navon IM, Zheng Jet al., 2019, Machine learning-based rapid response tools for regional air pollution modelling, Atmospheric Environment, Vol: 199, Pages: 463-473, ISSN: 1352-2310

A parameterised non-intrusive reduced order model (P-NIROM) based on proper orthogonal decomposition (POD) and machine learning methods has been firstly developed for model reduction of pollutant transport equations. Our motivation is to provide rapid response urban air pollution predictions and controls. The varying parameters in the P-NIROM are pollutant sources. The training data sets are obtained from the high fidelity modelling solutions (called snapshots) for selected parameters (pollutant sources, here) over the parameter space . From these training data sets, the machine learning method is used to generate the relationship between the reduced solutions and inputs (pollutant sources) over . Furthermore a set of hyper-surface functions associated with each POD basis function is constructed for representing the fluid dynamics over the reduced space. The accuracy of the P-NIROM is highly dependent on the quality of the training set, here obtained from the high fidelity model. Over existing machine learning methods, the P-NIROM algorithm proposed here has the advantages that (1) it is combined with NIROM, thus providing rapid and reasonably accurate solutions; and (2) it is a robust and efficient approach for representation of any parametrised partial differential equations as the model parameters/inputs vary. In this study, we demonstrate the way how to implement the P-NIROM for the pollutant transport equation (but not limited to due to its robustness). Its predictive capability is illustrated in a three-dimensional (3-D) simulation of power plant plumes over a large region in China, where the varying parameters are the emission intensity at three locations. Results indicate that in comparison to the high fidelity model, the CPU cost is reduced by factor up to five orders of magnitude while reasonable accuracy remains.

Journal article

Yang P, Xiang J, Fang F, Pavlidis D, Pain CCet al., 2019, Modelling of fluid-structure interaction for moderate reynolds number flows using an immersed-body method, COMPUTERS & FLUIDS, Vol: 179, Pages: 613-631, ISSN: 0045-7930

Journal article

Xiao D, Heaney CE, Mottet L, Fang F, Lin W, Navon IM, Guo Y, Matar OK, Robins AG, Pain CCet al., 2019, A reduced order model for turbulent flows in the urban environment using machine learning, Building and Environment, Vol: 148, Pages: 323-337, ISSN: 0360-1323

To help create a comfortable and healthy indoor and outdoor environment in which to live, there is a need to understand turbulent air flows within the urban environment. To this end, building on a previously reported method [1], we develop a fast-running Non-Intrusive Reduced Order Model (NIROM) for predicting the turbulent air flows found within an urban environment. To resolve larger scale turbulent fluctuations, we employ a Large Eddy Simulation (LES) model and solve the resulting computational model on unstructured meshes. The objective is to construct a rapid-running NIROM from these results that will have ‘similar’ dynamics to the original LES model. Based on Proper Orthogonal Decomposition (POD) and machine learning techniques, this Reduced Order Model (ROM) is six orders of magnitude faster than the high-fidelity LES model and we demonstrate how ‘similar’ it can be to the high-fidelity model by comparing statistical quantities such as the mean flows, Reynolds stresses and probability densities of the velocities. We also include validation of the high-fidelity model against data from wind tunnel experiments.This paper represents a key step towards the use of reduced order modelling for operational purposes with the tantalising possibility of it being used in place of Gaussian plume models, and the potential for greatly improved model fidelity and confidence.

Journal article

Hu R, Fang F, Salinas P, Pain C, StoDomingo ND, Mark Oet al., 2019, Numerical simulation of floods from multiple sources using an adaptive anisotropic unstructured mesh method, Advances in Water Resources, Vol: 123, Pages: 173-188, ISSN: 0309-1708

The coincidence of two or more extreme events (precipitation and storm surge, for example) may lead to severe floods in coastal cities. It is important to develop powerful numerical tools for improved flooding predictions (especially over a wide range of spatial scales - metres to many kilometres) and assessment of joint influence of extreme events. Various numerical models have been developed to perform high-resolution flood simulations in urban areas. However, the use of high-resolution meshes across the whole computational domain may lead to a high computational burden. More recently, an adaptive isotropic unstructured mesh technique has been first introduced to urban flooding simulations and applied to a simple flooding event observed as a result of flow exceeding the capacity of the culvert during the period of prolonged or heavy rainfall. Over existing adaptive mesh refinement methods (AMR, locally nested static mesh methods), this adaptive unstructured mesh technique can dynamically modify (both, coarsening and refining the mesh) and adapt the mesh to achieve a desired precision, thus better capturing transient and complex flow dynamics as the flow evolves.In this work, the above adaptive mesh flooding model based on 2D shallow water equations (named as Floodity) has been further developed by introducing (1) an anisotropic dynamic mesh optimization technique (anisotropic-DMO); (2) multiple flooding sources (extreme rainfall and sea-level events); and (3) a unique combination of anisotropic-DMO and high-resolution Digital Terrain Model (DTM) data. It has been applied to a densely urbanized area within Greve, Denmark. Results from MIKE 21 FM are utilized to validate our model. To assess uncertainties in model predictions, sensitivity of flooding results to extreme sea levels, rainfall and mesh resolution has been undertaken. The use of anisotropic-DMO enables us to capture high resolution topographic features (buildings, rivers and streets) only where and when

Journal article

Xiao D, Du J, Fang F, Pain C, Li Jet al., 2018, Parameterised non-intrusive reduced order methods for ensemble Kalman filter data assimilation, Computers and Fluids, Vol: 177, Pages: 69-77, ISSN: 0045-7930

This paper presents a novel Ensemble Kalman Filter (EnKF) data assimilation method based on a parameterised non-intrusive reduced order model (P-NIROM) which is independent of the original computational code. EnKF techniques involve the expensive calculations of ensembles. In this work, the recently developed P-NIROM Xiao et al. [40] is incorporated into EnKF to speed up the ensemble simulations. A reduced order flow dynamical model is generated from the solution snapshots, which are obtained from a number of the high fidelity full simulations over the specific parametric space RP. The varying parameter is the background error covariance σ ∈ RP. Using the Smolyak sparse grid method, a set of parameters in the Gaussian probability density function is selected as the training points. The proposed method uses a two-level interpolation method for constructing the P-NIROM using a Radial Basis Function (RBF) interpolation method. The first level interpolation approach is used for generating the solution snapshots and POD basis functions for any given background error covariance while the second level interpolation approach for forming a set of hyper-surfaces representing the reduced system.The EnKF in combination with P-NIROM (P-NIROM-EnKF) has been implemented within an unstructured mesh finite element ocean model and applied to a three dimensional wind driven circulation gyre case. The numerical results show that the accuracy of ensembles and updated solutions using the P-NIROM-EnKF is maintained while the computational cost is significantly reduced by several orders of magnitude in comparison to the full-EnKF.

Journal article

Li J, Zheng J, Zhu J, Fang F, Pain CC, Steppeler J, Navon IM, Xiao Het al., 2018, Performance of Adaptive Unstructured Mesh Modelling in Idealized Advection Cases over Steep Terrains, ATMOSPHERE, Vol: 9, ISSN: 2073-4433

Journal article

Xiao D, Fang F, Pain C, Salinas P, Navon IM, Wang Zet al., 2018, Non-intrusive model reduction for a 3D unstructured mesh control volume finite element reservoir model and its application to fluvial channels, International Journal of Oil, Gas and Coal Technology, Vol: 19, Pages: 316-339, ISSN: 1753-3309

non-intrusive model reduction computational method using hypersurfaces representation has been developed for reservoir simulation and further applied to 3D fluvial channel problems in this work. This is achieved by a combination of a radial basis function (RBF) interpolation and proper orthogonal decomposition (POD) method. The advantage of the method is that it is generic and non-intrusive, that is, it does not require modifications to the original complex source code, for example, a 3D unstructured mesh control volume finite element (CVFEM) reservoir model used here. The capability of this non-intrusive reduced order model (NIROM) based on hypersurfaces representation has been numerically illustrated in a horizontally layered porous media case, and then further applied to a 3D complex fluvial channel case. By comparing the results of the NIROM against the solutions obtained from the high fidelity full model, it is shown that this NIROM results in a large reduction in the CPU computation cost while much of the details are captured.

Journal article

Heaney C, Salinas P, Pain C, Fang F, Navon Met al., 2018, Well Optimisation With Goal-Based Sensitivity Maps Using Time Windows And Ensemble Perturbations, European conference on mathematics of oil recovery

Conference paper

Yang L, Lyu Z, Yang P, Pavlidis D, Fang F, Xiang J, Latham JP, Pain Cet al., 2018, Numerical Simulation of Attenuator Wave Energy Converter using One-Fluid Formulation, Proceedings of the 28th International Ocean and Polar Engineering Conference

Conference paper

Song J, Fan S, Lin W, Mottet L, Woodward H, Wykes MD, Arcucci R, Xiao D, Debay J-E, ApSimon H, Aristodemou E, Birch D, Carpentieri M, Fang F, Herzog M, Hunt GR, Jones RL, Pain C, Pavlidis D, Robins AG, Short CA, Linden PFet al., 2018, Natural ventilation in cities: the implications of fluid mechanics, BUILDING RESEARCH AND INFORMATION, Vol: 46, Pages: 809-828, ISSN: 0961-3218

Journal article

Heaney CE, Salinas P, Fang F, Pain CC, Navon IMet al., 2018, Goal-based sensitivity maps using time windows and ensemble perturbations

We present an approach for forming sensitivity maps (or sensitivites) usingensembles. The method is an alternative to using an adjoint, which can be verychallenging to formulate and also computationally expensive to solve. The mainnovelties of the presented approach are: 1) the use of goals, weighting theperturbation to help resolve the most important sensitivities, 2) the use oftime windows, which enable the perturbations to be optimised independently foreach window and 3) re-orthogonalisation of the solution through time, whichhelps optimise each perturbation when calculating sensitivity maps. These novelmethods greatly reduce the number of ensembles required to form the sensitivitymaps as demonstrated in this paper. As the presented method relies solely onensembles obtained from the forward model, it can therefore be applied directlyto forward models of arbitrary complexity arising from, for example,multi-physics coupling, legacy codes or model chains. It can also be applied tocompute sensitivities for optimisation of sensor placement, optimisation fordesign or control, goal-based mesh adaptivity, assessment of goals (e.g. hazardassessment and mitigation in the natural environment), determining the worth ofcurrent data and data assimilation. We analyse and demonstrate the efficiency of the approach by applying themethod to advection problems and also a non-linear heterogeneous multi-phaseporous media problem, showing, in all cases, that the number of ensemblesrequired to obtain accurate sensitivity maps is relatively low, in the order of10s.

Journal article

Hu R, Fang F, Salinas P, Pain Cet al., 2018, Unstructured mesh adaptivity for urban flooding modelling, Journal of Hydrology

Journal article

Heaney CE, Salinas P, Pain CC, Fang F, Navon IMet al., 2018, Well optimisation with goal-based sensitivity maps using time windows and ensemble perturbations

Knowledge of the sensitivity of a solution to small changes in the model parameters is exploited in many areas in computational physics and used to perform mesh adaptivity, or to correct errors based on discretisation and sub-grid-scale modelling errors, to perform the assimilation of data based on adjusting the most sensitive parameters to the model-observation misfit, and similarly to form optimised sub-grid-scale models. We present a goal-based approach for forming sensitivity (or importance) maps using ensembles. These maps are defined as regions in space and time of high relevance for a given goal, for example, the solution at an observation point within the domain. The presented approach relies solely on ensembles obtained from the forward model and thus can be used with complex models for which calculating an adjoint is not a practical option. This provides a simple approach for optimisation of sensor placement, goal based mesh adaptivity, assessment of goals and data assimilation. We investigate methods which reduce the number of ensembles used to construct the maps yet which retain reasonable fidelity of the maps. The fidelity comes from an integrated method including a goal-based approach, in which the most up-to-date importance maps are fed back into the perturbations to focus the algorithm on the key variables and domain areas. Also within the method smoothing is applied to the perturbations to obtain a multi-scale, global picture of the sensitivities; the perturbations are orthogonalised in order to generate a well-posed system which can be inverted; and time windows are applied (for time dependent problems) where we work backwards in time to obtain greater accuracy of the sensitivity maps. The approach is demonstrated on a multi-phase flow problem.

Conference paper

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