FPE Extra Issue 10, October 2016 
Issue 10, October 2016Towards Faster Wildland FireSpread ModelsBy: Alan Lattimer, Ph.D. and Brian Lattimer, Ph.D. Jensen Hughes, Inc.
From firefighter safety to the mitigation of damages in the wildlandurban interface, situational awareness is critical to the management of wildland fires. Accurate predictions of the fire spread generated in better than realtime are critical to maintaining this awareness. However, wildland fires are large, multiscale, complex phenomena that are affected by marginally under stood or known factors such as the weather and fuel loading. Due to the scale of the problem, firespread models must simplify the physics or the scale to meet the realtime requirements for generating a prediction. A third option for reducing the computation time is reducedorder modeling. This article introduces the basic reducedorder modeling technique for firespread models and discusses the challenges and opportunities moving forward. IntroductionThe modeling and simulation of wildland fires is a complex, multiscale problem. Several firespread models (such as FARSITE)^{1} have been developed to provide operational resource management during fire outbreaks.^{24} Physicsbased, largescale models for predicting wildland fires, such as FIRETEC and WFDS,^{511} are useful for understanding the processes that drive the spread of wildland fires, particularly into the wildlandurban interface (WUI). However, the realtime requirements and limited onscene computational resources often preclude their use as an operational tool. To reduce the time, levelset methods that track the fire front, such as WRFFire,^{1213} are often used to produce realtime predictions. The spread of the fire in these models is typically based on the quasiempirical mathematical firespread model given by Rothermel.^{14} ReducedOrder ModelingTo demonstrate the efficacy of reducedorder modeling, proper orthogonal decomposition (POD) in conjunction with the discrete empirical interpolation method (DEIM)^{1517} was used to predict the solution to the advectionreactiondiffusion equation given by Mandel et al.^{18} This technique also projects the nonlinearity, resulting in ROMs that decrease the online computational cost by 1001000 times. This ROM technique can be applied in other largescale fire applications, such as ventilation for mine fires. POD is the most commonly used modelreduction technique for nonlinear dynamics.^{1921 }This technique has previously been used for wildland firespread models with limited success with a speed up of approximately five times over the fullorder model (FOM).^{2223} By addressing the nonlinearity of the system using DEIM alongside POD, the ROM is significantly faster with little degradation in accuracy.
The details of how to create a reducedorder model are beyond the scope of this article, but the specifics of the mathematical model and how it is reduced can be found in Lattimer.^{24} The reader is also referred to several good resources in addition to the ones above on the specifics of POD.^{2528} The focus of this article is on POD, but Benner et al.^{29} and the references therein, are recommended as excellent resources for other reducedorder modeling strategies.
ResultsThe phenomenological model studied in this article is suggested by Mandel et al.^{18} as a model to predict flamefront propagation in wildland fires. The onedimensional (1D) and two dimensional (2D) versions of this coupled system were studied to assess the benefits of POD with DEIM over the standard POD approach. These approaches look to project the original system into a smaller system that captures the important features of the original system. Once the smaller system has been solved, we can project it back up to the original system size.
Table 1: Results for the ROM. Solution time for the FOM was 99.1 seconds
Figure 1: FOM versus POD and POD/DEIM ROM where rT = 200, rS = 150, and rDEIM = 250.
Figure 2: Solution times for the FOM and POD/DEIM models when running ten simulations.
As can be seen in Figure 1, the results for each of these models was almost identical. However, as Table 1 shows, the solution time for the POD model is just over five times faster than the FOM, and the POD/DEIM model was over a hundred times faster than the FOM for those shown in Figure 1. There is an offline computational cost to develop the ROM. The advantage is that this cost must only be paid once, and after that all models with similar input conditions can be run from the previously built ROM. To see how these time savings accrue, a graph showing the computational costs for running ten ROMs versus ten fullorder models is given in Figure 2.
Figure 3: FOM versus POD/DEIM ROM for the 2D firespread model.
The results for the 2D model are even better. The same methodology was followed for the 2D models as was done for the 1D models, the difference being that our domain is now a 1000 meter by 1000 meter grass field. The initial conditions have a fire started in the center of the field, and the model was again run for a 3000 second simulation. In this case, the fullorder model required 226 seconds to complete while the POD/DEIM ROM took 0.86 seconds. Thus, the ROM is approximately 260 times faster than the original model. Further, the original model required significantly more computational resources. Figure 3 shows a comparison of the two solutions which appear to be well within the operational needs for accuracy.
ConclusionsNonlinear wildland firespread models such as the one given by Mandel et al.^{18} can be challenging to reduce due to the nonlinearity of the model. By addressing this issue using DEIM, the computational time was reduced by another order of magnitude over POD alone without sacrificing accuracy. While this model is simplistic when compared to the stateoftheart firespread models, it does serve to show that reducedorder modeling can be an effective tool to reduce the computational costs without sacrificing the underlying physics and accuracy of the fullorder model. It is important again to note that reducedorder modeling is not a new firespread model. Rather, it is mathematical technique that can make existing models faster. In this way, reduced order modeling can be the tool that moves the existing models from running on highperformance computers to laptops, or, put another way, from the laboratory to the field. There is much research to still be done to ensure the efficacy of this technique on existing firespread solutions, but the initial results are encouraging. AcknowledgementsThis research was partially developed under Grant No. 200201459669, awarded by NIOSH. The findings and conclusions in this report are those of the authors and do not reflect the official policies of the Department of Health and Human Services; nor does mention of trade names, commercial practices, or organizations imply endorsement by the U.S. Government.
Alan Lattimer, Ph.D. and Brian Lattimer, Ph.D. are with Jensen Hughes, Inc. References^{[1] M. Finney. FARSITE: Fire area simulator: model development and evaluation. 2004. [2] A. L. Sullivan. Wildland surface fire spread modelling, 19902007. 1: Physical and quasi physical models. International Journal of Wildland Fire, 18(4):349, 2009. ISSN 10498001. doi: 10.1071/WF06143. [3] A. L. Sullivan. Wildland surface fire spread modelling, 19902007. 2: Empirical and quasi empirical models. International Journal of Wildland Fire, 18(4):369, 2009. ISSN 10498001. doi: 10.1071/WF06142. [4] A. L. Sullivan. Wildland surface fire spread modelling, 19902007. 3: Simulation and math ematical analogue models. International Journal of Wildland Fire, 18(4):387, 2009. ISSN 10498001. doi: 10.1071/WF06144. [5] W. Mell, S. Manzello, A. Maranghides, and D. Butry. The wildlandurban interface fire problemcurrent approaches and research needs. Journal of Wildland Fire, 2010. [6] W. Mell, M. Jenkins, J. Gould, and P. Cheney. A physicsbased approach to modelling grass land fires. Journal of Wildland Fire, 2007. [7] W. Mell, J. Charney, M. Jenkins, and P. Cheney. Numerical simulations of grassland fire behavior from the LANLFIRETEC and NISTWFDS models. Remote Sensing and, 2013. [8] F. Pimont, R. Parsons, E. Rigolot, F. de Coligny, J.L. Dupuy, P. Dreyfus, and R. R. Linn. Modeling fuels and fire effects in 3D: Model description and applications. Environmental Modelling & Software, 80:225–244, 2016. ISSN 13648152. doi: 10.1016/j.envsoft.2016.03.003. [9] R. Linn, J. Reisner, J. Colman, and J. Winterkamp. Studying wildfire behavior using FIRETEC. International Journal of Wildland Fire, 2002. [10] R. Linn and F. Harlow. FIRETEC: a transport description of wildfire behavior. 1997. [11] M. Clark, T. Fletcher, and R. Linn. A subgrid, mixturefractionbased thermodynamic equi librium model for gas phase combustion in FIRETEC: development and results. International Journal of Wildland Fire, 2010. [12] R. Rehm and R. McDermott. Firefront propagation using the level set method. 2009. [13] J. L. Coen, M. Cameron, J. Michalakes, E. G. Patton, P. J. Riggan, and K. M. Yedinak. WRFFire: coupled weatherwildland fire modeling with the weather research and forecasting model. Journal of Applied Meteorology and Climatology, 52(1):16–38, 2013. [14] R. Rothermel. A mathematical model for predicting fire spread in wildland fuels. 1972. [15] S. Chaturantabut and D. C. Sorensen. Discrete empirical interpolation for nonlinear model reduction. In Decision and Control, 2009 held jointly with the 2009 28th Chinese Control Conference. CDC/CCC 2009. Proceedings of the 48th IEEE Conference on, pages 4316– 4321. IEEE, 2009. [16] S. Chaturantabut and D. C. Sorensen. Nonlinear model reduction via discrete empirical inter polation. SIAM Journal on Scientific Computing, 32(5):2737–2764, 2010. [17] Z. Drmac and S. Gugercin. A new selection operator for the Discrete Empirical Interpolation Method–improved a priori error bound and extensions. arXiv preprint arXiv:1505.00370, 2015. [18] J. Mandel, L. S. Bennethum, J. D. Beezley, J. L. Coen, C. C. Douglas, M. Kim, and A. Vo dacek. A wildland fire model with data assimilation. Mathematics and Computers in Simula tion, 79(3):584–606, 2008. [19] K. Kunisch and S. Volkwein. Galerkin proper orthogonal decomposition methods for a gen eral equation in fluid dynamics. SIAM Journal on Numerical analysis, 40(2):492–515, 2002. [20] M. Hinze and S. Volkwein. Proper orthogonal decomposition surrogate models for nonlin ear dynamical systems: Error estimates and suboptimal control. In Dimension Reduction of LargeScale Systems, pages 261–306. Springer, 2005. [21] S. Volkwein. Proper orthogonal decomposition: Theory and reducedorder modelling. Lec ture Notes, University of Konstanz, 4:4, 2013. [22] B. R. Sharma, M. Kumar, and K. Cohen. SpatioTemporal Estimation of Wildfire Growth. In ASME 2013 Dynamic Systems and Control Conference, pages V002T25A005– V002T25A005. American Society of Mechanical Engineers, 2013. [23] E. Guelpa, A. Sciacovelli, V. Verda, and D. Ascoli. Model reduction approach for wildfire multiscenario analysis. In Advances in Forest Fire Research. Imprensa da Universidade de Coimbra, 2014. [24] A. M. Lattimer. Model Reduction of Nonlinear Fire Dynamics Models. PhD thesis, apr 2016. [25] T. BuiThanh, K. Willcox, O. Ghattas, and B. van Bloemen Waanders. Goaloriented, model constrained optimization for reduction of largescale systems. Journal of Computational Physics, 224(2):880–896, 2007. [26] K. Willcox, O. Ghattas, B. van Bloemen Waanders, and B. Bader. An optimization frame work for goaloriented, modelbased reduction of largescale systems. In Decision and Con trol, 2005 and 2005 European Control Conference. CDCECC’05. 44th IEEE Conference on, pages 2265–2271. IEEE, 2005. [27] K. Willcox and J. Peraire. Balanced model reduction via the proper orthogonal decomposition. AIAA journal, 40(11):2323–2330, 2002. [28] K. Willcox. Unsteady flow sensing and estimation via the gappy proper orthogonal decom position. Computers & fluids, 35(2):208–226, 2006. } ^{ } ^{[29] P. Benner, S. Gugercin, and K. Willcox. A survey of projectionbased model reduction meth ods for parametric dynamical systems. SIAM Review, 57(4):483–531, 2015. } Subscribe to FPE Extra To subscribe to receive FPE Extra via email each month, sign up here.
