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These notes started to educate ourselves and to collect some background for our future work, with the hope that perhaps they will be useful to others also. Many if not all results are more or less elementary or available in the literature, but we need to fill some holes (which are undoubtely statements so trivial that the authors we use do not consider them holes at all) or make straightforward extensions, and then we do the proofs in sufficient detail for reference. Topics include random fields and stochastic processes as random elements in Hilbert spaces, Karhunen-Lo\`{e}ve explansion and random orthonormal series, laws of large numbers, white noise, convergence of the Ensemble Kalman Filter (EnKF), and the ensemble Kalman Transform Filter (ETKF)., Comment: 69 pages, 3 figures, 62 references

Increases in wildfire activity and the resulting impacts have prompted the development of high-resolution wildfire behavior models for forecasting fire spread. Recent progress in using satellites to detect fire locations further provides the opportunity to use measurements to improve fire spread forecasts from numerical models through data assimilation. This work develops a method for inferring the history of a wildfire from satellite measurements, providing the necessary information to initialize coupled atmosphere-wildfire models from a measured wildfire state in a physics-informed approach. The fire arrival time, which is the time the fire reaches a given spatial location, acts as a succinct representation of the history of a wildfire. In this work, a conditional Wasserstein Generative Adversarial Network (cWGAN), trained with WRF-SFIRE simulations, is used to infer the fire arrival time from satellite active fire data. The cWGAN is used to produce samples of likely fire arrival times from the conditional distribution of arrival times given satellite active fire detections. Samples produced by the cWGAN are further used to assess the uncertainty of predictions. The cWGAN is tested on four California wildfires occurring between 2020 and 2022, and predictions for fire extent are compared against high resolution airborne infrared measurements. Further, the predicted ignition times are compared with reported ignition times. An average Sorensen's coefficient of 0.81 for the fire perimeters and an average ignition time error of 32 minutes suggest that the method is highly accurate.

Data likelihood of fire detection is the probability of the observed detection outcome given the state of the fire spread model. We derive fire detection likelihood of satellite data as a function of the fire arrival time on the model grid. The data likelihood is constructed by a combination of the burn model, the logistic regression of the active fires detections, and the Gaussian distribution of the geolocation error. The use of the data likelihood is then demonstrated by an estimation of the ignition point of a wildland fire by the maximization of the likelihood of MODIS and VIIRS data over multiple possible ignition points., Comment: 12 pages, 6 figures; VIII International Conference on Forest Fire Research, Coimbra, Portugal, November 2018

Observational data collected during experiments, such as the planned Fire and Smoke Model Evaluation Experiment (FASMEE), are critical for progressing and transitioning coupled fire-atmosphere models like WRF-SFIRE and WRF-SFIRE-CHEM into operational use. Historical meteorological data, representing typical weather conditions for the anticipated burn locations and times, have been processed to initialize and run a set of simulations representing the planned experimental burns. Based on an analysis of these numerical simulations, this paper provides recommendations on the experimental setup that include the ignition procedures, size and duration of the burns, and optimal sensor placement. New techniques are developed to initialize coupled fire-atmosphere simulations with weather conditions typical of the planned burn locations and time of the year. Analysis of variation and sensitivity analysis of simulation design to model parameters by repeated Latin Hypercube Sampling are used to assess the locations of the sensors. The simulations provide the locations of the measurements that maximize the expected variation of the sensor outputs with the model parameters., Comment: 35 pages, 4 tables, 28 figures

Assimilation of data into a fire-spread model is formulated as an optimization problem. The level set equation, which relates the fire arrival time and the rate of spread, is allowed to be satisfied only approximately, and we minimize a norm of the residual. Previous methods based on modification of the fire arrival time either used an additive correction to the fire arrival time, or made a position correction. Unlike additive fire arrival time corrections, the new method respects the dependence of the fire rate of spread on diurnal changes of fuel moisture and on weather changes, and, unlike position corrections, it respects the dependence of the fire spread on fuels and terrain as well. The method is used to interpolate the fire arrival time between two perimeters by imposing the fire arrival time at the perimeters as constraints., Comment: 13 pages, 6 figures, ICCS 2018. Minor update

Bayesian inverse problem on an infinite dimensional separable Hilbert space with the whole state observed is well posed when the prior state distribution is a Gaussian probability measure and the data error covariance is a cylindric Gaussian measure whose covariance has positive lower bound. If the state distribution and the data distribution are equivalent Gaussian probability measures, then the Bayesian posterior measure is not well defined. If the state covariance and the data error covariance commute, then the Bayesian posterior measure is well defined for all data vectors if and only if the data error covariance has positive lower bound, and the set of data vectors for which the Bayesian posterior measure is not well defined is dense if the data error covariance does not have positive lower bound., Comment: 13 pages

The asymptotic variance of the maximum likelihood estimate is proved to decrease when the maximization is restricted to a subspace that contains the true parameter value. Maximum likelihood estimation allows a systematic fitting of covariance models to the sample, which is important in data assimilation. The hierarchical maximum likelihood approach is applied to the spectral diagonal covariance model with different parameterizations of eigenvalue decay, and to the sparse inverse covariance model with specified parameter values on different sets of nonzero entries. It is shown computationally that using smaller sets of parameters can decrease the sampling noise in high dimension substantially., Comment: 18 pages, 6 figures; better resolution images

A new type of ensemble Kalman filter is developed, which is based on replacing the sample covariance in the analysis step by its diagonal in a spectral basis. It is proved that this technique improves the aproximation of the covariance when the covariance itself is diagonal in the spectral basis, as is the case, e.g., for a second-order stationary random field and the Fourier basis. The method is extended by wavelets to the case when the state variables are random fields, which are not spatially homogeneous. Efficient implementations by the fast Fourier transform (FFT) and discrete wavelet transform (DWT) are presented for several types of observations, including high-dimensional data given on a part of the domain, such as radar and satellite images. Computational experiments confirm that the method performs well on the Lorenz 96 problem and the shallow water equations with very small ensembles and over multiple analysis cycles., Comment: 15 pages, 4 figures

Ensemble methods, such as the ensemble Kalman filter (EnKF), the local ensemble transform Kalman filter (LETKF), and the ensemble Kalman smoother (EnKS) are widely used in sequential data assimilation, where state vectors are of huge dimension. Little is known, however, about the asymptotic behavior of ensemble methods. In this paper, we prove convergence in L^p of ensemble Kalman smoother to the Kalman smoother in the large-ensemble limit, as well as the convergence of EnKS-4DVAR, which is a Levenberg-Marquardt-like algorithm with EnKS as the linear solver, to the classical Levenberg-Marquardt algorithm in which the linearized problem is solved exactly.

Currently available satellite active fire detection products from the VIIRS and MODIS instruments on polar-orbiting satellites produce detection squares in arbitrary locations. There is no global fire/no fire map, no detection under cloud cover, false negatives are common, and the detection squares are much coarser than the resolution of a fire behavior model. Consequently, current active fire satellite detection products should be used to improve fire modeling in a statistical sense only, rather than as a direct input. We describe a new data assimilation method for active fire detection, based on a modification of the fire arrival time to simultaneously minimize the difference from the forecast fire arrival time and maximize the likelihood of the fire detection data. This method is inspired by contour detection methods used in computer vision, and it can be cast as a Bayesian inverse problem technique, or a generalized Tikhonov regularization. After the new fire arrival time on the whole simulation domain is found, the model can be re-run from a time in the past using the new fire arrival time to generate the heat fluxes and to spin up the atmospheric model until the satellite overpass time, when the coupled simulation continues from the modified state., Comment: 9 pages, VII International Conference on Forest Fire Research, Coimbra, Portugal, November 17-20, 2014

Fuel moisture has a major influence on the behavior of wildland fires and is an important underlying factor in fire risk assessment. We propose a method to assimilate dead fuel moisture content observations from remote automated weather stations (RAWS) into a time-lag fuel moisture model. RAWS are spatially sparse and a mechanism is needed to estimate fuel moisture content at locations potentially distant from observational stations. This is arranged using a trend surface model (TSM), which allows us to account for the effects of topography and atmospheric state on the spatial variability of fuel moisture content. At each location of interest, the TSM provides a pseudo-observation, which is assimilated via Kalman filtering. The method is tested with the time-lag fuel moisture model in the coupled weather-fire code WRF-SFIRE on 10-hr fuel moisture content observations from Colorado RAWS in 2013. We show using leave-one-out testing that the TSM compares favorably with inverse squared distance interpolation as used in the Wildland Fire Assessment System. Finally, we demonstrate that the data assimilation method is able to improve fuel moisture content estimates in unobserved fuel classes., Comment: 28 pages

In this study, we describe how WRF-Sfire is coupled with WRF-Chem to construct WRFSC, an integrated forecast system for wildfire and smoke prediction. The integrated forecast system has the advantage of not requiring a simple plume-rise model and assumptions about the size and heat release from the fire in order to determine fire emissions into the atmosphere. With WRF-Sfire, wildfire spread, plume and plume-top heights are predicted directly, at every WRF timestep, providing comprehensive meteorology and fire emissions to the chemical transport model WRF-Chem. Evaluation of WRFSC was based on comparisons between available observations to the results of two WRFSC simulations. The study found overall good agreement between forecasted and observed fire spread and smoke transport for the Witch-Guejito fire. Also the simulated PM2.5 (fine particulate matter) peak concentrations matched the observations. However, the NO and ozone levels were underestimated in the simulations and the peak concentrations were mistimed. Determining the terminal or plume-top height is one of the most important aspects of simulating wildfire plume transport, and the study found overall good agreement between simulated and observed plume-top heights, with some (10% or less) underestimation by the simulations. One of the most promising results of the study was the agreement between passive-tracer modeled plume-top heights for the Barker Canyon fire simulation and observations. This simulation took only 13h, with the first 24h forecast ready in almost 3h, making it a possible operational tool for providing emission profiles for external chemical transport models., Comment: 28 pages, 12 figures

Ensemble filters implement sequential Bayesian estimation by representing the probability distribution by an ensemble mean and covariance. Unbiased square root ensemble filters use deterministic algorithms to produce an analysis (posterior) ensemble with prescribed mean and covariance, consistent with the Kalman update. This includes several filters used in practice, such as the Ensemble Transform Kalman Filter (ETKF), the Ensemble Adjustment Kalman Filter (EAKF), and a filter by Whitaker and Hamill. We show that at every time index, as the number of ensemble members increases to infinity, the mean and covariance of an unbiased ensemble square root filter converge to those of the Kalman filter, in the case a linear model and an initial distribution of which all moments exist. The convergence is in $L^{p}$ and the convergence rate does not depend on the model dimension. The result holds in the infinitely dimensional Hilbert space as well., Comment: 14 pages, final draft

Fuel moisture is a major influence on the behavior of wildland fires and an important underlying factor in fire risk. We present a method to assimilate spatially sparse fuel moisture observations from remote automatic weather stations (RAWS) into the moisture model in WRF-SFIRE. WRF-SFIRE is a coupled atmospheric and fire behavior model which simulates the evolution of fuel moisture in idealized fuel species based on atmospheric state. The proposed method uses a modified trend surface model to estimate the fuel moisture field and its uncertainty based on currently available observations. At each grid point of WRF-SFIRE, this information is combined with the model forecast using a nonlinear Kalman filter, leading to an updated estimate of fuel moisture. We demonstrate the effectiveness of the method with tests in two real-world situations: a region in Southern California, where two large Santa Ana fires occurred recently, and on a domain enclosing Colorado., Comment: 4th Fire Behavior and Fuels Conference Raleigh, North Carolina, USA, February 18 - 22, 2013

Atmospheric pollution regulations have emerged as a dominant obstacle to prescribed burns. Thus, forecasting the pollution caused by wildland fires has acquired high importance. WRF and SFIRE model wildland fire spread in a two-way interaction with the atmosphere. The surface heat flux from the fire causes strong updrafts, which in turn change the winds and affect the fire spread. Fire emissions, estimated from the burning organic matter, are inserted in every time step into WRF-Chem tracers at the lowest atmospheric layer. The buoyancy caused by the fire then naturally simulates plume dynamics, and the chemical transport in WRF-Chem provides a forecast of the pollution spread. We discuss the choice of wood burning models and compatible chemical transport models in WRF-Chem, and demonstrate the results on case studies., Comment: Proceedings of 4th Fire Behavior and Fuels Conference, February 18-22, 2013, Raleigh, North Carolina, USA

We propose to use the ensemble Kalman smoother (EnKS) as linear least squares solver in the Gauss-Newton method for the large nonlinear least squares in incremental 4DVAR. The ensemble approach is naturally parallel over the ensemble members and no tangent or adjoint operators are needed. Further, adding a regularization term results in replacing the Gauss-Newton method, which may diverge, by^M the Levenberg-Marquardt method, which is known to be convergent. The regularization is implemented efficiently as an additional observation in the EnKS., Comment: 9 pages

We combine the adaptive and multilevel approaches to the BDDC and formulate a method which allows an adaptive selection of constraints on each decomposition level. We also present a strategy for the solution of local eigenvalue problems in the adaptive algorithm using the LOBPCG method with a preconditioner based on standard components of the BDDC. The effectiveness of the method is illustrated on several engineering problems. It appears that the Adaptive-Multilevel BDDC algorithm is able to effectively detect troublesome parts on each decomposition level and improve convergence of the method. The developed open-source parallel implementation shows a good scalability as well as applicability to very large problems and core counts., Comment: 37 pages, 10 figures, 13 tables

We describe two recent additions to WRF coupled with a fire spread model. Fire propagation is strongly dependent on fuel moisture, which in turn depends on the history of the atmosphere. We have implemented a equilibrium time-lag model of fuel moisture driven by WRF variables. The code allows the user to specify fuel parameters, with the defaults calibrated to the Canadian fire danger rating system for 10-hour fuel. The moisture model can run coupled with the atmosphere-fire model, or offline from WRF output to equilibrate the moisture over a period of time and to provide initial moisture conditions for a coupled atmosphere-fire-moisture simulation. The fire model also inserts smoke tracers into WRF-Chem to model the transport of fire emissions. The coupled model is available from OpenWFM.org. An earlier version of the fire model coupled with atmosphere is a part of WRF release., Comment: 2012 WRF Users Workshop, Boulder, CO, 25-29 June 2012, paper P51, 5 pages. Corrected a typo

We introduce several useful utilities in development for the creation and analysis of real wildland fire simulations using WRF and SFIRE. These utilities exist as standalone programs and scripts as well as extensions to other well known software. Python web scrapers automate the process of downloading and preprocessing atmospheric and surface data from common sources. Other scripts simplify the domain setup by creating parameter files automatically. Integration with Google Earth allows users to explore the simulation in a 3D environment along with real surface imagery. Postprocessing scripts provide the user with a number of output data formats compatible with many commonly used visualization suites allowing for the creation of high quality 3D renderings. As a whole, these improvements build toward a unified web application that brings a sophisticated wildland fire modeling environment to scientists and users alike., Comment: Submitted to proceedings of IGARSS 2012, 4 papers, 1 figure

This study uses in-situ measurements collected during the FireFlux field experiment to evaluate and improve the performance of coupled atmosphere-fire model WRF-Sfire. The simulation by WRF-Sfire of the experimental burn shows that WRF-Sfire is capable of providing realistic head fire rate-of-spread and the vertical temperature structure of the fire plume, and, up to 10 m above ground level, fire-induced surface flow and vertical velocities within the plume. The model captured the changes in wind speed and direction before, during, and after fire front passage, along with arrival times of wind speed, temperature, and updraft maximae, at the two instrumented flux towers used in FireFlux. The model overestimated vertical velocities and underestimated horizontal wind speeds measured at tower heights above the 10 m, and it is hypothesized that the limited model resolution over estimated the fire front depth, leading to too high a heat release and, subsequently, too strong an updraft. However, on the whole, WRF-Sfire fire plume behavior is consistent with the interpretation of FireFlux observations. The study suggests optimal experimental pre-planning, design, and execution of future field campaigns that are needed for further coupled atmosphere-fire model development and evaluation.

We present a methodology to change the state of the Weather Research Forecasting (WRF) model coupled with the fire spread code SFIRE, based on Rothermel's formula and the level set method, and with a fuel moisture model. The fire perimeter in the model changes in response to data while the model is running. However, the atmosphere state takes time to develop in response to the forcing by the heat flux from the fire. Therefore, an artificial fire history is created from an earlier fire perimeter to the new perimeter, and replayed with the proper heat fluxes to allow the atmosphere state to adjust. The method is an extension of an earlier method to start the coupled fire model from a developed fire perimeter rather than an ignition point. The level set method is also used to identify parameters of the simulation, such as the spread rate and the fuel moisture. The coupled model is available from openwfm.org, and it extends the WRF-Fire code in WRF release., Comment: ICCS 2012, 10 pages; corrected some DOI typesetting in the references

There are many wildfire behaviors of increasing relevance that are outside the forecast capabilities of even the most sophisticated operational fire spread and fire behavior model. The limitations of the operational models are due primarily to their inability to represent coupled fire-atmosphere interactions. Coupled wildfire-atmosphere models are physics-based fluid-dynamical prognostic models of wildfire spread and behavior that attempt an almost complete representation of fire-atmosphere interactions. This level of fidelity however means that these models cannot be used operationally. The reason is that, despite ever increasing computational resources, the complexity and range of processes and scales (1 mm to 100 km) involved in this modeling approach make computational costs prohibitively expensive. In this study we propose an intermediate approach. A physics-based coupled atmosphere-fire model is used to resolve the large-scale and local weather as well as the atmosphere-fire interactions, while combustion is represented simply using an existing operational surface fire behavior model. This model combination strikes a balance between fidelity and speed of execution. The feasibility of this approach is examined based on an analysis of a numerical simulation of two very large Santa Ana fires using WRF-Sfire, a coupled atmosphere-fire model available at the Open Wildland Fire Modeling Community (OpenWFM.org); an earlier version is available as WRF-Fire in WRF release. The study demonstrates that a wind and fire spread forecast of reasonable accuracy was obtained at an execution speed that would have made real-time wildfire forecasting of this event possible., Comment: Exploring the Mega-fire Reality 2011, November 14-17, 2011, Florida State University Conference Center, Tallahassee, Florida. Added full author names

In application of the Balancing Domain Decomposition by Constraints (BDDC) to a case with many substructures, solving the coarse problem exactly becomes the bottleneck which spoils scalability of the solver. However, it is straightforward for BDDC to substitute the exact solution of the coarse problem by another step of BDDC method with subdomains playing the role of elements. In this way, the algorithm of three-level BDDC method is obtained. If this approach is applied recursively, multilevel BDDC method is derived. We present a detailed description of a recently developed parallel implementation of this algorithm. The implementation is applied to an engineering problem of linear elasticity and a benchmark problem of Stokes flow in a cavity. Results by the multilevel approach are compared to those by the standard (two-level) BDDC method., Comment: 9 pages, 2 figures, 3 tables

We present an overview of a modeling environment, consisting of a coupled atmosphere-wildfire model, utilities for visualization, data processing, and diagnostics, open source software repositories, and a community wiki. The fire model, called SFIRE, is based on a fire-spread model, implemented by the level-set method, and it is coupled with the Weather Research Forecasting (WRF) model. A version with a subset of the features is distributed with WRF 3.3 as WRF-Fire. In each time step, the fire module takes the wind as input and returns the latent and sensible heat fluxes. The software architecture uses WRF parallel infrastructure for massively parallel computing. Recent features of the code include interpolation from an ideal logarithmic wind profile for nonhomogeneous fuels and ignition from a fire perimeter with an atmosphere and fire spin-up. Real runs use online sources for fuel maps, fine-scale topography, and meteorological data, and can run faster than real time. Visualization pathways allow generating images and animations in many packages, including VisTrails, VAPOR, MayaVi, and Paraview, as well as output to Google Earth. The environment is available from openwfm.org. New diagnostic variables were added to the code recently, including a new kind of fireline intensity, which takes into account also the speed of burning, unlike Byram's fireline intensity., Comment: 12 pages; Ninth Symposium on Fire and Forest Meteorology, Palm Springs, CA, October 2011

The current WRF-Fire model starts the fire from a given ignition point at a given time. We want to start the model from a given fire perimeter at a given time instead. However, the fuel balance and the state of the atmosphere depend on the history of the fire. The purpose of this work is to create an approximate artificial history of the fire based on the given fire perimeter and time and an approximate ignition point and time. Replaying the fire history then establishes a reasonable fuel balance and outputs heat fluxes into the atmospheric model, which allow the atmospheric circulation to develop. Then the coupled atmosphere-fire model takes over. In this preliminary investigation, the ignition times in the fire area are calculated based on the distance from the ignition point to the perimeter, assuming that the perimeter is convex or star-shaped. Simulation results for an ideal example show that the fire can continue in a natural way from the perimeter. Possible extensions include algorithms for more general perimeters and running the fire model backwards in time from the perimeter to create a more realistic history. The model used extends WRF-Fire and it is available from openwfm.org., Comment: 12th Annual WRF Users' Workshop, Boulder, CO, June 2011

We use a coupled atmosphere-fire model to simulate a fire that occurred on August 14--17, 2009, in the Harmanli region, Bulgaria. Data was obtained from GIS and satellites imagery, and from standard atmospheric data sources. Fuel data was classified in the 13 Anderson categories. For correct fire behavior, the spatial resolution of the models needed to be fine enough to resolve the essential micrometeorological effects. The simulation results are compared to available incident data. The code runs faster than real time on a cluster. The model is available from openwfm.org and it extends WRF-Fire from WRF 3.3 release., Comment: 8 pages, 2 tables, 5 figures. 8th International Conference on Large-Scale Scientific Computations, June 6-10, 2011, Sozopol, Bulgaria

We present a new type of the EnKF for data assimilation in spatial models that uses diagonal approximation of the state covariance in the wavelet space to achieve adaptive localization. The efficiency of the new method is demonstrated on an example., Comment: 4 pages, 4 figures

We describe the physical model, numerical algorithms, and software structure of WRF-Fire. WRF-Fire consists of a fire-spread model, implemented by the level-set method, coupled with the Weather Research and Forecasting model. In every time step, the fire model inputs the surface wind, which drives the fire, and outputs the heat flux from the fire into the atmosphere, which in turn influences the atmosphere. The level-set method allows submesh representation of the burning region and flexible implementation of various ignition modes. WRF-Fire is distributed as a part of WRF and it uses the WRF parallel infrastructure for parallel computing., Comment: Version 3.3, 41 pages, 2 tables, 12 figures. As published in Discussions, under review for Geoscientific Model Development

We describe the coupled atmosphere-wildfire model WRF-Fire, which is distributed as a part of WRF. The fire module is based on a fire-spread model, implemented by the level-set method. In each time step, the fire module takes the wind as input and returns the latent and sensible heat fluxes. We report on the software architecture and features of the software., Comment: 91st American Meteorological Society Annual Meeting, Seattle, WA, January 2011

A parallel implementation of the Balancing Domain Decomposition by Constraints (BDDC) method is described. It is based on formulation of BDDC with global matrices without explicit coarse problem. The implementation is based on the MUMPS parallel solver for computing the approximate inverse used for preconditioning. It is successfully applied to several problems of Stokes flow discretized by Taylor-Hood finite elements and BDDC is shown to be a promising method also for this class of problems., Comment: 27 pages, 5 figures, 7 tables

We explore the use of the optimal statistical interpolation (OSI) data assimilation method for the statistical tracking of emerging epidemics and to study the spatial dynamics of a disease. The epidemic models that we used for this study are spatial variants of the common susceptible-infectious-removed (S-I-R) compartmental model of epidemiology. The spatial S-I-R epidemic model is illustrated by application to simulated spatial dynamic epidemic data from the historic "Black Death" plague of 14th century Europe. Bayesian statistical tracking of emerging epidemic diseases using the OSI as it unfolds is illustrated for a simulated epidemic wave originating in Santa Fe, New Mexico., Comment: 15 pages, 5 figures. JSM 2010

WRF-Fire consists of the WRF (Weather Research and Forecasting Model) coupled with a fire spread model, based on the level-set method. We describe a preliminary application of WRF-Fire to a forest fire in Bulgaria, oportunities for research of forest fire models for Bulgaria, and plans for the development of an Environmental Decision Support Systems which includes computational modeling of fire behavior., Comment: 8 pages, 3 figures

The FFT EnKF data assimilation method is proposed and applied to a stochastic cell simulation of an epidemic, based on the S-I-R spread model. The FFT EnKF combines spatial statistics and ensemble filtering methodologies into a localized and computationally inexpensive version of EnKF with a very small ensemble, and it is further combined with the morphing EnKF to assimilate changes in the position of the epidemic., Comment: 11 pages, 3 figures. Submitted to ICCS 2010

We propose a new type of the Ensemble Kalman Filter (EnKF), which uses the Fast Fourier Transform (FFT) for covariance estimation from a very small ensemble with automatic tapering, and for a fast computation of the analysis ensemble by convolution, avoiding the need to solve a sparse system with the tapered matrix. The FFT EnKF is combined with the morphing EnKF to enable the correction of position errors, in addition to amplitude errors, and demonstrated on WRF-Fire, the Weather Research Forecasting (WRF) model coupled with a fire spread model implemented by the level set method., Comment: 8 pages

We combine the advantages of the adaptive and multilevel approaches, proposed previously by the authors, to propose a new method that preserves both, parallel scalability with increasing number of subdomains and excellent convergence properties. Performance of the method is illustrated on a two-dimensional problem of linear elasticity., Comment: Minor changes. 12 pages, 2 figures, 7 tables, to appear in DD19 proceedings

The objective of this paper is to explain the principles of the design of a coarse space in a simplified way and by pictures. The focus is on ideas rather than on a more historically complete presentation. Also, space limitation does not allow even to mention many important methods and papers that should be rightfully included., Comment: Minor changes. 8 pages, 4 figures, to appear in DD19 proceedings

The adaptive BDDC method is extended to the selection of face constraints in three dimensions. A new implementation of the BDDC method is presented based on a global formulation without an explicit coarse problem, with massive parallelism provided by a multifrontal solver. Constraints are implemented by a projection and sparsity of the projected operator is preserved by a generalized change of variables. The effectiveness of the method is illustrated on several engineering problems., Comment: 28 pages, 9 figures, 9 tables

The ensemble Kalman filter (EnKF) is a recursive filter suitable for problems with a large number of variables, such as discretizations of partial differential equations in geophysical models. The EnKF originated as a version of the Kalman filter for large problems (essentially, the covariance matrix is replaced by the sample covariance), and it is now an important data assimilation component of ensemble forecasting. EnKF is related to the particle filter (in this context, a particle is the same thing as an ensemble member) but the EnKF makes the assumption that all probability distributions involved are Gaussian. This article briefly describes the derivation and practical implementation of the basic version of EnKF, and reviews several extensions., Comment: 7 pages

Convergence of the ensemble Kalman filter in the limit for large ensembles to the Kalman filter is proved. In each step of the filter, convergence of the ensemble sample covariance follows from a weak law of large numbers for exchangeable random variables, the continuous mapping theorem gives convergence in probability of the ensemble members, and $L^p$ bounds on the ensemble then give $L^p$ convergence., Comment: Revised, 8 pages

An Ensemble Kalman Filter (EnKF, the predictor) is used make a large change in the state, followed by a Particle Filer (PF, the corrector) which assigns importance weights to describe non-Gaussian distribution. The weights are obtained by nonparametric density estimation. It is demonstrated on several numerical examples that the new predictor-corrector filter combines the advantages of the EnKF and the PF and that it is suitable for high dimensional states which are discretizations of solutions of partial differential equations., Comment: ICCS 2009, to appear; 9 pages; minor edits

After a short historical review, we present four popular substructuring methods: FETI-1, BDD, FETI-DP, BDDC, and derive the primal versions to the two FETI methods, called P-FETI-1 and P-FETI-DP, as proposed by Fragakis and Papadrakakis. The formulation of the BDDC method shows that it is the same as P-FETI-DP and the same as a preconditioner introduced by Cros. We prove the equality of eigenvalues of a particular case of the FETI-1 method and of the BDD method by applying a recent abstract result by Fragakis., Comment: 20 pages; typos corrected

A parallel implementation of the BDDC method using the frontal solver is employed to solve systems of linear equations from finite element analysis, and incorporated into a standard finite element system for engineering analysis by linear elasticity. Results of computation of stress in a hip replacement are presented. The part is made of titanium and loaded by the weight of human body. The performance of BDDC with added constraints by averages and with added corners is compared., Comment: Expanded, 22 pages, 8 figures

We are developing a wildland fire model based on semi-empirical relations that estimate the rate of spread of a surface fire and post-frontal heat release, coupled with WRF, the Weather Research and Forecasting atmospheric model. A level set method identifies the fire front. Data are assimilated using both amplitude and position corrections using a morphing ensemble Kalman filter. We will use thermal images of a fire for observations that will be compared to synthetic image based on the model state., Comment: 8 pages, 4 figures. ICCS 08

A wildland fire model based on semi-empirical relations for the spread rate of a surface fire and post-frontal heat release is coupled with the Weather Research and Forecasting atmospheric model (WRF). The propagation of the fire front is implemented by a level set method. Data is assimilated by a morphing ensemble Kalman filter, which provides amplitude as well as position corrections. Thermal images of a fire will provide the observations and will be compared to a synthetic image from the model state., Comment: 5 pages, 4 figures

Two wildland fire models are described, one based on reaction-diffusion-convection partial differential equations, and one based on semi-empirical fire spread by the level let method. The level set method model is coupled with the Weather Research and Forecasting (WRF) atmospheric model. The regularized and the morphing ensemble Kalman filter are used for data assimilation., Comment: Minor revision, except description of the model expanded. 29 pages, 9 figures, 53 references

BDDC method is the most advanced method from the Balancing family of iterative substructuring methods for the solution of large systems of linear algebraic equations arising from discretization of elliptic boundary value problems. In the case of many substructures, solving the coarse problem exactly becomes a bottleneck. Since the coarse problem in BDDC has the same structure as the original problem, it is straightforward to apply the BDDC method recursively to solve the coarse problem only approximately. In this paper, we formulate a new family of abstract Multispace BDDC methods and give condition number bounds from the abstract additive Schwarz preconditioning theory. The Multilevel BDDC is then treated as a special case of the Multispace BDDC and abstract multilevel condition number bounds are given. The abstract bounds yield polylogarithmic condition number bounds for an arbitrary fixed number of levels and scalar elliptic problems discretized by finite elements in two and three spatial dimensions. Numerical experiments confirm the theory., Comment: 26 pages, 3 figures, 2 tables, 20 references. Formal changes only

This is an introductory document surveying several results in polynomial approximation, known as the Bramble-Hilbert lemma., Comment: 3 pages, no figures; added 1D case for exposition

A wildfire model is formulated based on balance equations for energy and fuel, where the fuel loss due to combustion corresponds to the fuel reaction rate. The resulting coupled partial differential equations have coefficients that can be approximated from prior measurements of wildfires. An ensemble Kalman filter technique with regularization is then used to assimilate temperatures measured at selected points into running wildfire simulations. The assimilation technique is able to modify the simulations to track the measurements correctly even if the simulations were started with an erroneous ignition location that is quite far away from the correct one., Comment: 35 pages, 12 figures; minor revision January 2008. Original version available from http://www-math.cudenver.edu/ccm/reports