93 results found
Nóvoa A, Racca A, Magri L, 2024, Inferring unknown unknowns: Regularized bias-aware ensemble Kalman filter, Computer Methods in Applied Mechanics and Engineering, Vol: 418, ISSN: 0045-7825
Because of physical assumptions and numerical approximations, low-order models are affected by uncertainties in the state and parameters, and by model biases. Model biases, also known as model errors or systematic errors, are difficult to infer because they are ‘unknown unknowns’, i.e., we do not necessarily know their functional form a priori. With biased models, data assimilation methods may be ill-posed because either (i) they are ‘bias-unaware’ because the estimators are assumed unbiased, (ii) they rely on an a priori parametric model for the bias, or (iii) they can infer model biases that are not unique for the same model and data. First, we design a data assimilation framework to perform combined state, parameter, and bias estimation. Second, we propose a mathematical solution with a sequential method, i.e., the regularized bias-aware ensemble Kalman Filter (r-EnKF), which requires a model of the bias and its gradient (i.e., the Jacobian). Third, we propose an echo state network as the model bias estimator. We derive the Jacobian of the network, and design a robust training strategy with data augmentation to accurately infer the bias in different scenarios. Fourth, we apply the r-EnKF to nonlinearly coupled oscillators (with and without time-delay) affected by different forms of bias. The r-EnKF infers in real-time parameters and states, and a unique bias. The applications that we showcase are relevant to acoustics, thermoacoustics, and vibrations; however, the r-EnKF opens new opportunities for combined state, parameter and bias estimation for real-time and on-the-fly prediction in nonlinear systems.
Racca A, Doan NAK, Magri L, 2023, Predicting turbulent dynamics with the convolutional autoencoder echo state network, Journal of Fluid Mechanics, Vol: 975, ISSN: 0022-1120
The dynamics of turbulent flows is chaotic and difficult to predict. This makes the design of accurate reduced-order models challenging. The overarching objective of this paper is to propose a nonlinear decomposition of the turbulent state to predict the flow based on a reduced-order representation of the dynamics. We divide the turbulent flow into a spatial problem and a temporal problem. First, we compute the latent space, which is the manifold onto which the turbulent dynamics live. The latent space is found by a series of nonlinear filtering operations, which are performed by a convolutional autoencoder (CAE). The CAE provides the decomposition in space. Second, we predict the time evolution of the turbulent state in the latent space, which is performed by an echo state network (ESN). The ESN provides the evolution in time. Third, by combining the CAE and the ESN, we obtain an autonomous dynamical system: The CAE-ESN. This is the reduced-order model of the turbulent flow. We test the CAE-ESN on the two-dimensional Kolmogorov flow and the three-dimensional minimal flow unit. We show that the CAE-ESN: (i) finds a latent-space representation of the turbulent flow that has of the degrees of freedom than the physical space; (ii) time-accurately and statistically predicts the flow at different Reynolds numbers; and (iii) takes computational time to predict the flow with respect to solving the governing equations. This work opens possibilities for nonlinear decomposition and reduced-order modelling of turbulent flows from data.
Ozan DE, Magri L, 2023, Hard-constrained neural networks for modeling nonlinear acoustics, Physical Review Fluids, Vol: 8
In this computational paper, we model acoustic dynamics in space and time from synthetic sensor data. The tasks are (i) to predict and extrapolate the spatiotemporal dynamics and (ii) to reconstruct the acoustic state from partial observations. To achieve this, we develop acoustic neural networks. These are networks that learn from sensor data, while being constrained by prior knowledge on acoustic and wave physics. The prior knowledge is constrained as a soft constraint, which informs the training, and as a hard constraint (Galerkin neural networks), which constrains parts of the network's architecture as an inductive bias. First, we show that standard feedforward neural networks are unable to extrapolate in time, even in the simplest case of periodic oscillations. This motivates the constraints on the prior knowledge. Second, we constrain the prior knowledge on acoustics in increasingly effective ways by (i) employing periodic activations (periodically activated neural networks), (ii) informing the training of the networks with a penalty term that favors solutions that fulfill the governing equations (soft constrained), (iii) constraining the architecture in a physically motivated solution space (hard constrained), and (iv) a combination of these. Third, we apply the networks on two test cases for two tasks in nonlinear regimes, from periodic to chaotic oscillations. The first test case is a twin experiment, in which the data are produced by a prototypical time-delayed model. In the second test case, the data are generated by a higher-fidelity model with mean-flow effects and a kinematic model for the flame source. We find that (i) constraining the physics in the architecture improves interpolation while requiring smaller network sizes, (ii) extrapolation in time is achieved by periodic activations, and (iii) velocity can be reconstructed accurately from only pressure measurements with a combination of physics-based hard and soft constraints. In acoustics and ther
The prediction of the drop size distribution (DSD) resulting from liquid atomization is key to the optimization of multiphase flows from gas-turbine propulsion through agriculture to healthcare. Obtaining high-fidelity data of liquid atomization, either experimentally or numerically, is expensive, which makes the exploration of the design space difficult. First, to tackle these challenges, we propose a framework to predict the DSD of a liquid spray based on data as a function of the spray angle, the Reynolds number, and the Weber number. Second, to guide the design of liquid atomizers, the model accurately predicts the volume of fluid contained in drops of specific sizes while providing uncertainty estimation. To do so, we propose a Gaussian process regression (GPR) model, which infers the DSD and its uncertainty form the knowledge of its integrals and of its first moment, i.e., the mean drop diameter. Third, we deploy multiple GPR models to estimate these quantities at arbitrary points of the design space from data obtained from a large number of numerical simulations of a flat fan spray. The kernel used for reconstructing the DSD incorporates prior physical knowledge, which enables the prediction of sharply peaked and heavy-tailed distributions. Fourth, we compare our method with a benchmark approach, which estimates the DSD by interpolating the frequency polygon of the binned drops with a GPR. We show that our integral approach is significantly more accurate, especially in the tail of the distribution (i.e., large, rare drops), and it reduces the bias of the density estimator by up to 10 times. Finally, we discuss physical aspects of the model's predictions and interpret them against experimental results from the literature. This work opens opportunities for modeling drop size distribution in multiphase flows from data.
Özalp E, Margazoglou G, Magri L, 2023, Reconstruction, forecasting, and stability of chaotic dynamics from partial data., Chaos, Vol: 33
The forecasting and computation of the stability of chaotic systems from partial observations are tasks for which traditional equation-based methods may not be suitable. In this computational paper, we propose data-driven methods to (i) infer the dynamics of unobserved (hidden) chaotic variables (full-state reconstruction); (ii) time forecast the evolution of the full state; and (iii) infer the stability properties of the full state. The tasks are performed with long short-term memory (LSTM) networks, which are trained with observations (data) limited to only part of the state: (i) the low-to-high resolution LSTM (LH-LSTM), which takes partial observations as training input, and requires access to the full system state when computing the loss; and (ii) the physics-informed LSTM (PI-LSTM), which is designed to combine partial observations with the integral formulation of the dynamical system's evolution equations. First, we derive the Jacobian of the LSTMs. Second, we analyze a chaotic partial differential equation, the Kuramoto-Sivashinsky, and the Lorenz-96 system. We show that the proposed networks can forecast the hidden variables, both time-accurately and statistically. The Lyapunov exponents and covariant Lyapunov vectors, which characterize the stability of the chaotic attractors, are correctly inferred from partial observations. Third, the PI-LSTM outperforms the LH-LSTM by successfully reconstructing the hidden chaotic dynamics when the input dimension is smaller or similar to the Kaplan-Yorke dimension of the attractor. The performance is also analyzed against noisy data. This work opens new opportunities for reconstructing the full state, inferring hidden variables, and computing the stability of chaotic systems from partial data.
Jain A, Giusti A, Magri L, 2023, Indirect noise from weakly reacting inhomogeneities, Journal of Fluid Mechanics, Vol: 965, Pages: 1-26, ISSN: 0022-1120
Indirect noise is a significant contributor to aircraft engine noise, which needs to be minimized in the design of aircraft engines. Indirect noise is caused by the acceleration of flow inhomogeneities through a nozzle. High-fidelity simulations showed that some flow inhomogeneities can be chemically reacting when they leave the combustor and enter the nozzle (Giusti et al., Trans. ASME J. Engng Gas Turbines Power, vol. 141, issue 1, 2019). The state-of-the-art models, however, are limited to chemically non-reacting (frozen) flows. In this work, first, we propose a low-order model to predict indirect noise in nozzle flows with reacting inhomogeneities. Second, we identify the physical sources of sound, which generate indirect noise via two physical mechanisms: (i) chemical reaction generates compositional perturbations, thereby adding to compositional noise; and (ii) exothermic reaction generates entropy perturbations. Third, we numerically compute the nozzle transfer functions for different frequency ranges (Helmholtz numbers) and reaction rates (Damköhler numbers) in subsonic flows with hydrogen and methane inhomogeneities. Fourth, we extend the model to supersonic flows. We find that hydrogen inhomogeneities have a larger impact on indirect noise than methane inhomogeneities. Both the Damköhler number and the Helmholtz number markedly influence the phase and magnitude of the transmitted and reflected waves, which affect sound generation and thermoacoustic stability. This work provides a physics-based low-order model which can open new opportunities for predicting noise emissions and instabilities in aeronautical gas turbines with multi-physics flows.
Jain A, Magri L, 2023, Compositional noise in nozzles with dissipation, JOURNAL OF FLUID MECHANICS, Vol: 963, ISSN: 0022-1120
Jain A, Magri L, 2023, Sound Generation in Multicomponent Nozzle Flows With Dissipation, JOURNAL OF ENGINEERING FOR GAS TURBINES AND POWER-TRANSACTIONS OF THE ASME, Vol: 145, ISSN: 0742-4795
Margazoglou G, Magri L, 2023, Stability analysis of chaotic systems from data, NONLINEAR DYNAMICS, Vol: 111, Pages: 8799-8819, ISSN: 0924-090X
Magri L, Schmid PJ, Moeck JP, 2023, Linear Flow Analysis Inspired by Mathematical Methods from Quantum Mechanics, ANNUAL REVIEW OF FLUID MECHANICS, Vol: 55, Pages: 541-574, ISSN: 0066-4189
Conti ZX, Choudhary R, Magri L, 2023, A physics-based domain adaptation framework for modeling and forecasting building energy systems, DATA-CENTRIC ENGINEERING, Vol: 4
Kelshaw D, Rigas G, Magri L, 2022, Physics-informed CNNs for super-resolution of sparse observations on dynamical systems, 36th conference on Neural Information Processing Systems (NeurIPS)
In the absence of high-resolution samples, super-resolution of sparseobservations on dynamical systems is a challenging problem with wide-reachingapplications in experimental settings. We showcase the application ofphysics-informed convolutional neural networks for super-resolution of sparseobservations on grids. Results are shown for the chaotic-turbulent Kolmogorovflow, demonstrating the potential of this method for resolving finer scales ofturbulence when compared with classic interpolation methods, and thuseffectively reconstructing missing physics.
Racca A, Magri L, 2022, Data-driven prediction and control of extreme events in a chaotic flow, PHYSICAL REVIEW FLUIDS, Vol: 7, ISSN: 2469-990X
Novoa A, Magri L, 2022, Real-time thermoacoustic data assimilation, JOURNAL OF FLUID MECHANICS, Vol: 948, ISSN: 0022-1120
Schaefer F, Magri L, Polifke W, 2022, A hybrid adjoint network model for thermoacoustic optimization, Journal of Engineering for Gas Turbines and Power: Transactions of the ASME, Vol: 144, Pages: 1-9, ISSN: 0742-4795
A method is proposed that allows the computation of the continuous adjoint of a thermoacoustic network model based on the discretized direct equations. This hybrid approach exploits the self-adjoint character of the duct element, which allows all jump conditions to be derived from the direct scattering matrix. In this way, the need to derive the adjoint equations for every element of the network model is eliminated. This methodology combines the advantages of the discrete and continuous adjoint, as the accuracy of the continuous adjoint is achieved while maintaining the flexibility of the discrete adjoint. It is demonstrated how the obtained adjoint system may be utilized to optimize a thermoacoustic configuration by determining the optimal damper setting for an annular combustor.
Jain A, Magri L, 2022, A physical model for indirect noise in non-isentropic nozzles: transfer functions and stability, JOURNAL OF FLUID MECHANICS, Vol: 935, ISSN: 0022-1120
Ozan DE, Magri L, 2022, Physics-aware learning of nonlinear limit cycles and adjoint limit cycles, INTER NOISE 2022, Publisher: Institute of Noise Control Engineering, Pages: 1191-1199, ISSN: 0736-2935
Thermoacoustic oscillations occur when the heat released by a flame is sufficiently in phase with the acoustic pressure. Under this condition, the linear instability can saturate to a nonlinear self-excited oscillation with a large amplitude. A typical nonlinear regime is a limit cycle, which is characterised by a periodic orbit in the thermoacoustic phase space. In this paper, we develop a physics-aware data-driven method to predict periodic solutions using forward neural networks. The physics is constrained in two ways. First, the training is informed by a physical residual, which penalises solutions that violate the conservation of mass, momentum, and energy. Second, periodicity is imposed by introducing periodic activation functions in the neural network. We test the algorithm on synthetic data generated from a nonlinear time-delayed model of a Rijke tube. We extend our study to learning the adjoint variables of the Rijke system. Adjoint methods offer a cheap and easy way to calculate the gradients with respect to design parameters, We find that (i) periodic solutions of thermoacoustic systems can be accurately learned with this method, (ii) for periodic data, periodic activations outperform conventional activations in terms of extrapolation capability beyond the training range, and (iii) exploiting the physical constraints, fewer data is sufficient to achieve a good performance. This work opens up possibilities for the prediction of nonlinear thermoacoustics by combining physical knowledge and data.
Huhn F, Magri L, 2022, Gradient-free optimization of chaotic acoustics with reservoir computing, PHYSICAL REVIEW FLUIDS, Vol: 7, ISSN: 2469-990X
Jain A, Magri L, 2022, Indirect noise is sensitive to the compact-nozzle assumptions
In aircraft engine combustors, incomplete mixing and air cooling give rise to flow inhomogeneities. When accelerated through the nozzle guide vane downstream of the combustor, these inhomogeneities give rise to acoustic waves. This is commonly known as indirect combustion noise. When these sound waves are reflected off the outlet and travel upstream of the combustor, they can lead to thermoacoustic oscillations. To predict indirect noise, models proposed in the literature assume the wavelength of impinging disturbances to be large with respect to the nozzle spatial extent (compact nozzle assumptions). However, in reality, the assumption might not hold; hence, a mismatch between the model and experimental results can be observed. The overarching objective of this study is to show that the acoustic transfer functions are sensitive to the compact-nozzle assumption. We show this on a realistic model of nozzle flow, which contains dissipation (non-isentropic) effects. First, we show that indirect-noise predictions are sensitive to a small change in the Helmholtz number for a nearly compact nozzle, particularly in the subsonic flow regime. Second, we show that modelling dissipation is key to mitigating the unphysical sensitivity of the compact-nozzle assumption. This study highlights the importance of the non-compact and non-isentropic assumptions for the accurate prediction of indirect noise transfer functions and thermoacoustic instability in aeronautical gas turbines.
Jain A, Magri L, 2022, SOUND GENERATION IN MULTICOMPONENT NOZZLE FLOWS WITH DISSIPATION
Low emission aircraft engines burn in a lean regime, which makes the combustor susceptible to unsteady combustion. Along with improper mixing and air cooling, the unsteady combustion process gives rise to flow inhomogeneities. The acceleration of these inhomogeneities in the nozzle downstream of the combustor generates indirect combustion noise. If the acoustic waves that are reflected off the nozzle are sufficiently in phase with the heat released by the flame, thermoacoustic instabilities can occur. The generation and transmission of sound through the nozzle guide vane are typically modelled with a compact and isentropic nozzle model. Because the flow is non-isentropic due to losses from wall friction and recirculation zones, in the literature, a mismatch is observed between experimental and theoretical predictions in subsonic-choked regimes. In this work, we propose a low-order physical model to predict indirect noise in a multi-component nozzle flow with dissipation using conservation laws whilst modelling non-isentropicity using a friction factor. The model is generalized for finite-length (non-compact) arbitrary geometry nozzles. We show that the friction factor can account for wall friction and two (or three) dimensional effects, such as flow recirculation in a cross-averaged sense. We analyse the model numerically for both subsonic and supersonic nozzles, emphasizing the importance of non-isentropic and non-compact assumptions with compositional inhomogeneities. Further, we show the effect of the nozzle geometry. The results are validated with existing experimental data from the literature.
Nóvoa A, Racca A, Magri L, 2022, Bias-aware thermoacoustic data assimilation
The occurrence and amplitude of nonlinear thermoacoustic instabilities can be quickly estimated with low-order models. Low-order models, however, contain model errors, i.e., the equations do not capture all the physical mechanisms. From a statistical inference point of view, we say that low-order models are biased. We propose a data-assimilation methodology that can simultaneously infer the acoustic state, model parameters and model error from reference data. We propose reservoir computing (echo state networks) to represent the model bias. The echo state network is combined with an ensemble square-root Kalman filter to perform, in real time, (i) the estimation of the acoustic pressure and velocity, (ii) the inference of the flame parameters, and (iii) the learning of the model bias. The proposed methodology is tested on a time-delayed low-order model, with synthetic experimental data from a higher-order model with a flame. This work opens up new possibilities for assimilating data for cheap low-order models to self-correct on-the-fly.
Racca A, Magri L, 2022, Statistical Prediction of Extreme Events from Small Datasets, COMPUTATIONAL SCIENCE - ICCS 2022, PT III, Vol: 13352, Pages: 707-713, ISSN: 0302-9743
Schaefer F, Magri L, Polifke W, 2021, A Hybrid Adjoint Network Model for Thermoacoustic Optimization, ASME Turbo Expo 2021
We propose a method that allows the computation of the continuous adjoint network system based on the discretized direct equations. This hybrid approach exploits the self-adjoint character of the duct element, which allows all jump conditions to be derived from the direct scattering matrix. We, thus, eliminate the need to derive the adjoint equations for every element of the network model. This methodology combines the advantages of the discrete and continuous adjoint, as the accuracy of the continuous adjoint is achieved whilst maintaining the flexibility of the discrete adjoint. We show how the obtained adjoint system may be utilized to optimize the thermoacoustic model by determining the optimal damper setting for an annular combustor.
Doan NAK, Polifke W, Magri L, 2021, Short- and long-term prediction of a chaotic flow: A physics-constrained reservoir computing approach
We propose a physics-constrained machine learning method-based on reservoir computing- to time-accurately predict extreme events and long-term velocity statistics in a model of turbulent shear flow. The method leverages the strengths of two different approaches: empirical modelling based on reservoir computing, which it learns the chaotic dynamics from data only, and physical modelling based on conservation laws, which extrapolates the dynamics when training data becomes unavailable. We show that the combination of the two approaches is able to accurately reproduce the velocity statistics and to predict the occurrence and amplitude of extreme events in a model of self-sustaining process in turbulence. In this flow, the extreme events are abrupt transitions from turbulent to quasi-laminar states, which are deterministic phenomena that cannot be traditionally predicted because of chaos. Furthermore, the physics-constrained machine learning method is shown to be robust with respect to noise. This work opens up new possibilities for synergistically enhancing data-driven methods with physical knowledge for the time-accurate prediction of chaotic flows.
Doan NAK, Polifke W, Magri L, 2021, Auto-Encoded Reservoir Computing for Turbulence Learning
We present an Auto-Encoded Reservoir-Computing (AE-RC) approach to learn the dynamics of a 2D turbulent flow. The AE-RC consists of an Autoencoder, which discovers an efficient manifold representation of the flow state, and an Echo State Network, which learns the time evolution of the flow in the manifold. The AE-RC is able to both learn the time-accurate dynamics of the flow and predict its first-order statistical moments. The AE-RC approach opens up new possibilities for the spatio-temporal prediction of turbulence with machine learning.
Racca A, Magri L, 2021, Automatic-differentiated Physics-Informed Echo State Network (API-ESN)
We propose the Automatic-differentiated Physics-Informed Echo State Network (API-ESN). The network is constrained by the physical equations through the reservoir’s exact time-derivative, which is computed by automatic differentiation. As compared to the original Physics-Informed Echo State Network, the accuracy of the time-derivative is increased by up to seven orders of magnitude. This increased accuracy is key in chaotic dynamical systems, where errors grows exponentially in time. The network is showcased in the reconstruction of unmeasured (hidden) states of a chaotic system. The API-ESN eliminates a source of error, which is present in existing physics-informed echo state networks, in the computation of the time-derivative. This opens up new possibilities for an accurate reconstruction of chaotic dynamical states.
Racca A, Magri L, 2021, Robust Optimization and Validation of Echo State Networks for learning chaotic dynamics
An approach to the time-accurate prediction of chaotic solutions is by learning temporal patterns from data. Echo State Networks (ESNs), which are a class of Reservoir Computing, can accurately predict the chaotic dynamics well beyond the predictability time. Existing studies, however, also showed that small changes in the hyperparameters may markedly affect the network’s performance. The aim of this paper is to assess and improve the robustness of Echo State Networks for the time-accurate prediction of chaotic solutions. The goal is three-fold. First, we investigate the robustness of routinely used validation strategies. Second, we propose the Recycle Validation, and the chaotic versions of existing validation strategies, to specifically tackle the forecasting of chaotic systems. Third, we compare Bayesian optimization with the traditional Grid Search for optimal hyperparameter selection. Numerical tests are performed on two prototypical nonlinear systems that have both chaotic and quasiperiodic solutions. Both model-free and model-informed Echo State Networks are analysed. By comparing the network’s robustness in learning chaotic versus quasiperiodic solutions, we highlight fundamental challenges in learning chaotic solutions. The proposed validation strategies, which are based on the dynamical systems properties of chaotic time series, are shown to outperform the state-of-the-art validation strategies. Because the strategies are principled-they are based on chaos theory such as the Lyapunov time-they can be applied to other Recurrent Neural Networks architectures with little modification. This work opens up new possibilities for the robust design and application of Echo State Networks, and Recurrent Neural Networks, to the time-accurate prediction of chaotic systems.
De Domenico F, Rolland EO, Rodrigues J, et al., 2021, Compositional and entropy indirect noise generated in subsonic non-isentropic nozzles, Journal of Fluid Mechanics, ISSN: 0022-1120
© 2021 The Author(s),. Published by Cambridge University Press. Indirect noise generated by the acceleration of synthetic compositional and entropic perturbations through non-isentropic nozzles is measured experimentally. A physics-based analytical low-order model to evaluate the indirect noise generated by non-isentropic compact nozzles is developed and validated with experimental measurements. A one-dimensional model for describing the waves generated by the addition of mass, momentum, energy and species to a steady flow in an entropy and composition wave generator is presented. The transfer functions describing the multiple reflections of acoustic waves in an enclosed environment are derived. This analytical framework allows unambiguous identification and isolation of the experimental direct and indirect noise generated by the injection of helium, methane, argon or carbon dioxide into a flow duct. Experimental data show that entropic and compositional noise make a significant contribution to the overall pressure traces acquired in the entropy generator. Moreover, it is demonstrated that the isentropic modelling assumption is inadequate to capture the experimental behaviour, while the analytical model for non-isentropic nozzles successfully describes the direct and indirect noise transfer functions. The disregard for the compositional contribution and the unjustified use of the isentropic assumption can provide significantly inaccurate noise predictions. This work shows that compositional noise, as well as non-isentropicity in the system, should be considered in future thermoacoustic and combustion noise models.
Qadri UA, Magri L, Ihme M, et al., 2021, Using adjoint-based optimization to enhance ignition in non-premixed jets., Proc Math Phys Eng Sci, Vol: 477, Pages: 20200472-20200472, ISSN: 1364-5021
Gradient-based optimization is used to reliably and optimally induce ignition in three examples of laminar non-premixed mixture configurations. Using time-integrated heat release as a cost functional, the non-convex optimization problem identified optimal energy source locations that coincide with the stoichiometric local mixture fraction surface for short optimization horizons, while for longer horizons, the hydrodynamics plays an increasingly important role and a balance between flow and chemistry features determines non-trivial optimal ignition locations. Rather than identifying a single optimal ignition location, the results of this study show that there may be several equally good ignition locations in a given flow configuration.
Chandramoorthy N, Magri L, Wang Q, 2020, Variational optimization and data assimilation in chaotic time-delayed systems with automatic-differentiated shadowing sensitivity
In this computational paper, we perform sensitivity analysis of long-time (or ensemble) averages in the chaotic regime using the shadowing algorithm. We introduce automatic differentiation to eliminate the tangent/adjoint equation solvers used in the shadowing algorithm. In a gradient-based optimization, we use the computed shadowing sensitivity to minimize different long-time averaged functionals of a chaotic time-delayed system by optimal parameter selection. In combined state and parameter estimation for data assimilation, we use the computed sensitivity to predict the optimal trajectory given information from a model and data from measurements beyond the predictability time. The algorithms are applied to a thermoacoustic model. Because the computational framework is rather general, the techniques presented in this paper may be used for sensitivity analysis of ensemble averages, parameter optimization and data assimilation of other chaotic problems, where shadowing methods are applicable.
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