08:30
Uncertainty quantification in data assimilation based on the four-dimensional variational method
Hiromichi Nagao | The University of Tokyo | Japan
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Authors:
Hiromichi Nagao | The University of Tokyo | Japan
Shin-ichi Ito | The University of Tokyo | Japan
Data assimilation (DA) is a computational technique that integrates numerical simulation models and observation data based on Bayesian statistics. DA plays an important role in the modern weather forecasting, and is widely applied in various fields such as seismology, biology and materials science. The four-dimensional variational method (4DVar) is often used for DA based on massive simulation models. A big disadvantage in the conventional 4DVar is that it optimizes only parameters and states but never evaluates their uncertainties. In this paper, we propose a new 4DVar that provides optimum estimates and their uncertainties within reasonable computation time and resource constraints. The uncertainties are given as several diagonal elements of an inverse Hessian matrix, which is the covariance matrix of a normal distribution that approximates the target posterior probability density function in the neighborhood of the optimum. The key technique in the proposed method is the second-order adjoint method, which allows us to directly evaluate the diagonal elements of the inverse Hessian matrix without computing all of its elements. This drastically reduces the number of computations and the amount of memory. The proposed method is validated through numerical tests using an Allen-Cahn type simulation model. We confirm that the proposed method correctly reproduces the parameter and initial state assumed in advance, and successfully evaluates the uncertainty of the parameter.
08:50
Two-Stage Data Assimilation of Isolated Large Fractures in Reservoir Simulation Based on Ensemble Kalman Filters
Michael Liem | Institute of Fluid Dynamics, ETH Zurich | Switzerland
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Michael Liem | Institute of Fluid Dynamics, ETH Zurich | Switzerland
Patrick Jenny | Institute of Fluid Dynamics, ETH Zurich | Switzerland
The prime uncertainty in reservoir simulations lies in the permeability field, as only few measurements of reservoir properties are available. When fractures are present, their location, orientation and size greatly influence the resulting flow field. Therefore, it is important to estimate those parameters as precisely as possible. Ensemble Kalman filters (EnKF) are widely used for history matching (or data assimilation) in the context of sub-surface flows in order to estimate parameters, reduce uncertainty and improve simulation results.
This work considers isolated large fractures which can appear e.g. during reservoir stimulation of a geothermal system. We simulate the reservoir with an embedded discrete fracture model (EDFM). It is assumed that large fractures appear one after the other and that we get prior probabilistic knowledge of the fracture parameters (location, orientation and size) from seismic measurements. We reduce the uncertainty of those fracture parameters with an iterative Ensemble Kalman Filter (EnKF) using empirical measurement data; here from a reference simulation. A two-stage data assimilation approach is devised; in the first stage, during fracture formation, pressure and flow at inlet and outlet are used as measurements. In the second stage, once all fractures are created, a tracer is injected at the inlet and its concentrations at the outlets are used as measurements.
09:10
Discrete adjoint based Data Assimilation for RANS Closure Models: The effect of the coupling between adjoint variables
Pasha Piroozmand | ETH Zurich | Switzerland
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Authors:
Pasha Piroozmand | ETH Zurich | Switzerland
Oliver Brenner | Institute of Fluid Dynamics, ETH Zurich | Switzerland
Patrick Jenny | Institute of Fluid Dynamics, ETH Zurich | Switzerland
Capability of discrete adjoint method has been demonstrated in many aerodynamic shape optimization problems. However, the method is less used for CFD data-assimilation purposes. Here, we utilize the method to minimize the discrepancy between a RANS simulation output and the experimental or observational sparse available reference data by tweaking the RANS model parameters. The main computational advantage of the adjoint-method compare to other counterparts like sequential Kalman filter methods is that the computational cost of the procedure is almost independent of the number of the parameters to be tuned. The adjoint equations are set-up based on the discrete residual equations of the forward problem. Since the adjoint variables are coupled (as in state variables in the state equations), in order to compute accurate sensitivity fields and ,therefore, better minimization convergence, they are solved with coupled linear solvers in the open-source CFD toolbox foam-extend-4.0. The effect of the coupling between the adjoint velocity and adjoint pressure fields are analysed and compared with segregated methods that do not consider the coupling terms. The method is developed for an incompressible k-epsilon model with one variant parameter field with the same size of the mesh and is tested on the period-hill benchmark case.
09:30
Adjoint-based computation of the exact Hessian-vector multiplication
Yuto Miyatake | Osaka University | Japan
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Authors:
Shin-ichi Ito | The University of Tokyo | Japan
Takeru Matsuda | The University of Tokyo | Japan
Yuto Miyatake | Osaka University | Japan
We consider a function of the numerical solution of an initial value problem, and its Hessian matrix with respect to the initial data. Such a Hessian matrix often arises when we are concerned with the adjoint method in the context of data assimilation. More specifically, it appears when we try to quantify the uncertainty for the initial state estimation or to improve the computational efficiency of the estimation. In these contexts, a linear system whose coefficient matrix is a Hessian needs to be solved. Since the Hessian matrix is symmetric, the conjugate gradient (CG) method seems a suitable choice, which requires a Hessian-vector multiplication. It is known that the Hessian-vector multiplication can be approximated by numerically integrating the so-called second-order adjoint system backwardly; however, the symmetry of the Hessian is not inherited in general, and the convergence of the CG method cannot be guaranteed. This issue is problematic, especially when it is difficult to obtain sufficiently accurate numerical solutions for the original system and second-order adjoint system. In this talk, we show an algorithm that computes the intended Hessian-vector multiplication exactly. The key idea is to give a concise derivation of second-order adjoint systems and to apply a particular numerical method to the second-order adjoint system. In the discussion, symplectic partitioned Runge-Kutta methods play an important role.
09:50
Predicting the Eulerian kinetic energy spectrum from Lagrangian drifters using combined data assimilation and parameter estimation
Mustafa Mohamad | Courant Institute for Mathematical Sciences | United States
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Mustafa Mohamad | Courant Institute for Mathematical Sciences | United States
Andrew Majda | Courant Institute for Mathematical Sciences | United States
The assimilation and prediction of a flow field given a stream of measurements provided by passively advected Lagrangian drifters is discussed. We quantify recovery of the Eulerian energy spectra from observations of Lagrangian drifters by special Lagrangian data assimilation algorithms, based on conditionally Gaussian Kalman filters. Prediction of the Eulerian energy slope is demonstrated through combined assimilation and parameter estimation, and recovery skill of the spectrum in various regimes is demonstrated. The method is applied to high-dimensional quasi-geostrophic models and also compared to existing methods in the literature.
10:10
Improving well-posedness and robustness to data noisiness for electrical impedance tomography inverse problem
Ying Liang | The Chinese University of Hong Kong | Hong Kong
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Ying Liang | The Chinese University of Hong Kong | Hong Kong
Jun Zou | The Chinese University of Hong Kong | Hong Kong
The objective of this work is to reconstruct a tomographic image from electrode measurements on the skin surface. Mathematically, it is formulated as an inverse problem of recovery of electrical conductivity of the interior of a domain from boundary measurements of current and potential, where the measurements are subject to noise. Due to the nonlinearity, ill-posedness and noisiness of the inverse problem, it is non trivial to develop a method for accurate tomographic image reconstruction. In this talk, we discuss recent development of various formulations in providing better approximations and stability with respect to data in the electrical impedance tomographic image reconstruction.
10:30
Solving Inverse Problems Using Bayesian Optimization of Composite Functions
Raul Astudillo | Cornell University | United States
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Raul Astudillo | Cornell University | United States
Peter Frazier | Cornell University | United States
We consider the use of Bayesian optimization (BO) for solving inverse problems over low-dimensional parameter spaces with time-consuming-to-evaluate non-convex derivative-free forward models. BO is a standard method for black-box optimization and has been often used to solve these problems. In this talk, we discuss a novel approach that exploits the structure of objective functions arising in such inverse problems to substantially improve sampling efficiency. More specifically, we leverage that the objective is often a composition of the form f(x)=g(h(x)), where h(x) is the forward model’s predictions under parameters x, and g is a simple cheap-to-evaluate function, such as the sum of squares, measuring the loss between these predictions and observed data. Our approach models h using a multi-output Gaussian process and chooses where to sample using the expected improvement evaluated on the implied non-Gaussian posterior on f, which we call expected improvement for composite functions (EI-CF). Although EI-CF cannot be computed in closed form, we provide a stochastic gradient estimator that allows its efficient maximization. We also show that our approach recovers a globally optimal solution as sampling effort grows to infinity, generalizing previous convergence results for classical expected improvement. Numerical experiments show that our approach substantially outperforms standard BO benchmarks, reducing simple regret by several orders of magnitude.