2,981 research outputs found
Second order adjoints for solving PDE-constrained optimization problems
Inverse problems are of utmost importance in many fields of science and engineering. In the
variational approach inverse problems are formulated as PDE-constrained optimization problems,
where the optimal estimate of the uncertain parameters is the minimizer of a certain cost
functional subject to the constraints posed by the model equations. The numerical solution
of such optimization problems requires the computation of derivatives of the model output
with respect to model parameters. The first order derivatives of a cost functional (defined
on the model output) with respect to a large number of model parameters can be calculated
efficiently through first order adjoint sensitivity analysis. Second order adjoint models
give second derivative information in the form of matrix-vector products between the Hessian
of the cost functional and user defined vectors. Traditionally, the construction of second
order derivatives for large scale models has been considered too costly. Consequently, data
assimilation applications employ optimization algorithms that use only first order derivative
information, like nonlinear conjugate gradients and quasi-Newton methods.
In this paper we discuss the mathematical foundations of second order adjoint sensitivity
analysis and show that it provides an efficient approach to obtain Hessian-vector products. We
study the benefits of using of second order information in the numerical optimization process
for data assimilation applications. The numerical studies are performed in a twin experiment
setting with a two-dimensional shallow water model. Different scenarios are considered with
different discretization approaches, observation sets, and noise levels. Optimization algorithms
that employ second order derivatives are tested against widely used methods that require
only first order derivatives. Conclusions are drawn regarding the potential benefits and the
limitations of using high-order information in large scale data assimilation problems
Spectral diagonal ensemble Kalman filters
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 figure
A random map implementation of implicit filters
Implicit particle filters for data assimilation generate high-probability
samples by representing each particle location as a separate function of a
common reference variable. This representation requires that a certain
underdetermined equation be solved for each particle and at each time an
observation becomes available. We present a new implementation of implicit
filters in which we find the solution of the equation via a random map. As
examples, we assimilate data for a stochastically driven Lorenz system with
sparse observations and for a stochastic Kuramoto-Sivashinski equation with
observations that are sparse in both space and time
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