5 research outputs found
Ensemble Kalman filter for multiscale inverse problems
We present a novel algorithm based on the ensemble Kalman filter to solve
inverse problems involving multiscale elliptic partial differential equations.
Our method is based on numerical homogenization and finite element
discretization and allows to recover a highly oscillatory tensor from
measurements of the multiscale solution in a computationally inexpensive
manner. The properties of the approximate solution are analysed with respect to
the multiscale and discretization parameters, and a convergence result is shown
to hold. A reinterpretation of the solution from a Bayesian perspective is
provided, and convergence of the approximate conditional posterior distribution
is proved with respect to the Wasserstein distance. A numerical experiment
validates our methodology, with a particular emphasis on modelling error and
computational cost
A probabilistic finite element method based on random meshes: Error estimators and Bayesian inverse problems
We present a novel probabilistic finite element method (FEM) for the solution
and uncertainty quantification of elliptic partial differential equations based
on random meshes, which we call random mesh FEM (RM-FEM). Our methodology
allows to introduce a probability measure on standard piecewise linear FEM. We
present a posteriori error estimators based uniquely on probabilistic
information. A series of numerical experiments illustrates the potential of the
RM-FEM for error estimation and validates our analysis. We furthermore
demonstrate how employing the RM-FEM enhances the quality of the solution of
Bayesian inverse problems, thus allowing a better quantification of numerical
errors in pipelines of computations
Drift Estimation of Multiscale Diffusions Based on Filtered Data
We study the problem of drift estimation for two-scale continuous time
series. We set ourselves in the framework of overdamped Langevin equations, for
which a single-scale surrogate homogenized equation exists. In this setting,
estimating the drift coefficient of the homogenized equation requires
pre-processing of the data, often in the form of subsampling; this is because
the two-scale equation and the homogenized single-scale equation are
incompatible at small scales, generating mutually singular measures on the path
space. We avoid subsampling and work instead with filtered data, found by
application of an appropriate kernel function, and compute maximum likelihood
estimators based on the filtered process. We show that the estimators we
propose are asymptotically unbiased and demonstrate numerically the advantages
of our method with respect to subsampling. Finally, we show how our filtered
data methodology can be combined with Bayesian techniques and provide a full
uncertainty quantification of the inference procedure
Drift Estimation of Multiscale Diffusions Based on Filtered Data
We study the problem of drift estimation for two-scale continuous time series. We set ourselves in the framework of overdamped Langevin equations, for which a single-scale surrogate homogenized equation exists. In this setting, estimating the drift coefficient of the homogenized equation requires pre-processing of the data, often in the form of subsampling; this is because the two-scale equation and the homogenized single-scale equation are incompatible at small scales, generating mutually singular measures on the path space. We avoid subsampling and work instead with filtered data, found by application of an appropriate kernel function, and compute maximum likelihood estimators based on the filtered process. We show that the estimators we propose are asymptotically unbiased and demonstrate numerically the advantages of our method with respect to subsampling. Finally, we show how our filtered data methodology can be combined with Bayesian techniques and provide a full uncertainty quantification of the inference procedure
MATHICSE Technical Report : Ensemble Kalman filter for multiscale inverse problems
We present a novel algorithm based on the ensemble Kalman filter to solve inverse problems involving multiscale elliptic partial differential equations. Our method is based on numerical homogenization and finite element discretization and allows to recover a highly oscillatory tensor from measurements of the multiscale solution in a computationally inexpensive manner. The properties of the approximate solution are analysed with respect to the multiscale and discretization parameters, and a convergence result is shown to hold. A reinterpretation of the solution from a Bayesian perspective is provided, and convergence of the approximate conditional posterior distribution is proved with respect to the Wasserstein distance. A numerical experiment validates our methodology, with a particular emphasis on modelling error and computational cost