The purpose of this paper is to provide a comprehensive presentation and interpretation of the Ensemble Kalman Filter (EnKF) and its numerical implementation. The EnKF has a large user group, and numerous publications have discussed applications and theoretical aspects of it. This paper reviews the important results from these studies and also presents new ideas and alternative interpretations which further explain the success of the EnKF. In addition to providing the theoretical framework needed for using the EnKF, there is also a focus on the algorithmic formulation and optimal numerical implementation. A program listing is given for some of the key subroutines. The paper also touches upon specific issues such as the use of nonlinear measurements, in situ profiles of temperature and salinity, and data which are available with high frequency in time. An ensemble based optimal interpolation (EnOI) scheme is presented as a cost-effective approach which may serve as an alternative to the EnKF in some applications. A fairly extensive discussion is devoted to the use of time correlated model errors and the estimation of model bias.

The Ensemble Kalman Filter: Theoretical formulation and practical implementation

This book provides a thorough introduction to the mathematical and algorithmic aspects of certified reduced basis methods for parametrized partial differential equations. Central aspects ranging from model construction, error estimation and computational efficiency to empirical interpolation methods are discussed in detail for coercive problems. More advanced aspects associated with time-dependent problems, non-compliant and non-coercive problems and applications with geometric variation are also discussed as examples.

Certified Reduced Basis Methods for Parametrized Partial Differential Equations

高等学校計算数学学報 = Numerical mathematics

Non-intrusive reduced order modeling of nonlinear problems using neural networks

Thank you very much for reading numerical approximation of partial differential equations. Maybe you have knowledge that, people have search hundreds times for their chosen novels like this numerical approximation of partial differential equations, but end up in harmful downloads. Rather than reading a good book with a cup of tea in the afternoon, instead they are facing with some harmful virus inside their laptop.

Numerical Approximation Of Partial Differential Equations

https://re.public.polimi.it/bitstream/11311/871764/2/1-s2.0-S0045782514003223-main.pdf

Isogeometric numerical dispersion analysis for two-dimensional elastic wave propagation

An improvement on geometrical parameterizations by transfinite maps

Solving inverse problems in a complex, geologically realistic, and discrete model space and from a sparse set of observations is a very challenging task. Extensive exploration by Markov chain Monte Carlo (McMC) methods often results in considerable computational efforts. Most optimization methods, on the other hand, are limited to linear (continuous) model spaces and the minimization of an objective function, what often proves to be insufficient. To overcome these problems, we propose a new ensemble-based exploration scheme for geostatistical prior models generated by a multiple-point statistics (MPS) tool. The principle of our method is to expand an existing set of models by using posterior facies information for conditioning new MPS realizations. The algorithm is independent of the physical parametrization. It is tested on a simple synthetic inverse problem. When compared to two existing McMC methods (iterative spatial resampling (ISR) and Interrupted Markov chain Monte Carlo (IMcMC)), the required number of forward model runs was divided by a factor of 8–12.

https://doc.rero.ch/record/306943/files/Jaeggli_C-Posterior_population_expansion-20180119.pdf

Posterior population expansion for solving inverse problems

In the field of groundwater hydrology and more generally geophysics, solving inverse problems in a complex, geologically realistic, and discrete model space often requires the usage of Monte Carlo methods. In a previous paper we introduced PoPEx, a sampling strategy, able to handle such constraints efficiently. Unfortunately, the predictions suffered from a slight bias. In the present work, we propose a series of major modifications of PoPEx. The computational cost of the algorithm is reduced and the underlying uncertainty quantification is improved. Advanced machine learning techniques are combined with an adaptive importance sampling strategy to define a highly efficient and ergodic method that produces unbiased and rapidly convergent predictions. The proposed algorithm may be used for solving a broad range of inverse problems in many different fields. It only requires to obtain a forward problem solver, an inverse problem description and a conditional simulation tool that samples from the prior distribution. Furthermore, its parallel implementation scales perfectly. This means that the required computational time can be decreased almost arbitrarily, such that it is only limited by the available computing resources. The performance of the method is demonstrated using the inversion of a synthetic tracer test problem in an alluvial aquifer. The prior geological knowledge is modeled using multiple-point statistics. The problem consists of the identification of $2 \cdot 10^4$ parameters corresponding to $4$ geological facies values. It is used to show empirically the convergence of the PoPEx method.

/pdf/parallelized-adaptive-importance-sampling-for-solving-5cdzql8vcb.pdf

Parallelized Adaptive Importance Sampling for Solving Inverse Problems

In this paper, we carry out a numerical dispersion analysis for the linear elastodynamics equations approximated by means of NURBS– based Isogeometric Analysis in the framework of the Galerkin method; specifically, we consider the analysis of harmonic plane waves in an isotropic and homogeneous elastic medium. We compare and discuss the errors associated to the compressional and shear wave velocities and we provide the anisotropic curves for numerical approximations obtained by considering B–splines and NURBS basis functions of different regularity, namely globally C– and Cp−1–continuous, being p the polynomial degree. We conclude our analysis by numerically simulating the seismic wave propagation in a sinusoidal shaped valley with discontinuous elastic parameters across an internal interface.

/pdf/isogeometric-numerical-dispersion-analysis-for-elastic-wave-4ollfejcdt.pdf

Christoph Jäggli

Papers

Isogeometric numerical dispersion analysis for two-dimensional elastic wave propagation

An improvement on geometrical parameterizations by transfinite maps

Posterior population expansion for solving inverse problems

Parallelized Adaptive Importance Sampling for Solving Inverse Problems

Isogeometric numerical dispersion analysis for elastic wave propagation