arXiv subjects

Johannes Keller

Publications and source records attributed to Johannes Keller.

12 recordsLinked to original sources

arXiv2021

Investigating the Pilot Point Ensemble Kalman Filter for geostatistical inversion and data assimilation

Parameter estimation has a high importance in the geosciences. The ensemble Kalman filter (EnKF) allows parameter estimation for large, time-dependent systems. For large systems, the EnKF is applied using small ensembles, which may lead to spurious correlations and, ultimately, to filter divergence. We present a thorough evaluation of the pilot point ensemble Kalman filter (PP-EnKF), a variant of the ensemble Kalman filter for parameter estimation. In this evaluation, we explicitly state the update equations of the PP-EnKF, discuss the differences of this update equation compared to the update equations of similar EnKF methods, and perform an extensive performance comparison. The performance of the PP-EnKF is tested and compared to the performance of seven other EnKF methods in two model setups, a tracer setup and a well setup. In both setups, the PP-EnKF performs well, ranking better than the classical EnKF. For the tracer setup, the PP-EnKF ranks third out of eight methods. At the same time, the PP-EnKF yields estimates of the ensemble variance that are close to EnKF results from a very large-ensemble reference, suggesting that it is not affected by underestimation of the ensemble variance. In a comparison of the ensemble variances, the PP-EnKF ranks first and third out of eight methods. Additionally, for the well model and ensemble size 50, the PP-EnKF yields correlation structures significantly closer to a reference than the classical EnKF, an indication of the method's skill to suppress spurious correlations for small ensemble sizes.

stat.AP↗

arXiv2020

Sensitivity and depth of investigation from Monte Carlo ensemble statistics

For many geophysical measurements, such as direct current or electromagnetic induction methods, information fades away with depth. This has to be taken into account when interpreting models estimated from such measurements. For that reason, a measurement sensitivity analysis and determining the depth of investigation are standard steps during geophysical data processing. In deterministic gradient-based inversion, the most used sensitivity measure, the differential sensitivity, is readily available since these inversions require the computation of Jacobian matrices. In contrast, differential sensitivity may not be readily available in Monte Carlo inversion methods, since these methods do not necessarily include a linearization of the forward problem. Instead, a prior ensemble is used to simulate an ensemble of forward responses. Then, the prior ensemble is updated according to Bayesian inference. We propose to use the covariance between the prior ensemble and the forward response ensemble for constructing sensitivity measures. In Monte Carlo approaches, the estimation of this covariance does not require additional computations of the forward model. Normalizing this covariance by the variance of the prior ensemble, one obtains a simplied regression coefficient. We investigate differences between this simplified regression coefficient and differential sensitivity using simple forward models. For linear forward models, the simplied regression coefficient is equal to differential sensitivity, except for the influences of the sampling error and of the correlation structure of the prior distribution. In the non-linear case, the behaviour of the simplified regression coefficient as sensitivity measure is analysed for a simple non-linear forward model and a frequency-domain electromagnetic forward model. [...]

physics.geo-ph↗

arXiv2020

A Geometric Heat-Flow Theory of Lagrangian Coherent Structures

We consider Lagrangian coherent structures (LCSs) as the boundaries of material subsets whose advective evolution is metastable under weak diffusion. For their detection, we first transform the Eulerian advection-diffusion equation to Lagrangian coordinates, in which it takes the form of a time-dependent diffusion or heat equation. By this coordinate transformation, the reversible effects of advection are separated from the irreversible joint effects of advection and diffusion. In this framework, LCSs express themselves as (boundaries of) metastable sets under the Lagrangian diffusion process. In the case of spatially homogeneous isotropic diffusion, averaging the time-dependent family of Lagrangian diffusion operators yields Froyland's dynamic Laplacian. In the associated geometric heat equation, the distribution of heat is governed by the dynamically induced intrinsic geometry on the material manifold, to which we refer as the geometry of mixing. We study and visualize this geometry in detail, and discuss connections between geometric features and LCSs viewed as diffusion barriers in two numerical examples. Our approach facilitates the discovery of connections between some prominent methods for coherent structure detection: the dynamic isoperimetry methodology, the variational geometric approaches to elliptic LCSs, a class of graph Laplacian-based methods and the effective diffusivity framework used in physical oceanography.

math.DS↗

arXiv2019

Polyanalytic Toeplitz operators: isomorphisms, symbolic calculus and approximation of Weyl operators

We discuss an extension of Toeplitz quantization based on polyanalytic functions. We derive isomorphism theorem for polyanalytic Toeplitz operators between weighted Sobolev-Fock spaces of polyanalytic functions, which are images of modulation spaces under polyanalytic Bargmann transforms. This generalizes well-known results from the analytic setting. Finally, we derive an asymptotic symbol calculus and present an asymptotic expansion of complex Weyl operators in terms of polyanalytic Toeplitz operators.

math-ph↗

arXiv2019

Probabilistic One-Dimensional Inversion of Frequency-Domain Electromagnetic Data Using a Kalman Ensemble Generator

Frequency-domain electromagnetic (FDEM) data of the subsurface are determined by electrical conductivity and magnetic susceptibility. We apply a Kalman Ensemble generator (KEG) to one-dimensional probabilistic multi-layer inversion of FDEM data to derive conductivity and susceptibility simultaneously. The KEG provides an efficient alternative to an exhaustive Bayesian framework for FDEM inversion, including a measure for the uncertainty of the inversion result. Additionally, the method provides a measure for the depth below which the measurement is insensitive to the parameters of the subsurface. This so-called depth of investigation is derived from ensemble covariances. A synthetic and a field data example reveal how the KEG approach can be applied to FDEM data and how FDEM calibration data and prior beliefs can be combined in the inversion procedure. For the field data set, many inversions for one-dimensional subsurface models are performed at neighbouring measurement locations. Assuming identical prior models for these inversions, we save computational time by re-using the initial KEG ensemble across all measurement locations.

physics.geo-ph↗

arXiv2018

Comparing seven variants of the Ensemble Kalman Filter: How many synthetic experiments are needed?

The Ensemble Kalman Filter (EnKF) is a popular estimation technique in the geosciences. It is used as a numerical tool for state vector prognosis and parameter estimation. The EnKF can, for example, help to evaluate the geothermal potential of an aquifer. In such applications, the EnKF is often used with small or medium ensemble sizes. It is therefore of interest to characterize the EnKF behavior for these ensemble sizes. For seven ensemble sizes (50, 70, 100, 250, 500, 1000, 2000) and seven EnKF-variants (Damped, Iterative, Local, Hybrid, Dual, Normal Score and Classical EnKF), we computed 1000 synthetic parameter estimation experiments for two set-ups: a 2D tracer transport problem and a 2D flow problem with one injection well. For each model, the only difference among synthetic experiments was the generated set of random permeability fields. The 1000 synthetic experiments allow to calculate the pdf of the RMSE of the characterization of the permeability field. Comparing mean RMSEs for different EnKF-variants, ensemble sizes and flow/transport set-ups suggests that multiple synthetic experiments are needed for a solid performance comparison. In this work, 10 synthetic experiments were needed to correctly distinguish RMSE differences between EnKF-variants smaller than 10%. For detecting RMSE differences smaller than 2%, 100 synthetic experiments were needed for ensemble sizes 50, 70, 100 and 250. The overall ranking of the EnKF-variants is strongly dependent on the physical model set-up and the ensemble size.

stat.AP↗

arXiv2018

The spectrogram expansion of Wigner functions

Wigner functions generically attain negative values and hence are not probability densities. We prove an asymptotic expansion of Wigner functions in terms of Hermite spectrograms, which are probability densities. The expansion provides exact formulas for the quantum expectations of polynomial observables. In the high frequency regime it allows to approximate quantum expectation values up to any order of accuracy in the high frequency parameter. We present a Markov Chain Monte Carlo method to sample from the new densities and illustrate our findings by numerical experiments.

math-ph↗

arXiv2016

An invariant class of wave packets for the Wigner transform

Generalised Hagedorn wave packets appear as exact solutions of Schrödinger equations with quadratic, possibly complex, potential, and are given by a polynomial times a Gaussian. We show that the Wigner transform of generalised Hagedorn wave packets is a wave packet of the same type in phase space. The proofs build on a parametrisation via Lagrangian frames and a detailed analysis of the polynomial prefactors, including a novel Laguerre connection. Our findings directly imply the recently found tensor product structure of the Wigner transform of Hagedorn wave packets.

math-ph↗

arXiv2016

Quantum expectations via spectrograms

We discuss a new phase space method for the computation of quantum expectation values in the high frequency regime. Instead of representing a wavefunction by its Wigner function, which typically attains negative values, we define a new phase space density by adding a first-order Hermite spectrogram term as a correction to the Husimi function. The new phase space density yields accurate approximations of the quantum expectation values as well as allows numerical sampling from non-negative densities. We illustrate the new method by numerical experiments in up to $128$ dimensions.

quant-ph↗

arXiv2015

A new Phase Space Density for Quantum Expectations

We introduce a new density for the representation of quantum states on phase space. It is constructed as a weighted difference of two smooth probability densities using the Husimi function and first-order Hermite spectrograms. In contrast to the Wigner function, it is accessible by sampling strategies for positive densities. In the semiclassical regime, the new density allows to approximate expectation values to second order with respect to the high frequency parameter and is thus more accurate than the uncorrected Husimi function. As an application, we combine the new phase space density with Egorov's theorem for the numerical simulation of time-evolved quantum expectations by an ensemble of classical trajectories. We present supporting numerical experiments in different settings and dimensions.

math-ph↗

arXiv2014

Quasi-classical description of molecular dynamics based on Egorov's theorem

Egorov's theorem on the classical propagation of quantum observables is related to prominent quasi-classical descriptions of quantum molecuar dynamics as the linearized semiclassical initial value representation (LSC-IVR), the Wigner phase space method or the statistical quasiclassical method. The error estimates show that different accuracies are achievable for the computation of expectation values and position densities. Numerical experiments for a Morse model of diatomic iodine and confined Henon-Heiles systems in various dimensions illustrate the theoretical results.

physics.chem-ph↗

arXiv2013

Propagation of Quantum Expectations with Husimi Functions

We analyse the dynamics of expectation values of quantum observables for the time-dependent semiclassical Schrödinger equation. To benefit from the positivity of Husimi functions, we switch between observables obtained from Weyl and Anti-Wick quantization. We develop and prove a second order Egorov type propagation theorem with Husimi functions by establishing transition and commutator rules for Weyl and Anti-Wick operators. We provide a discretized version of our theorem and present numerical experiments for Schrödinger equations in dimensions two and six that validate our results.

math.NA↗