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Ensemble Kalman filter for neural network based one-shot inversion
No full textWe study the use of novel techniques arising in machine learning for inverse problems. Our approach replaces the complex forward model by a neural network, which is trained simultaneously in a one-shot sense when estimating the unknown parameters from data, i.e. the neural network is trained only for the unknown parameter. By establishing a link to the Bayesian approach to inverse problems, an algorithmic framework is developed which ensures the feasibility of the parameter estimate w.r. to the forward model. We propose an efficient, derivative-free optimization method based on variants of the ensemble Kalman inversion. Numerical experiments show that the ensemble Kalman filter for neural network based one-shot inversion is a promising direction combining optimization and machine learning techniques for inverse problems
Instantaneous rock transformations in the deep crust driven by reactive fluid flow
Get PDFFluid–rock interactions are a fundamental component of geodynamic processes. They link mass and energy transfer with largescale
tectonic deformation and drive mineral deposit formation, carbon sequestration and rheological changes of the lithosphere.
Spatial evidence indicates that fluid–rock interactions operate on length scales that range from the grain boundary to
tectonic plates, but the timescales of regional fluid–rock interactions remain essentially unconstrained. Here we present observations
from an exceptionally well-exposed fossil hydrothermal system from an ophiolite sequence in northern Norway that
we use to inform a multielement advection–diffusion–reaction transport model. We calculated the velocity of the fluid-driven
reaction fronts and found that they can propagate at up to 10 cm per year, equivalent to the fastest tectonic plate motion and
mid-ocean-ridge spreading rates. Propagation through the low-permeability rocks of the mid-crust is facilitated by a transient,
reaction-induced permeability increase. We conclude that large-scale fluid-mediated rock transformations in continental collision
and subduction zones occur on timescales of tens of years when reactive fluids are present. We infer that natural carbon
sequestration, ore deposit formation and transient and long-term petrophysical changes of the crust proceed instantaneously,
from a geological perspective
Extending Transition Path Theory: Periodically Driven and Finite-Time Dynamics
No full textGiven two distinct subsets A, B in the state space of some dynamical system, transition
path theory (TPT) was successfully used to describe the statistical behavior of
transitions from A to B in the ergodic limit of the stationary system.We derive generalizations
of TPT that remove the requirements of stationarity and of the ergodic limit
and provide this powerful tool for the analysis of other dynamical scenarios: periodically
forced dynamics and time-dependent finite-time systems. This is partially
motivated by studying applications such as climate, ocean, and social dynamics. On
simple model examples, we show how the new tools are able to deliver quantitative
understanding about the statistical behavior of such systems.We also point out explicit
cases where the more general dynamical regimes show different behaviors to their stationary
counterparts, linking these tools directly to bifurcations in non-deterministic systems
NUMERICAL HOMOGENIZATION OF FRACTAL INTERFACE PROBLEMS
Get PDFWe consider the numerical homogenization of a class of fractal elliptic interface
problems inspired by related mechanical contact problems from the geosciences. A particular
feature is that the solution space depends on the actual fractal geometry. Our main
results concern the construction of projection operators with suitable stability and approximation
properties. The existence of such projections then allows for the application of
existing concepts from localized orthogonal decomposition (LOD) and successive subspace
correction to construct first multiscale discretizations and iterative algebraic solvers with
scale-independent convergence behavior for this class of problems
Affine Invariant Interacting Langevin Dynamics for Bayesian Inference
Get PDFWe propose a computational method (with acronym ALDI) for sampling from a given target distribution
based on first-order (overdamped) Langevin dynamics which satisfies the property of affine
invariance. The central idea of ALDI is to run an ensemble of particles with their empirical covariance
serving as a preconditioner for their underlying Langevin dynamics. ALDI does not require
taking the inverse or square root of the empirical covariance matrix, which enables application to
high-dimensional sampling problems. The theoretical properties of ALDI are studied in terms of
nondegeneracy and ergodicity. Furthermore, we study its connections to diffusion on Riemannian
manifolds and Wasserstein gradient flows. Bayesian inference serves as a main application area for
ALDI. In case of a forward problem with additive Gaussian measurement errors, ALDI allows for a
gradient-free approximation in the spirit of the ensemble Kalman filter. A computational comparison
between gradient-free and gradient-based ALDI is provided for a PDE constrained Bayesian inverse
problem
On the convergence of the Laplace approximation and noise-level-robustness of Laplace-based Monte Carlo methods for Bayesian inverse problems
Get PDFThe Bayesian approach to inverse problems provides a rigorous framework for the
incorporation and quantification of uncertainties in measurements, parameters and
models. We are interested in designing numerical methods which are robust w.r.t. the
size of the observational noise, i.e., methods which behave well in case of concentrated
posterior measures. The concentration of the posterior is a highly desirable situation
in practice, since it relates to informative or large data. However, it can pose a computational
challenge for numerical methods based on the prior measure. We propose to
employ the Laplace approximation of the posterior as the base measure for numerical
integration in this context. The Laplace approximation is a Gaussian measure centered
at the maximum a-posteriori estimate and with covariance matrix depending on the
logposterior density. We discuss convergence results of the Laplace approximation
in terms of the Hellinger distance and analyze the efficiency of Monte Carlo methods
based on it. In particular, we show that Laplace-based importance sampling and
Laplace-based quasi-Monte-Carlo methods are robust w.r.t. the concentration of the
posterior for large classes of posterior distributions and integrands whereas prior-based
importance sampling and plain quasi-Monte Carlo are not. Numerical experiments are
presented to illustrate the theoretical findings
Kernel-Based Approximation of the Koopman Generator and Schrödinger Operator
No full textAbstract
Many dimensionality and model reduction techniques rely on estimating dominant eigenfunctions of associated dynamical operators from data. Important examples include the Koopman operator and its generator, but also the Schrödinger operator. We propose a kernel-based method for the approximation of differential operators in reproducing kernel Hilbert spaces and show how eigenfunctions can be estimated by solving auxiliary matrix eigenvalue problems. The resulting algorithms are applied to molecular dynamics and quantum chemistry examples. Furthermore, we exploit that, under certain conditions, the Schrödinger operator can be transformed into a Kolmogorov backward operator corresponding to a drift-diffusion process and vice versa. This allows us to apply methods developed for the analysis of high-dimensional stochastic differential equations to quantum mechanical systems
Approach to equilibrium and nonequilibrium stationary distributions of interacting many-particle systems that are coupled to different heat baths
No full textA Hamiltonian-based model of many harmonically interacting massive particles that are subject to linear friction and coupled to heat baths at different temperatures is used to study the dynamic approach to equilibrium and nonequilibrium stationary states. An equilibrium system is here defined as a system whose stationary distribution equals the Boltzmann distribution, the relation of this definition to the conditions of detailed balance and vanishing probability current is discussed both for underdamped as well as for overdamped systems. Based on the exactly calculated dynamic approach to the stationary distribution, the functional that governs this approach, which is called the free entropy Sfree(t), is constructed. For the stationary distribution Sfree(t) becomes maximal and its time derivative, the free entropy production ˙Sfree(t), is minimal and vanishes. Thus, Sfree(t) characterizes equilibrium as well as nonequilibrium stationary distributions by their extremal and stability properties. For an equilibrium system, i.e., if all heat baths have the same temperature, the free entropy equals the negative free energy divided by temperature and thus corresponds to the Massieu function which was previously introduced in an alternative formulation of statistical mechanics. Using a systematic perturbative scheme for calculating velocity and position correlations in the overdamped massless limit, explicit results for few particles are presented: For two particles localization in position and momentum space is demonstrated in the nonequilibrium stationary state, indicative of a tendency to phase separate. For three elastically interacting particles heat flows from a particle coupled to a cold reservoir to a particle coupled to a warm reservoir if the third reservoir is sufficiently hot. This does not constitute a violation of the second law of thermodynamics, but rather demonstrates that a particle in such a nonequilibrium system is not characterized by an effective temperature which equals the temperature of the heat bath it is coupled to. Active particle models can be described in the same general framework, which thereby allows us to characterize their entropy production not only in the stationary state but also in the approach to the stationary nonequilibrium state. Finally, the connection to nonequilibrium thermodynamics formulations that include the reservoir entropy production is discussed
Detecting regime transitions of the nocturnal and Polar near-surface temperature inversion
No full textMany natural systems undergo critical transitions, i.e. sudden shifts from one dynamical regime to another. In the climate system, the atmospheric boundary layer can experience sudden transitions between fully turbulent states and quiescent, quasi-laminar states. Such rapid transitions are observed in Polar regions or at night when the atmospheric boundary layer is stably stratified, and they have important consequences in the strength of mixing with the higher levels of the atmosphere. To analyze the stable boundary layer, many approaches rely on the identification of regimes that are commonly denoted as weakly and very stable regimes. Detecting transitions between the regimes is crucial for modeling purposes.
In this work a combination of methods from dynamical systems and statistical modeling is applied to study these regime transitions and to develop an early-warning signal that can be applied to non-stationary field data. The presented metric aims at detecting nearing transitions by statistically quantifying the deviation from the dynamics expected when the system is close to a stable equilibrium. An idealized stochastic model of near-surface inversions is used to evaluate the potential of the metric as an indicator of regime transitions. In this stochastic system, small-scale perturbations can be amplified due to the nonlinearity, resulting in transitions between two possible equilibria of the temperature inversion. The simulations show such noise-induced regime transitions, successfully identified by the indicator. The indicator is further applied to time series data from nocturnal and Polar meteorological measurements