A5052090559- Gaussian Processes and Bayesian Inference
- Probabilistic and Robust Engineering Design
- Markov Chains and Monte Carlo Methods
- Statistical Methods and Inference
- Stochastic processes and financial applications
- Statistical and numerical algorithms
- Statistical Mechanics and Entropy
- Numerical methods in inverse problems
- Geochemistry and Geologic Mapping
- Soil Geostatistics and Mapping
- Mathematical Approximation and Integration
- Bayesian Methods and Mixture Models
- Functional Brain Connectivity Studies
- Statistical Methods and Bayesian Inference
- Reservoir Engineering and Simulation Methods
- Target Tracking and Data Fusion in Sensor Networks
- Advanced Thermodynamics and Statistical Mechanics
- Probability and Statistical Research
- Augmented Reality Applications
- Distributed Control Multi-Agent Systems
- Flexible and Reconfigurable Manufacturing Systems
- Complex Systems and Time Series Analysis
- Probability and Risk Models
- Computability, Logic, AI Algorithms
- Evolution and Genetic Dynamics
University of Canterbury
2023-2024
Friedrich-Alexander-Universität Erlangen-Nürnberg
2017-2022
Freie Universität Berlin
2022
University of Michigan
2021
University of Augsburg
2016-2017
Abstract The Bayesian approach to inverse problems provides a rigorous framework for the incorporation and quantification of uncertainties in measurements, parameters models. We are interested designing numerical methods which robust w.r.t. size observational noise, i.e., behave well case concentrated posterior measures. concentration is highly desirable situation practice, since it relates informative or large data. However, can pose computational challenge based on prior measure. propose...
The Ensemble Kalman methodology in an inverse problems setting can be viewed as iterative scheme, which is a weakly tamed discretization scheme for certain stochastic differential equation (SDE). Assuming suitable approximation result, dynamical properties of the SDE rigorously pulled back via discrete to original inversion. results this paper make step towards closing gap missing result by proving strong convergence simplified model scalar SDE. We focus here on toy with similar one arising...
The ensemble Kalman inversion is widely used in practice to estimate unknown parameters from noisy measurement data. Its low computational costs, straightforward implementation, and non-intrusive nature makes the method appealing various areas of application. We present a complete analysis with perturbed observations for fixed size when applied linear inverse problems. well-posedness convergence results are based on continuous time scaling limits method. resulting coupled system stochastic...
Abstract In this paper we propose polarized consensus-based dynamics in order to make optimization (CBO) and sampling (CBS) applicable for objective functions with several global minima or distributions many modes, respectively. For this, “polarize” the a localizing kernel resulting model can be viewed as bounded confidence opinion formation presence of common objective. Instead being attracted weighted mean original methods, which prevents detection more than one minimum mode, our method...
We prove that maximum a posteriori estimators are well-defined for diagonal Gaussian priors $\mu$ on $\ell^p$ under common assumptions the potential $\Phi$. Further, we show connections to Onsager--Machlup functional and provide corrected strongly simplified proof in Hilbert space case $p=2$, previously established by Dashti et al (2013) Kretschmann (2019). These corrections do not generalize setting $1 \leq p < \infty$, which requires novel convexification result difference between...
.We propose an approach based on function evaluations and Bayesian inference to extract higher-order differential information of objective functions from a given ensemble particles. Pointwise evaluation some potential V in contains implicit about first- or derivatives, which can be made explicit with little computational effort (ensemble-based gradient inference). We suggest using this for the improvement established ensemble-based numerical methods optimization sampling such as...
.The ensemble Kalman inversion (EKI) method is a for the estimation of unknown parameters in context (Bayesian) inverse problems. The approximates underlying measure by an particles and iteratively applies update to evolve (the approximation the) prior into posterior measure. For convergence analysis EKI it common practice derive continuous version, replacing iteration with stochastic differential equation. In this paper we validate approach showing that converges paths time equation...
.The ensemble Kalman inversion (EKI) for the solution of Bayesian inverse problems type \(y = A u +\varepsilon\) , with \(u\) being an unknown parameter, \(y\) a given datum, and \(\varepsilon\) measurement noise, is powerful tool usually derived from sequential Monte Carlo point view. It describes dynamics particles \(\{u^j(t)\}_{j=1}^J\) whose initial empirical measure sampled prior, evolving over artificial time \(t\) toward approximate problem, \(t=1\) emulating posterior, \(t\to...
In this paper we propose polarized consensus-based dynamics in order to make optimization (CBO) and sampling (CBS) applicable for objective functions with several global minima or distributions many modes, respectively. For this, ``polarize'' the a localizing kernel resulting model can be viewed as bounded confidence opinion formation presence of common objective. Instead being attracted weighted mean original methods, which prevents detection more than one minimum mode, our method every...
The main idea of nested sampling is to substitute the high-dimensional likelihood integral over parameter space $Ω$ by an unit line $[0,1]$ employing a push-forward with respect suitable transformation. For this substitution, it often implicitly or explicitly assumed that samples from prior are uniformly distributed along after having been mapped We show assumption wrong, especially in case function plateaus. Nevertheless, we substitution enacted works because more interesting reasons which...
Abstract This manuscript derives locally weighted ensemble Kalman methods from the point of view ensemble-based function approximation. is done by using pointwise evaluations to build up a local linear or quadratic approximation function, tapering off effect distant particles via weighting. introduces candidate method (the Ensemble method) for combining strengths particle filter (ability cope with nonlinear maps and non-Gaussian distributions) (no degeneracy), can also be applied...
It has long been posited that there is a connection between the dynamical equations describing evolutionary processes in biology and sequential Bayesian learning methods. This manuscript describes new research which this precise rigorously established continuous time setting. Here we focus on partial differential equation known as Kushner-Stratonovich evolution of posterior density time. Of particular importance piecewise smooth approximation observation path from discrete filtering...
This manuscript derives locally weighted ensemble Kalman methods from the point of view ensemble-based function approximation. is done by using pointwise evaluations to build up a local linear or quadratic approximation function, tapering off effect distant particles via weighting. introduces possible candidate (the Ensemble method) for combining strengths particle filter (ability cope with nonlinear maps and non-Gaussian distributions) (no degeneracy), can also be applied optimisation...
The replicator-mutator equation is a model for populations of individuals carrying different traits, with fitness function mediating their ability to replicate, and stochastic mutation. We derive analytical solutions the in continuous time traits quadratic function. Using these results we can explain quantify (without need numerical in-silico simulations) series evolutionary phenomena, particular flying kite effect, survival flattest, population sustain itself while tracking an optimal...
In a Bayesian inverse problem setting, the solution consists of posterior measure obtained by combining prior belief, information about forward operator, and noisy observational data. This is most often given in terms density with respect to reference high-dimensional (or infinite-dimensional) Banach space. Although Monte Carlo sampling methods provide way querying posterior, necessity evaluating operator many times (which will be costly PDE solver) prohibits this practice. For reason,...
We study the two-dimensional snake-like pattern that arises in phase separation of alloys described by spinodal decomposition Cahn-Hilliard model. These are somewhat universal patterns due to an overlay eigenfunctions Laplacian with a similar wave-number. Similar structures appear other models like reaction-diffusion systems describing animal coats' or vegetation desertification. Our main result studies random functions given cosine Fourier series independent Gaussian coefficients, dominate...