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We consider inverse problems estimating distributed parameters from indirect noisy observations through discretization of continuum models described by partial differential or integral equations. It is well understood that errors arising from the discretization can be detrimental for ill-posed inverse problems, as discretization error behaves as correlated noise. While this problem can be avoided with a discretization fine enough to decrease the modeling error level below that of the exogenous noise that is addressed, e.g. by regularization, the computational resources needed to deal with the additional degrees of freedom may increase so much as to require high performance computing environments. Following an earlier idea, we advocate the notion of the discretization as one of the unknowns of the inverse problem, which is updated iteratively together with the solution. In this approach, the discretization, defined in terms of an underlying metric, is refined selectively only where the representation power of the current mesh is insufficient. In this paper we allow the metrics and meshes to be anisotropic, and we show that this leads to significant reduction of memory allocation and computing time.

 

Bayesian particle filters (PFs) are a viable alternative to sampling methods such as Markov chain Monte Carlo methods to estimate model parameters and related uncertainties when the forward model is a dynamical system, and the data are time series that depend on the state vector. PF techniques are particularly attractive when the dimensionality of the state space is large and the numerical solution of the dynamical system over the time interval corresponding to the data is time consuming. Moreover, information contained in the PF solution can be used to infer on the sensitivity of the unknown parameters to different temporal segments of the data. This, in turn, can guide the design of more efficient and effective data collection procedures. In this article the PF method is applied to the problem of estimating cell membrane permeability to gases from pH measurements on or near the cell membrane. The forward model in this case comprises a spatially distributed system of coupled reaction–diffusion differential equations. The high dimensionality of the state space and the need to account for the micro-environment created by the pH electrode measurement device are additional challenges that are addressed by the solution method.

 

ABSTRACT Stellar streams are sensitive probes of the Galactic potential. The likelihood of a stream model given stream data is often assessed using simulations. However, comparing to simulations is challenging when even the stream paths can be hard to quantify. Here we present a novel application of self-organizing maps and first-order Kalman filters to reconstruct a stream’s path, propagating measurement errors and data sparsity into the stream path uncertainty. The technique is Galactic-model independent, non-parametric, and works on phase-wrapped streams. With this technique, we can uniformly analyse and compare data with simulations, enabling both comparison of simulation techniques and ensemble analysis with stream tracks of many stellar streams. Our method is implemented in the public Python package TrackStream, available at https://github.com/nstarman/trackstream. 

Abstract Dictionary learning, aiming at representing a signal in terms of the atoms of a dictionary, has gained popularity in a wide range of applications, including, but not limited to, image denoising, face recognition, remote sensing, medical imaging and feature extraction. Dictionary learning can be seen as a possible data-driven alternative to solve inverse problems by identifying the data with possible outputs that are either generated numerically using a forward model or the results of earlier observations of controlled experiments. Sparse dictionary learning is particularly interesting when the underlying signal is known to be representable in terms of a few vectors in a given basis. In this paper, we propose to use hierarchical Bayesian models for sparse dictionary learning that can capture features of the underlying signals, e.g. sparse representation and nonnegativity. The same framework can be employed to reduce the dimensionality of an annotated dictionary through feature extraction, thus reducing the computational complexity of the learning task. Computed examples where our algorithms are applied to hyperspectral imaging and classification of electrocardiogram data are also presented.