Delineating Parameter Unidentifiabilities in Complex Models

Scientists use mathematical modelling to understand and predict the properties of complex physical systems. In highly parameterised models there often exist relationships between parameters over which model predictions are identical, or nearly so. These are known as structural or practical unidentifiabilities, respectively. They are hard to diagnose and make reliable parameter estimation from data impossible. They furthermore imply the existence of an underlying model simplification. We describe a scalable method for detecting unidentifiabilities, and the functional relations defining them, for generic models. This allows for model simplification, and appreciation of which parameters (or functions thereof) cannot be estimated from data. Our algorithm can identify features such as redundant mechanisms and fast timescale subsystems, as well as the regimes in which such approximations are valid. We base our algorithm on a novel quantification of regional parametric sensitivity: multiscale sloppiness. Traditionally, the link between parametric sensitivity and the conditioning of the parameter estimation problem is made locally, through the Fisher Information Matrix. This is valid in the regime of infinitesimal measurement uncertainty. We demonstrate the duality between multiscale sloppiness and the geometry of confidence regions surrounding parameter estimates made where measurement uncertainty is non-negligible. Further theoretical relationships are provided linking multiscale sloppiness to the Likelihood-ratio test. From this, we show that a local sensitivity analysis (as typically done) is insufficient for determining the reliability of parameter estimation, even with simple (non)linear systems. Our algorithm provides a tractable alternative. We finally apply our methods to a large-scale, benchmark Systems Biology model of NF-$\kappa$B, uncovering previously unknown unidentifiabilities

A TV flow based local scale estimate and its application to texture discrimination

This paper presents a local region based scale measure, which exploits properties of a certain type of nonlinear diffusion, the so-called total variation (TV) flow. During the signal evolution by means of TV flow, pixels change their value with a speed that is inversely proportional to the size of the region they belong to. From this evolution speed one can derive a local scale estimate based on regions instead of derivative filters. Main motivation for such a scale measure is its application to texture discrimination, in particular the construction of an alternative to Gabor filters. When the scale estimate is combined with the components of the structure tensor, which provides orientation information, it yields a texture feature space of only four dimensions. Like Gabor features, this sparse feature space discriminates textures by means of their orientation and scale, yet the representation of orientation and scale is less redundant. The quality of the feature space containing the new scale measure is evaluated in texture segmentation experiments by comparing results to those achieved with Gabor filters. It turns out that one can gain a total speedup of factor 2 without loosing any quality concerning the discrimination of textures

Curvature based corner detector for discrete, noisy and multi-scale contours

International audienceEstimating curvature on digital shapes is known to be a difficult problem even in high resolution images 10,19. Moreover the presence of noise contributes to the insta- bility of the estimators and limits their use in many computer vision applications like corner detection. Several recent curvature estimators 16,13,15, which come from the dis- crete geometry community, can now process damaged data and integrate the amount of noise in their analysis. In this paper, we propose a comparative evaluation of these estimators, testing their accuracy, efficiency, and robustness with respect to several type of degradations. We further compare the best one with the visual curvature proposed by Liu et al. 14, a recently published method from the computer vision community. We finally propose a novel corner detector, which is based on curvature estimation, and we provide a comprehensive set of experiments to compare it with many other classical cor- ner detectors. Our study shows that this corner detector has most of the time a better behavior than the others, while requiring only one parameter to take into account the noise level. It is also promising for multi-scale shape description

A scalable parallel finite element framework for growing geometries. Application to metal additive manufacturing

This work introduces an innovative parallel, fully-distributed finite element framework for growing geometries and its application to metal additive manufacturing. It is well-known that virtual part design and qualification in additive manufacturing requires highly-accurate multiscale and multiphysics analyses. Only high performance computing tools are able to handle such complexity in time frames compatible with time-to-market. However, efficiency, without loss of accuracy, has rarely held the centre stage in the numerical community. Here, in contrast, the framework is designed to adequately exploit the resources of high-end distributed-memory machines. It is grounded on three building blocks: (1) Hierarchical adaptive mesh refinement with octree-based meshes; (2) a parallel strategy to model the growth of the geometry; (3) state-of-the-art parallel iterative linear solvers. Computational experiments consider the heat transfer analysis at the part scale of the printing process by powder-bed technologies. After verification against a 3D benchmark, a strong-scaling analysis assesses performance and identifies major sources of parallel overhead. A third numerical example examines the efficiency and robustness of (2) in a curved 3D shape. Unprecedented parallelism and scalability were achieved in this work. Hence, this framework contributes to take on higher complexity and/or accuracy, not only of part-scale simulations of metal or polymer additive manufacturing, but also in welding, sedimentation, atherosclerosis, or any other physical problem where the physical domain of interest grows in time

Fast, Exact and Multi-Scale Inference for Semantic Image Segmentation with Deep Gaussian CRFs

In this work we propose a structured prediction technique that combines the virtues of Gaussian Conditional Random Fields (G-CRF) with Deep Learning: (a) our structured prediction task has a unique global optimum that is obtained exactly from the solution of a linear system (b) the gradients of our model parameters are analytically computed using closed form expressions, in contrast to the memory-demanding contemporary deep structured prediction approaches that rely on back-propagation-through-time, (c) our pairwise terms do not have to be simple hand-crafted expressions, as in the line of works building on the DenseCRF, but can rather be `discovered' from data through deep architectures, and (d) out system can trained in an end-to-end manner. Building on standard tools from numerical analysis we develop very efficient algorithms for inference and learning, as well as a customized technique adapted to the semantic segmentation task. This efficiency allows us to explore more sophisticated architectures for structured prediction in deep learning: we introduce multi-resolution architectures to couple information across scales in a joint optimization framework, yielding systematic improvements. We demonstrate the utility of our approach on the challenging VOC PASCAL 2012 image segmentation benchmark, showing substantial improvements over strong baselines. We make all of our code and experiments available at {https://github.com/siddharthachandra/gcrf}Comment: Our code is available at https://github.com/siddharthachandra/gcr