We apply a Bayesian parameter estimation technique to a chemical kinetic mechanism for n-propylbenzene oxidation in a shock tube to propagate errors in experimental data to errors in Arrhenius parameters and predicted species concentrations. We find that, to apply the methodology successfully, conventional optimization is required as a preliminary step. This is carried out in two stages: First, a quasi-random global search using a Sobol low-discrepancy sequence is conducted, followed by a local optimization by means of a hybrid gradient-descent/Newton iteration method. The concentrations of 37 species at a variety of temperatures, pressures, and equivalence ratios are optimized against a total of 2378 experimental observations. We then apply the Bayesian methodology to study the influence of uncertainties in the experimental measurements on some of the Arrhenius parameters in the model as well as some of the predicted species concentrations. Markov chain Monte Carlo algorithms are employed to sample from the posterior probability densities, making use of polynomial surrogates of higher order fitted to the model responses. We conclude that the methodology provides a useful tool for the analysis of distributions of model parameters and responses, in particular their uncertainties and correlations. Limitations of the method are discussed. For example, we find that using second-order response surfaces and assuming
Matrix factorization (or low-rank matrix completion) with missing data is a key computation in many computer vision and machine learning tasks, and is also related to a broader class of nonlinear optimization problems such as bundle adjustment. The problem has received much attention recently, with renewed interest in variable-projection approaches, yielding dramatic improvements in reliability and speed. However, on a wide class of problems, no one approach dominates, and because the various approaches have been derived in a multitude of different ways, it has been difficult to unify them. This paper provides a unified derivation of a number of recent approaches, so that similarities and differences are easily observed. We also present a simple meta-algorithm which wraps any existing algorithm, yielding 100% success rate on many standard datasets. Given 100% success, the focus of evaluation must turn to speed, as 100% success is trivially achieved if we do not care about speed. Again our unification allows a number of generic improvements applicable to all members of the family to be isolated, yielding a unified algorithm that outperforms our re-implementation of existing algorithms, which in some cases already outperform the original authors' publicly available codes.
Bundle adjustment is used in structure-from-motion pipelines as final refinement stage requiring a sufficiently good initialization to reach a useful local mininum. Starting from an arbitrary initialization almost always gets trapped in a poor minimum. In this work we aim to obtain an initialization-free approach which returns global minima from a large proportion of purely random starting points. Our key inspiration lies in the success of the Variable Projection (VarPro) method for affine factorization problems, which have close to 100% chance of reaching a global minimum from random initialization. We find empirically that this desirable behaviour does not directly carry over to the projective case, and we consequently design and evaluate strategies to overcome this limitation. Also, by unifying the affine and the projective camera settings, we obtain numerically better conditioned reformulations of original bundle adjustment algorithms.
Variable Projection (VarPro) is a framework to solve optimization problems efficiently by optimally eliminating a subset of the unknowns. It is in particular adapted for Separable Nonlinear Least Squares (SNLS) problems, a class of optimization problems including low-rank matrix factorization with missing data and affine bundle adjustment as instances. VarPro-based methods have received much attention over the last decade due to the experimentally observed large convergence basin for certain problem classes, where they have a clear advantage over standard methods based on Joint optimization over all unknowns. Yet no clear answers have been found in the literature as to why VarPro outperforms others and why Joint optimization, which has been successful in solving many computer vision tasks, fails on this type of problems. Also, the fact that VarPro has been mainly tested on small to medium-sized datasets has raised questions about its scalability. This paper intends to address these unsolved puzzles.
Self-calibration of a magnetometer usually requires controlled magnetic environment as the calibration output can be affected by field distortions from nearby magnetic objects. In this paper, we develop a 2-stage method which can accurately selfcalibrate magnetometer from measurements containing anomalous readings due to local magnetic disturbances. The method proceeds by robustly fitting an ellipsoid to measurement data via L1-norm convex optimization, yielding initial model variables that are less prone to magnetic disruptions. These are then served as a starting point for robust nonlinear least squares optimization, which refines the magnetometer model to minimize sensor estimation errors while suppressing heavy anomalies. Synthetic and real experimental results are provided to demonstrate improved accuracy of the proposed method in presence of outliers. We additionally show empirically that the method is directly applicable to self-calibration of 3-axis accelerometers.
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