The geometry of nonlinear least squares with applications to sloppy models and optimization

Parameter estimation by nonlinear least squares minimization is a common problem with an elegant geometric interpretation: the possible parameter values of a model induce a manifold in the space of data predictions. The minimization problem is then to find the point on the manifold closest to the data. We show that the model manifolds of a large class of models, known as sloppy models, have many universal features; they are characterized by a geometric series of widths, extrinsic curvatures, and parameter-effects curvatures. A number of common difficulties in optimizing least squares problems are due to this common structure. First, algorithms tend to run into the boundaries of the model manifold, causing parameters to diverge or become unphysical. We introduce the model graph as an extension of the model manifold to remedy this problem. We argue that appropriate priors can remove the boundaries and improve convergence rates. We show that typical fits will have many evaporated parameters. Second, bare model parameters are usually ill-suited to describing model behavior; cost contours in parameter space tend to form hierarchies of plateaus and canyons. Geometrically, we understand this inconvenient parametrization as an extremely skewed coordinate basis and show that it induces a large parameter-effects curvature on the manifold. Using coordinates based on geodesic motion, these narrow canyons are transformed in many cases into a single quadratic, isotropic basin. We interpret the modified Gauss-Newton and Levenberg-Marquardt fitting algorithms as an Euler approximation to geodesic motion in these natural coordinates on the model manifold and the model graph respectively. By adding a geodesic acceleration adjustment to these algorithms, we alleviate the difficulties from parameter-effects curvature, improving both efficiency and success rates at finding good fits.Comment: 40 pages, 29 Figure

A framework for quantification and physical modeling of cell mixing applied to oscillator synchronization in vertebrate somitogenesis

In development and disease, cells move as they exchange signals. One example is found in vertebrate development, during which the timing of segment formation is set by a ‘segmentation clock’, in which oscillating gene expression is synchronized across a population of cells by Delta-Notch signaling. Delta-Notch signaling requires local cell-cell contact, but in the zebrafish embryonic tailbud, oscillating cells move rapidly, exchanging neighbors. Previous theoretical studies proposed that this relative movement or cell mixing might alter signaling and thereby enhance synchronization. However, it remains unclear whether the mixing timescale in the tissue is in the right range for this effect, because a framework to reliably measure the mixing timescale and compare it with signaling timescale is lacking. Here, we develop such a framework using a quantitative description of cell mixing without the need for an external reference frame and constructing a physical model of cell movement based on the data. Numerical simulations show that mixing with experimentally observed statistics enhances synchronization of coupled phase oscillators, suggesting that mixing in the tailbud is fast enough to affect the coherence of rhythmic gene expression. Our approach will find general application in analyzing the relative movements of communicating cells during development and disease.Fil: Uriu, Koichiro. Kanazawa University; JapónFil: Bhavna, Rajasekaran. Max Planck Institute of Molecular Cell Biology and Genetics; Alemania. Max Planck Institute for the Physics of Complex Systems; AlemaniaFil: Oates, Andrew C.. Francis Crick Institute; Reino Unido. University College London; Reino UnidoFil: Morelli, Luis Guillermo. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Parque Centenario. Instituto de Investigación en Biomedicina de Buenos Aires - Instituto Partner de la Sociedad Max Planck; Argentina. Max Planck Institute for Molecular Physiology; Alemania. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Física; Argentin

Influence of time-dependent restrained strains in the shear response of RC frames

The final publication is available at Springer via http://dx.doi.org/10.1617/s11527-016-0875-8Time-dependent strains, when restrained, can lead to important tensile forces and damage, affecting, among other aspects, the shear response and ultimate load carrying capacity of shear-critical RC frames. This paper presents a detailed study of this problematic by means of an extension of a shear-sensitive fibre beam model to time dependent behaviour of concrete. The model is firstly validated with experimental tests on diagonally pre-cracked beams under sustained loads. From these analyses, the contributions of shear distortions and bending curvatures to the total long-term deflection of the beams are discerned. Afterwards, the model is applied to study the influence of restraining strains due to long-term creep and shrinkage in the service and ultimate shear response of frames. In contrast with flexural resistant mechanisms, delayed strains may influence the latter shear resistance of integral structures by reducing the concrete contribution to shear resistance and leading to a sooner activation of the transversal reinforcement. These aspects can be relevant in assessing existing structures and this model, due to its relative simplicity, can be advantageous for practical applications.Peer ReviewedPostprint (author's final draft

Circuit quantum acoustodynamics with surface acoustic waves

The experimental investigation of quantum devices incorporating mechanical resonators has opened up new frontiers in the study of quantum mechanics at a macroscopic level$^{1,2}$. Superconducting microwave circuits have proven to be a powerful platform for the realisation of such quantum devices, both in cavity optomechanics$^{3,4}$, and circuit quantum electro-dynamics (QED)$^{5,6}$. While most experiments to date have involved localised nanomechanical resonators, it has recently been shown that propagating surface acoustic waves (SAWs) can be piezoelectrically coupled to superconducting qubits$^{7,8}$, and confined in high-quality Fabry-Perot cavities up to microwave frequencies in the quantum regime$^{9}$, indicating the possibility of realising coherent exchange of quantum information between the two systems. Here we present measurements of a device in which a superconducting qubit is embedded in, and interacts with, the acoustic field of a Fabry-Perot SAW cavity on quartz, realising a surface acoustic version of cavity quantum electrodynamics. This quantum acoustodynamics (QAD) architecture may be used to develop new quantum acoustic devices in which quantum information is stored in trapped on-chip surface acoustic wavepackets, and manipulated in ways that are impossible with purely electromagnetic signals, due to the $10^{5}$ times slower speed of travel of the mechanical waves.Comment: 12 pages, 9 figures, 1 tabl