A radial focus+context visualization for multi-dimensional functions

A lens system can be represented as a point in the N-dimensional parameter space. There is a set of quantities called merit functions associated with points in this parameter space that define the performance of the lens system. Ideally this set can be reduced to a single quantity called the merit function. Lens design iterations consist of transferring from one point in parameter space to another where the merit function indicates an improvement. By repeating this process the designer may eventually reach a satisfactory solution. Even when the functions involved are theoretically continuous, the transferring in parameter space may never be strictly continuous because it is a numerical process. Yet, it should be possible to take arbitrary short steps on going from one point to another. Long steps, or leaps, can abruptly lead to undesirable regions from where it may be difficult to return to the previous, more desirable region. There are three common methods to set up the merit function: Levenberg’s damped least squares (DLS), G. Spencer’s constrained damped least squares (CDLS) and E. Glatzel’s automatic adaptive correction (AAC). This paper reviews the pros and cons of these methods. A discontinuity found in Spencer’s method will be discussed in detail.

Hough Transform (HT) can provide a significant solution to the problems associated with line detection in an image. Line detection is used to detect the presence of lines in an image, at a particular orientation. The importance of line detection is used for detecting sharp changes in image brightness. HT is a popular technique in computer vision. The HT for line detection replaces the original problem of detecting collinear points in an image space by a simpler task of finding peak points in a parameter space. The proposed method used in this paper is to implement Generalized Hough Transform (GHT) on an image, since in GHT any straight line in the image is represented by a single point in the (ρ θ) parameter space and any part of this straight line is transformed into the same point. It can compute the HT of 256 × 256 test images. It can able to detect the lines present in the image and also it can extract feature points in an image.

Binary black hole mergers are of great interest to the astrophysics community, not least because of their promise to test general relativity in the highly dynamic, strong field regime. Detections of gravitational waves from these sources by LIGO and Virgo have garnered widespread media and public attention. Among these sources, precessing systems (with misaligned black-hole spin/orbital angular momentum) are of particular interest because of the rich dynamics they offer. However, these systems are, in turn, more complex compared to nonprecessing systems, making them harder to model or develop intuition about. Visualizations of numerical simulations of precessing systems provide a means to understand and gain insights about these systems. However, since these simulations are very expensive, they can only be performed at a small number of points in parameter space. We present binaryBHexp, a tool that makes use of surrogate models of numerical simulations to generate on-the-fly interactive visualizations of precessing binary black holes. These visualizations can be generated in a few seconds, and at any point in the 7-dimensional parameter space of the underlying surrogate models. With illustrative examples, we demonstrate how this tool can be used to learn about precessing binary black hole systems.

In order for the results of timing analysis to be useful, they must provide insight and guidance on how the circuit may be improved so as to fix any reported timing problems. A limitation of many recent variability-aware timing analysis techniques is that, while they report delay distributions, or verify multiple corners, they do not provide the required guidance for re-design. We propose an efficient block-based parameterized timing analysis technique that can accurately capture circuit delay at every point in the parameter space, by reporting all paths that can become critical. Using an efficient pruning algorithm, only those potentially critical paths are carried forward, while all other paths are discarded during propagation. This allows one to examine local robustness to parameters in different regions of the parameter space, not by considering differential sensitivity at a point (which would be useless in this context) but by knowledge of the paths that can become critical at nearby points in parameter space. We give a formal definition of this problem and propose a technique for solving it that improves on the state of the art, both in terms of theoretical computational complexity and in terms of run time on various test circuits.

Observations of the cosmic 21-cm power spectrum (PS) are starting to enable precision Bayesian inference of galaxy properties and physical cosmology, during the first billion years of our Universe. Here we investigate the impact of common approximations about the likelihood used in such inferences, including: (i) assuming a Gaussian functional form; (ii) estimating the mean from a single realization; and (iii) estimating the (co)variance at a single point in parameter space. We compare ‘classical’ inference that uses an explicit likelihood with simulation-based inference (SBI) that estimates the likelihood from a training set. Our forward models include: (i) realizations of the cosmic 21-cm signal computed with 21cmFAST by varying ultraviolet (UV) and X-ray galaxy parameters together with the initial conditions; (ii) realizations of the telescope noise corresponding to a $1000 \, \mathrm{h}$ integration with the low-frequency component of the Square Kilometre Array (SKA1-Low); and (iii) the excision of Fourier modes corresponding to a foreground-dominated horizon ‘wedge’. We find that the 1D PS likelihood is well described by a Gaussian accounting for covariances between wave modes and redshift bins (higher order correlations are small). However, common approaches of estimating the forward-modelled mean and (co)variance from a random realization or at a single point in parameter space result in biased and overconstrained posteriors. Our best results come from using SBI to fit a non-Gaussian likelihood with a Gaussian mixture neural density estimator. Such SBI can be performed with up to an order of magnitude fewer simulations than classical, explicit likelihood inference. Thus SBI provides accurate posteriors at a comparably low computational cost.

The well-known difficulty of controlling many synthesis parameters in performance, for exploration and expression, is addressed. Inspired by interactive evolution, random vectors in parameter space are assigned to an array of pressure sensitive pads. Vectors are scaled with pressure and added to define the current point in parameter space. Vectors can be scaled globally, allowing exploration of the whole space or minute timberal expression. The vector origin can be shifted at any time, allowing exploration of subspaces. In essence, this amounts to mutation-based interactive evolution with continuous interpolation between population members. With a suitable sound engine, the system forms a surprisingly expressive performance instrument, used by the electronic free impro duo pantoMorf in concerts and recording sessions over the last year.

Computational modelling of biological processes poses multiple challenges in each stage of the modelling exercise. Some significant challenges include identifiability, precisely estimating parameters from limited data, informative experiments and anisotropic sensitivity in the parameter space. One of these challenges’ crucial but inconspicuous sources is the possible presence of large regions in the parameter space over which model predictions are nearly identical. This property, known as sloppiness, has been reasonably well-addressed in the past decade, studying its possible impacts and remedies. However, certain critical unanswered questions concerning sloppiness, particularly related to its quantification and practical implications in various stages of system identification, still prevail. In this work, we systematically examine sloppiness at a fundamental level and formalise two new theoretical definitions of sloppiness. Using the proposed definitions, we establish a mathematical relationship between the parameter estimates’ precision and sloppiness in linear predictors. Further, we develop a novel computational method and a visual tool to assess the goodness of a model around a point in parameter space by identifying local structural identifiability and sloppiness and finding the most sensitive and least sensitive parameters for non-infinitesimal perturbations. We demonstrate the working of our method in benchmark systems biology models of various complexities. The pharmacokinetic HIV infection model analysis identified a new set of biologically relevant parameters that can be used to control the free virus in an active HIV infection.

When searching for new gravitational-wave or electromagnetic sources, the $n$ signal parameters (masses, sky location, frequencies,...) are unknown. In practice, one hunts for signals at a discrete set of points in parameter space, with a computational cost that is proportional to the number of these points. If that is fixed, the question arises, where should the points be placed in parameter space? The current literature advocates selecting the set of points (called a "template bank") whose Wigner-Seitz (also called Vorono\"i) cells have the smallest covering radius ($\equiv$ smallest maximal mismatch). Mathematically, such a template bank is said to have "minimum thickness". Here, for realistic populations of signal sources, we compute the fraction of potential detections which are "lost" because the template bank is discrete. We show that at fixed computational cost, the minimum thickness template bank does not maximize the expected number of detections. Instead, the most detections are obtained for a bank which minimizes a particular functional of the mismatch. For closely spaced templates, the fraction of lost detections is proportional to a scale-invariant "quantizer constant" G, which measures the average squared distance from the nearest template, i.e., the average expected mismatch. This provides a straightforward way to characterize and compare the effectiveness of different template banks. The template bank which minimizes G is mathematically called the "optimal quantizer", and maximizes the expected number of detections. We review optimal quantizer and minimum thickness template banks that are built as n-dimensional lattices, showing that even the best of these offer only a marginal advantage over template banks based on the humble cubic lattice.

A laboratory investigation is presented with the aim of studying systematically the diversity of flows resulting from the interaction of a baroclinic flow with periodic bottom topography. Flow visualisation and particle-tracking experiments were performed using a differentially heated rotating annulus of fluid with sinusoidal bottom topography. Multi-directional regime scans with varying initial locations in parameter space revealed a rich variety of flow regimes, including those with near-axisymmetric flow at upper levels and a weak stationary wave forced by the topography which is evanescent at upper levels, and flows with a strong forced wave at all heights both with and without travelling baroclinic waves. A range of time-dependent and vacillating flows was also found. The parametric occurrence of the principal regimes agrees well with a number of earlier studies, which covered a more limited range and at lower resolution in parameter space. It is clear from these results that multiple flow states at the same point in parameter space are very common in baroclinically unstable flows with topography. The parametric extent of the influence of the topographically forced stationary waves was shown to be qualitatively consistent with the well-known Charney–Drazin criterion for vertical wave propagation.

Spectroscopic ellipsometry and some other optical metrology techniques, such as reflectometry and scatterometry, are model-based optical measurements and, therefore, require appropriate modeling to determine the geometric and material properties of substrates, thin films, and multilayer structures. Parametric sensitivity analysis (SA) provides essential assistance in the model-building process to quantify the relative importance of model parameters for model output and to identify those with high/little influence. SA can be performed in a variety of ways, and this article discusses an application of the Morris or elementary effect (EE) method, a screening type SA procedure, to spectroscopic ellipsometry modeling. The method is a global SA technique and uses a stepping of m parameters along certain so-called “trajectories” or sequences of points in parameter space, randomly constructed in order to maximally fill the volume of the m-dimensional parameter space. However, it was thought that the EE method relies greatly on a sampling strategy or a way of selecting “optimized trajectories” in the parameter space, i.e., a necessary number of trajectories chosen to be well spread over the space to properly cover the entire realistic ranges of all input factors. Here, we use two sampling methods for selecting trajectories with possibly different distributions and investigate their effects on the estimation of various sensitivity measures in spectroscopic ellipsometry data modeling. The SA results indicate that the performance of the sampling strategy should not be judged only by maximizing the trajectory spread but also include some additional convergence criteria for the sensitivity measures.

The input-decentralization procedure proposed by Siljak and Vukcevic is analyzed in this note. It is shown that the procedure is valid on an open and dense subset of the parameter space of all large-scale systems with controllable subsystems. The examples which have appeared recently for which the proposed procedure fails are isolated points in the entire parameter space.

This paper describes the problem of spatial selection of devices operating in the same frequency range. The paper considers the possibility of focusing electromagnetic fields at several given points in space. The proposed idea of focusing can be an additional way to separate devices that operate in the same frequency range. A system consisting of space, radiating antennas and focusing points is represented as a multipole with the number of inputs equal to the number of radiating antennas. And with as many outputs as there are focus points. A coordinate system has been introduced that makes it possible to calculate the distances between radiation and focusing points. A method for calculating the transmission coefficients between emission points and reception points is presented. An expression is obtained, which is a system of linear algebraic equations, which makes it possible to calculate the necessary amplitudes and phases of signals supplied to radiating antennas. A simulation model containing 28 radiating antennas was presented. 9 focus points were set, 5 of which should have maxima of the electromagnetic field. The simulation results confirmed the theoretical calculations. Tasks for further research are described.

This paper presents an approach for finding regions Of specified gain margins (GM) and phase margins (PM) in the parameter space of a single-input single-output (SISO) control system with adjustable parameters. The method uses the Nyquist stability criterion of encirclement of a point in the polar plot. The necessary of the specified GM or PM regions in the parameter space are fully treated by mapping a point related to specified GM or PM in the polar plane into the parameter space. A general problem with previous mapping techniques is identified where points in the parameter space do not necessarily satisfy the required GM or PM. A searchbased technique is proposed to address this problem. Three examples illustrate the theoretical development. I. Introduction G AIN margin (GM) and phase margin (PM) are perhaps the two most important and widespread accepted criteria in analysis and design of practical control systems. These criteria are related to the open-loop transfer function of systems. In the robust stabilization problem of control systems, however, parameter space methods generally consider the closed-loop characteristic equation of the system.13 Parameter space techniques were initially studied by Vishnegradesky for a third-order system using a graphical technique that was popularized by Nimark and was referred to as D-partition or D-decomposition46 in its early development. This method actually establishes a direct correlation between the variable parameters of the closed-loop characteristic equation and the various important stability regions, e.g., open left half of the complex plane (OLHP). The method is directly suitable for graphical representation, in which case it is convenient to consider two parameters at a time. The idea of correlating the free parameters of the open-loop transfer function of a system and the relative stability regions appears to have been initially given in Ref. 7. Unfortunately , the method was not developed in detail, and the idea of isolating the GM region in the parameter plane was not correct. A recent improvement to the idea was presented by introducing the gain-phase margin tester into the system,8 which attempted to isolate specified GM and PM regions in the parameter space to allow a choice of the adjustable parameters. However, it is argued here that in this latter approach all the possible of the relative stability regions are not in fact considered, and some and possibly all points in the relative stability region do not necessarily satisfy the actual specified relative stability margin. In particular, the so called singular boundaries of the D-partition method6 need to be considered. Finally, arising through frequency loci with re-entrant or convoluted characteristics also need to be considered with care. In this paper, a method for accounting for the

Since all of adaptive models converge to the same point in parameter space, classical second level multiple model adaptive control method cannot deal with linear time-varying system. In order to deal with this drawback, this paper proposes a modified second level adaptive control method based on combining the advantages of adaptive model and fixed model for a class linear discrete time-varying plant where only a little prior information which is known. Firstly, multiple adaptive models are selected in the space of parameters, and the parameters are identifying and the controller is designed based on second level adaptive method. Secondly, when one of sub-model of the adaptive models are closed to the real parameters of the system, all of these adaptive models are fixed, and then the controller is designed based on multiple fixed models. Lastly, simulations have demonstrated that this method is effectively to a class of linear time-varying system.

Two slabs of dielectrics with different values of refractive index are synthesized to form a composite structure and the whole system satisfies parity-time-reciprocal scaling symmetry. Four coherent-perfect-absorber-laser points are induced by modulating the incident wavelength of lightwaves and the loss/gain factor of materials. The locations of coherent-perfect-absorber-laser points in parameter space, which manifests the multi-wavelengths of coherent-perfect-absorber-lasers could be regulated by the incidence angle. Especially, a value of π phase shift in the transmitted/reflected light wave has been observed as well. This research has potential applications in multi-wavelength absorbers/lasers and half-wave phase shifters.