Numerical Toy Research Articles

Vesicle condensation induced by synapsin: condensate size, geometry, and vesicle shape deformations

We study the formation of vesicle condensates induced by the protein synapsin, as a cell-free model system mimicking vesicle pool formation in the synapse. The system can be considered as an example of liquid–liquid phase separation (LLPS) in biomolecular fluids, where one phase is a complex fluid itself consisting of vesicles and a protein network. We address the pertinent question why the LLPS is self-limiting and stops at a certain size, i.e., why macroscopic phase separation is prevented. Using fluorescence light microscopy, we observe different morphologies of the condensates (aggregates) depending on the protein-to-lipid ratio. Cryogenic electron microscopy then allows us to resolve individual vesicle positions and shapes in a condensate and notably the size and geometry of adhesion zones between vesicles. We hypothesize that the membrane tension induced by already formed adhesion zones then in turn limits the capability of vesicles to bind additional vesicles, resulting in a finite condensate size. In a simple numerical toy model we show that this effect can be accounted for by redistribution of effective binding particles on the vesicle surface, accounting for the synapsin-induced adhesion zone.Graphic abstract

Co-Learning Bayesian Optimization.

Bayesian optimization (BO) is well known to be sample efficient for solving black-box problems. However, BO algorithms may get stuck in suboptimal solutions even with plenty of samples. Intrinsically, such a suboptimal problem of BO can attribute to the poor surrogate accuracy of the trained Gaussian process (GP), particularly that in the regions where the optimal solutions locate. Hence, we propose to build multiple GP models instead of a single GP surrogate to complement each other, thus resolving the suboptimal problem of BO. Nevertheless, according to the bias-variance tradeoff equation, the individual prediction errors can increase when increasing the diversity of models, which may lead even worse overall surrogate accuracy. On the other hand, based on the theory of the Rademacher complexity, it has been proven that exploiting the agreement of models on unlabeled information can reduce the complexity of hypothesis space, therefore achieving the required surrogate accuracy with fewer samples. Such value of model agreement has been extensively demonstrated for co-training style algorithms to boost model accuracy with a small portion of samples. Inspired by the above, we propose a novel BO algorithm labeled as co-learning BO (CLBO), which exploits both model diversity and agreement on unlabeled information to improve the overall surrogate accuracy with limited samples, therefore achieving more efficient global optimization. Through tests on five numerical toy problems and three engineering benchmarks, the effectiveness of the proposed CLBO has been well demonstrated.

Joint training of a predictor network and a generative adversarial network for time series forecasting: A case study of bearing prognostics

The lack of bearing run-to-failure data has been one of the challenges in developing and practically implementing robust bearing prognostics models. This paper proposes a new Generative Adversarial Network (GAN) based prognostics method for RUL prediction. We propose a novel joint training strategy to integrate the training process of a bearing health predictor within the GAN architecture. GAN uses available time series degradation data to generate synthetic degradation data that enhances the predictor’s learning and forecast performance, thus improving the RUL prediction accuracy. We demonstrate the utility and performance of the proposed method through two examples. The first numerical toy case study of forecasting polynomial-like time series shows that the proposed Jointly Trained Health Predictor (HP-JT) method produces smaller one- and multi-step-ahead prediction errors than a traditional health predictor (HP). In the second case study, we design a cross-validation study utilizing an open-source bearing dataset to evaluate the model’s performance in RUL prediction. Compared to HP, the proposed method decreases the bearing RUL prediction average error by 29.4% in a five-fold cross-validation study. We further compare the model with standard data augmentation techniques such as adding noise and using a variational autoencoder (VAE). The results from the case studies show that the proposed method can generate time series representing the real-data distribution.

Chaos in a melting pot

Abstract

Bayesian-entropy gaussian process for constrained metamodeling

A novel Bayesian-Entropy Gaussian Process (BEGP) is proposed for constrained metamodeling. Gaussian Process (GP) regression is a flexible and robust tool for surrogate modeling using observation data. For many engineering problems, available information other than observations may be known, such as physical constraints, boundary conditions, and empirical knowledge. Based on the Bayesian-Entropy (BE) principle, this paper introduces a novel framework for encoding extra information in addition to point data in constructing a GP regression model. The extra information is treated as constraints on the mean prediction of GP. The BE method can rigorously incorporate extra information as constraints into classical Bayesian framework. The constraint term is added into the posterior distribution of the hyperparameters when training the GP model. BEGP serves as an information fusion tool to enhance the extrapolation behavior of the GP model by incorporating additional knowledge about the problem. The proposed methodology is demonstrated on a numerical toy example and a structural analysis example highlighting the ability to smoothly connect two local GPs and incorporate boundary conditions as extra constraints. The BEGP shows the ability of incorporating physics constraints to enhance prediction and extrapolation behaviors. Finally, conclusions and future work are drawn based on the proposed study.

Black hole shadows, photon rings, and lensing rings

The presence of a bright "photon ring" surrounding a dark "black hole shadow" has been discussed as an important feature of the observational appearance of emission originating near a black hole. We clarify the meaning and relevance of these heuristics with analytic calculations and numerical toy models. The standard usage of the term "shadow" describes the appearance of a black hole illuminated from all directions, including from behind the observer. A backlit black hole casts a somewhat larger shadow. Neither shadow heuristic is particularly relevant to understanding the appearance of emission originating near the black hole, where the emission profile and gravitational redshift play the dominant roles in determining the observed size of the central dark area. A photon ring results from light rays that orbit near the black hole before escaping to infinity, where they arrive near a ring-shaped "critical curve" on the image plane. Although the brightness can become arbitrarily large near this critical curve in the case of optically thin emitting matter, we show that the enhancement is only logarithmic, and hence is of no relevance to present observations. For optically thin emission from a geometrically thin or thick disk, photons that make only a fraction of an orbit will generically give rise to a much wider "lensing ring," which is a demagnified image of the back of the disk, superimposed on top of the direct emission. The lensing ring is centered at a radius ~5% larger than the photon ring and its width is ~0.5-1M. It can be relatively brighter by a factor of 2-3 and thus could provide a significant feature in high resolution images. Nevertheless, the characteristic features of the observed image are dominated by the location and properties of the emitting matter near the black hole. We comment on the recent M87* Event Horizon Telescope observations and mass measurement.

Endemic SIR model in random media with applications

We consider an averaging principle for the endemic SIR model in a semi-Markov random media. Under stationary conditions of a semi- Markov media we show that the perturbed endemic SIR model converges to the classic endemic SIR model with averaged coefficients. Numerical toy examples and their interpretations are also presented for two-state Markov and semi-Markov chains. We also discuss two numerical examples involving real data: 1) Dengue Fever Disease (Indonesia and Malaysia (2009)) and 2) Cholera Outbreak in Zimbabwe (2008-2009). Novelty of the paper consists in studying of an endemic SIR model in semi-Markov random media and in implementations and interpretations of the results through numerical toy examples and discussion of numerical examples with real data.

Endemic SIR Model in Random Media with Applications

We consider an averaging principle for the endemic SIR model in a semi-Markov random media. Under stationary conditions of a semi- Markov media we show that the perturbed endemic SIR model converges to the classic endemic SIR model with averaged coefficients. Numerical toy examples and their interpretations are also presented for two-state Markov and semi-Markov chains. We also discuss two numerical examples involving real data: 1) Dengue Fever Disease (Indonesia and Malaysia (2009)) and 2) Cholera Outbreak in Zimbabwe (2008-2009). Novelty of the paper consists in studying of an endemic SIR model in semi-Markov random media and in implementations and interpretations of the results through numerical toy examples and discussion of numerical examples with real data.

Total decay and transition rates from LQCD

We present a new technique for extracting total transition rates into final states with any number of hadrons from lattice QCD. The method involves constructing a finite-volume Euclidean four-point function whose corresponding infinite-volume spectral function gives access to the decay and transition rates into all allowed final states. The inverse problem of calculating the spectral function is solved via the Backus-Gilbert method, which automatically includes a smoothing procedure. This smoothing is in fact required so that an infinite-volume limit of the spectral function exists. Using a numerical toy example we find that reasonable precision can be achieved with realistic lattice data. In addition, we discuss possible extensions of our approach and, as an example application, prospects for applying the formalism to study the onset of deep-inelastic scattering. More details are given in the published version of this work, Ref. [1].

Exploiting the hidden symmetry of spinning black holes: conservation laws and numerical tests

The Kerr black hole is stationary and axisymmetric, which leads to conservation of energy and azimuthal angular momentum along the orbits of free test particles in its vicinity, but also to conservation laws for the evolution of continuum matter fields. However, the Kerr space-time possesses an additional "hidden symmetry" which exhibits itself in an unexpected conserved quantity along geodesics known as the Carter constant. We investigate the possibility of using this hidden symmetry to obtain conservation laws and other identities which could be used to test astrophysical simulations of the evolution of matter fields near spinning black holes. After deriving such identities, we set up a simple numerical toy model on which we demonstrate how they can detect the violations of evolution equations in a numerical simulation. Even though one of the expressions we derive is in the form of a conservation law, we end up recommending an equivalent but simpler expression that is not in the form of a conservation law for practical implementation.

Generic inference of inflation models by non-Gaussianity and primordial power spectrum reconstruction

We present a generic inference method for inflation models from observational data by the usage of higher-order statistics of the curvature perturbation on uniform density hypersurfaces. This method is based on the calculation of the posterior for the primordial non-Gaussianity parameters fNL and gNL, which in general depend on specific parameters of inflation and reheating models, and enables to discriminate among the still viable inflation models. To keep analyticity as far as possible to dispense with numerically expensive sampling techniques a saddle-point approximation is introduced, whose precision is validated for a numerical toy example. The mathematical formulation is done in a generic way so that the approach remains applicable to cosmic microwave background data as well as to large scale structure data. Additionally, we review a few currently interesting inflation models and present numerical toy examples thereof in two and three dimensions to demonstrate the efficiency of the higher-order statistics method. A second quantity of interest is the primordial power spectrum. Here, we present two Bayesian methods to infer it from observational data, the so called critical filter and an extension thereof with smoothness prior, both allowing for a non-parametric spectrum reconstruction. These methods are able to reconstruct the spectra of the observed perturbations and the primordial ones of curvature perturbation even in case of non-Gaussianity and partial sky coverage. We argue that observables like T- and B-modes permit to measure both spectra. This also allows to infer the level of non-Gaussianity generated since inflation.

Generic inference of inflation models by local non-Gaussianity

AbstractThe presence of multiple fields during inflation might seed a detectable amount of non-Gaussianity in the curvature perturbations, which in turn becomes observable in present data sets like the cosmic microwave background (CMB) or the large scale structure (LSS). Within this proceeding we present a fully analytic method to infer inflationary parameters from observations by exploiting higher-order statistics of the curvature perturbations. To keep this analyticity, and thereby to dispense with numerically expensive sampling techniques, a saddle-point approximation is introduced whose precision has been validated for a numerical toy example. Applied to real data, this approach might enable to discriminate among the still viable models of inflation.

Probabilistic fatigue damage prognosis using maximum entropy approach

A general framework for probabilistic fatigue damage prognosis using maximum entropy concept is proposed and developed in this paper. The fatigue damage is calculated using a physics-based crack growth model. Due to the stochastic nature of fatigue crack propagation process, uncertainties arising from the underlying physical model, parameters of the model and the response variable measurement noise are considered and integrated into this framework. Incorporating all those uncertainties, a maximum relative entropy (MRE) approach is proposed to update the statistical description of model parameters and narrow down the prognosis deviations. A Markov Chain Monte Carlo (MCMC) simulation is then employed to generate samples from updated posterior probability distributions and provide statistical information for the maximum relative entropy updating procedure. A numerical toy problem is given to demonstrate the proposed MRE prognosis methodology. Experimental data for aluminum alloys are used to validate model predictions under uncertainty. Following this, a detailed comparison between the proposed MRE approach and the classical Bayesian updating method is performed to illustrate advantages of the proposed prognosis framework.

Stability analysis of product ?-methods for Abel integral equations of the second kind

Stability analysis of product θ-methods for Abel integral equations of the second kind, based on the test equation $$y(t) = 1 - \lambda \int\limits_0^1 {\frac{{y(s)}}{{(t - s)^\alpha }}} ds,t \geqq 0,$$ 0 0 and we investigate whether this property is inherited by numerical approximations toY.