Bayesian cyclic bounds for periodic parameter estimation

In this study, we investigate the impact of pore distribution and size on the mechanical response of elastic porous materials. Two-dimensional porous media with convex porosity are considered (e.g., foams and sponges). These structures are numerically generated by subtracting randomly dispersed circular holes from a solid square domain. The in-plane position vector of pore centers is given by a uniform probability density function (PDF). The pore diameters are, instead, drawn from a non-uniform two-parameter continuous probability distribution. The influence of the void ratio, i.e., the volume fraction of voids, and the parameters of the diameter PDF are studied over a wide range of values, including the percolation threshold, where the effective elastic modulus of the material abruptly drops to zero. This transition takes place at a critical value of the void ratio, which turns out to be unaffected by the parameters of the pore size distribution. Our results demonstrate that the elastic response of these structures is primarily governed by the void ratio, with the correlation length playing a secondary role. While the pore size distribution may affect the correlation length, it has a minimal impact on the foam’s overall behavior.

This thesis aims at developing efficient mathematical methods for solving hierarchical dynamic optimization problems. The main motivation is to model processes in nature, for which there is evidence to assume that they run optimally. We describe models of such processes by optimal control problems (called optimal control models (OCMs)). However, an OCM typically includes unknown parameters that cannot be derived entirely on a theoretical basis, which is in particular the case for the cost function. Therefore, we develop parameter estimation techniques to estimate the unknowns in an OCM from observation data of the process. Mathematically, this leads to a hierarchical dynamic optimization problem with a parameter estimation problem on the upper level and an optimal control problem on the lower level. We focus on multi-stage equality and inequality constrained optimal control problems based on nonlinear ordinary differential equations. The main goal of this thesis is to derive numerically efficient mathematical methods for solving hierarchical dynamic optimization problems, and to use these methods to estimate parameters in high-dimensional OCMs from real-world measurement data. We develop parameter-dependent OCMs for the gait of cerebral palsy patients and able-bodied subjects. The unknown parameters in the OCMs are then estimated from real-world motion capture data provided by the Heidelberg MotionLab of the Orthopedic University Clinic Heidelberg by using the mathematical methods developed within this work. The main novelties and contributions of this thesis to the field of hierarchical dynamic optimization are summarized herein. - We establish a novel mathematical method, a so-called direct all-at-once approach, for solving hierarchical dynamic optimization problems based on the direct multiple shooting method and first-order optimality conditions. - Furthermore, we propose an efficient numerical algorithm for large-scale hierarchical dynamic optimization problems, which fully exploits the structures inherited from both the hierarchical setting and the discretization. - Pontryagin's maximum principle is used to analyze solution properties of hierarchical dynamic optimization problems like second-order optimality conditions of the lower-level problem. - In addition, we propose and discuss alternative methods for hierarchical dynamic optimization that are based on derivative-free optimization and a bundle approach. These methods keep the hierarchical problem setting and do not reformulate the lower-level problem using first-order optimality conditions. - We establish a novel lifting method for regularizing mathematical programs with complementarity constraints, which is discussed and numerically investigated by means of a well-known collection of benchmark problems. - Proofs of regularity and convergence results for sequential quadratic programming methods applied to lifted mathematical programs with complementarity constraints are provided. - Efficient state-of-the-art implementations of all mathematical methods derived in this thesis, as well as a benchmark collection of hierarchical dynamic optimization problems are presented. - High-dimensional optimal control gait models for cerebral palsy patients and able-bodied subjects are developed. The mathematical methods derived in this thesis are used to estimate the unknown model parameters from real-world motion capture data provided by the Heidelberg MotionLab of the Orthopedic University Clinic Heidelberg. The theoretical and practical results presented in this thesis can be considered an initial motivating step towards answering open questions in current medical research in fields like treatment planning, classification of gaits, or the evaluation of surgeries by means of hierarchical dynamic optimization.

In 1967, Bühlmann has shown that the credibility formula was the best linearized approximation to the exact Bayesian forecast.His result for the credibility factor can be found back by means of some Bayesian inference techniques. Introducing a uniform prior probability density function for the credibility factor provides us with a method for estimating z, a correction term to the Bühlmann's result is obtained. It is shown how prior boundary conditions can be introduced.

PoCET: a Polynomial Chaos Expansion Toolbox for Matlab

Dithering improves the linearity of A/D converters (ADCs). Wagdy and Goff (1994) have developed a criterion for assessing the deviation of ADC transfer characteristics from the unity-gain line. This criterion is based on a closed-form formula for the deviation factor D derived from the characteristic function of the ADC quantization error function. This approach was extended further by Wagdy (1996) to include nonideal ADCs. The results in that work pertain to additive dither, but a closed-form formula for subtractive dither may be very difficult to derive. In this paper, the ADC is modeled and dither is simulated using a software random number generator. First, additive dithering with continuous (analog) uniform (rectangular) probability density function (RPDF) is considered. The cases of an ADC with no errors, single-bit error, or multi-bit errors are investigated. Results are in excellent agreement with the previous ones, which validates the formulae therein. Second, the author presents new results using discrete (digital) rectangular dither (DRPDF). This dither type is generated via a D/A converter (DAC) whose number of bits, k, is greater by del than the ADC number of bits, m. For various values of del, subtractively-dithered ADCs outperform additively-dithered ones as far as D is concerned. Also, the effect of increasing del on D for an ADC is investigated for the subtractive dither case, thus providing some quantitative results which are useful for the designer.

We investigate subcritical resonant heat transfer in a heated periodic grooved channel by modulating the flow with an oscillation of random amplitude. This excitation effectively destabilizes the flow at relatively low Reynolds number and establishes strong communication between the grooved flow and the Tollmien–Schlichting channel waves, as revealed by various statistical quantities we analysed. Both single-frequency and multi-frequency responses are considered, and an optimal frequency value is obtained in agreement with previous deterministic studies. In particular, we employ a new approach, the multi-element generalized polynomial chaos (ME-gPC) method, to model the stochastic velocity and temperature fields for uniform and Beta probability density functions (PDFs) of the random amplitude. We present results for the heat transfer enhancement parameter $E$ for which we obtain mean values, lower and upper bounds as well as PDFs. We first study the dependence of the mean value of $E$ on the magnitude of the random amplitude for different Reynolds numbers, and we demonstrate that the deterministic results are embedded in the stochastic simulation results. Of particular interest are the PDFs of $E$, which are skewed with their peaks increasing towards larger values of $E$ as the Reynolds number increases. We then study the effect of multiple frequencies described by a periodically correlated random process. We find that the mean value of $E$ is increased slightly while the variance decreases substantially in this case, an indication of the robustness of this excitation approach. The stochastic modelling approach offers the possibility of designing ‘smart’ PDFs of the stochastic input that can result in improved heat transfer enhancement rates.

The classical intermittency theory developed forty years ago considers a uniform reinjection probability density function (RPD). However, in the last ten years, studies have found a more general RPD, which depends on the type of prereinjection points, extreme or infinite derivative points. Recently, we have introduced an analytical scheme to estimate the non-uniform RPD when the pre-reinjection points are extreme. Here, this theoretical scheme is extended to embrace pre-reinjection points with infinite derivative. The proposed theoretical method, to get the RPD, needs only the explicit expression of the map, and it is general and direct to use. For different intermittency types and different non-linearity, the theoretical predictions show a very high agreement with the numerical results.

International audience

The three main tools in the single image restoration theory are the maximum likelihood (ML) estimator, the maximum a posteriori probability (MAP) estimator, and the set theoretic approach using projection onto convex sets (POCS). This paper utilizes the above known tools to propose a unified methodology toward the more complicated problem of superresolution restoration. In the superresolution restoration problem, an improved resolution image is restored from several geometrically warped, blurred, noisy and downsampled measured images. The superresolution restoration problem is modeled and analyzed from the ML, the MAP, and POCS points of view, yielding a generalization of the known superresolution restoration methods. The proposed restoration approach is general but assumes explicit knowledge of the linear space- and time-variant blur, the (additive Gaussian) noise, the different measured resolutions, and the (smooth) motion characteristics. A hybrid method combining the simplicity of the ML and the incorporation of nonellipsoid constraints is presented, giving improved restoration performance, compared with the ML and the POCS approaches. The hybrid method is shown to converge to the unique optimal solution of a new definition of the optimization problem. Superresolution restoration from motionless measurements is also discussed. Simulations demonstrate the power of the proposed methodology.

The problem of statistical pattern recognition with noisy or imprecise feature measurements is considered. An exact analytical expression is found for the probability of misclassification under this condition, for multiclass multivariate systems. The probability of error exceeds that of the ideal case for the special case of two classes, the a priori conditional probability density functions are assumed to be normal, along with the two cases of feature measurement error, namely normal and uniform probability density functions. Monotonicity of the misclassification probability with measurement error variance is shown. Numerical results are presented for both cases over a workable range of parameters. The study is useful in practical pattern recognition problems.

This paper presents a novel class of self-organizing sensing agents that adaptively learn an anisotropic, spatio-temporal Gaussian process using noisy measurements and move in order to improve the quality of the estimated covariance function. This approach is based on a class of anisotropic covariance functions of Gaussian processes introduced to model a broad range of spatio-temporal physical phenomena. The covariance function is assumed to be unknown a priori. Hence, it is estimated by the maximum a posteriori probability (MAP) estimator. The prediction of the field of interest is then obtained based on the MAP estimate of the covariance function. An optimal sampling strategy is proposed to minimize the information-theoretic cost function of the Fisher Information Matrix. Simulation results demonstrate the effectiveness and the adaptability of the proposed scheme.

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A node in a switch architecture is examined. The node receives messages, stores and processes them locally, and transmits them to other nodes. Both message transmission and processing times depend on the message length, which is a random variable. The node is modeled as a queue with a single server where the interarrival and service times are related, and different cases are studied where this relation ranges from direct dependence to moderate correlation. Exponential and uniform probability density functions (pdf's) are examined, and the results are compared to the independent model. A relationship is developed for the system time pdf of a message, which is used to find the system time pdf and the system state probabilities. These quantities are directly related to buffer sizes and transmission delays within the architecture. >

Combined heat power dispatch with improved operation remains a popular preference in microgrid having uncertain and non-convex nature of the system. This research work deploys an efficient energy management system driven by a novel stochastic transitions tuned unscented Gaussian distribution (ST-UnGD) for optimal energy transaction. The proposed scheme has been introduced with uncertainty modeling of available generations and load using probabilistic transition-based timed petri net. Echo algorithm has been used to influence prioritized transition for appropriate number of token distribution. The control over the process during transition has been supervised with UnGD. A uniform probability density function has been created using Gaussian distribution to characterize random variables for effective time series scenario generation. The mean and variance which helps in sorting ten competent scenarios from one thousand scenarios, has been computed with the proposed strategy and compared with other methods. The improvement has been noticed with standard deviation of 0.0162% after 40 trials run with average operational cost of 254.211€ when token distribution was updated by 100, 200, 300, 400, and 500 firings. A statistical analysis by parametric and non-parametric test confirmed the effectiveness of the proposed algorithm showing appropriate confidence level in the mentioned case study.

In recent decades, analysis of structures considering variability of some parameters for more reliable design has attracted the attention of researchers. In this paper, the stochastic analysis of a cantilever deep beam made of large deformable neo-hookean material is carried out. For this purpose, the meshless local Petrov-Galerkin (MLPG) method is developed to obtain the geometrically non-linear equilibrium equations. The radial point interpolation method is used for generating the shape functions. The incremental iterative Newton-Raphson method with suitable load steps is used to solve the non-linear governing equations. The results of deterministic analysis obtained with proposed method are compared with the finite element results and good agreement is achieved. The initial elasticity modulus of neo-hookean material is considered to be uncertain variable. To generate random field of uncertain variable with normal, lognormal and uniform probability density functions (PDFs), the Monte Carlo Simulation (MCS) technique was employed. The sufficient number of simulations for convergence the results was determined experimentally. The effect of elasticity modulus, PDF and coefficients of variation (COV) on maximum vertical displacement, PDF and COV of results are studied in details. Comparing the stochastic and deterministic results shows that the uncertainty in mechanical properties has significant effect on results.