Collecting enough samples is difficult in real applications. Several interval-based non-probabilistic reliability methods have been reported. The key of these to estimate system index. In this paper, a new method proposed calculate Kriging model used replace time-consuming simulations, and the efficient global optimization determine training samples. A refinement learning function best componen...

We now move on to a very interesting and powerful technique in combinatorics, called the probabilistic method. There is an excellent (really excellent!) textbook written by Noga Alon and Joel Spencer [AS16]. In fact, most materials that we will cover are from [AS16]. The method works as follows. Our goal is to prove the existence of certain object or structure with desired properties. We start ...

Modern treatments of structured Principal Component Analysis often focus on the estimation of a single component under various assumptions or priors, such as sparsity and smoothness, and then the procedure is extended to multiple components by sequential estimation interleaved with deflation. While prior work has highlighted the importance of proper deflation for ensuring the quality of the est...

We consider the probabilistic numerical scheme for fully nonlinear partial differential equations suggested in [Comm. Pure Appl. Math. 60 (2007) 1081–1110] and show that it can be introduced naturally as a combination of Monte Carlo and finite difference schemes without appealing to the theory of backward stochastic differential equations. Our first main result provides the convergence of the d...

Sonel mapping is a Monte-Carlo-based acoustical modeling technique that approximates the acoustics of an environment while accounting for diffuse and specular reflections as well as diffraction effects. Through the use of a probabilistic Russian roulette strategy to determine the type of interaction between a sound and any objects/surfaces it may encounter, sonel mapping avoids excessively larg...

Fatigue crack growth is a random process and various uncertainties affect the remaining useful life of engineering materials and structures. A general probabilistic life prediction methodology for accurate and efficient fatigue prognosis is proposed in this paper. The proposed methodology is based-on an inverse first-order reliability method (FORM) to evaluate the fatigue life at an arbitrary r...

Material flow modeling constitutes an important approach to predicting and understanding the flows of materials through the anthroposphere into the environment. The new “Dynamic Probabilistic Material Flow Analysis (DPMFA)” method, combining dynamic material flow modeling with probabilistic modeling, is presented in this paper. Material transfers that lead to particular environmental stocks are...

These notes accompany a lecture given in the summer of 2006 at the Center for Combinatorics at Nankai University. They are provided as a reference (and especially bibliography) for students new to the probabilistic method. The goal in these notes is twofold. First we wish to demonstrate the basics of the probabilistic method, covering both the expectation and second moment applications. Second,...

We use a probabilistic method to evaluate the limit of ∑n x=1 φ(x)x r−1 n−(r+1), where φ(x) is the Euler totient function and r is a nonnegative integer. We extend the probabilistic method to evaluate two other generalized types of series that involve Euler’s totient function. In addition to the probabilistic method, an analytic approach is presented to evaluate the series when the exponent par...