scholarly journals Existence of entropy solutions for some nonlinear problems in Orlicz spaces

2002 ◽  
Vol 7 (2) ◽  
pp. 85-102 ◽  
Author(s):  
A. Benkirane ◽  
J. Bennouna
Keyword(s):  
Sobolev Spaces ◽  
Orlicz Spaces ◽  
Elliptic Problems ◽  
Nonlinear Problems ◽  
Entropy Solutions ◽  
Nonlinear Elliptic Problems ◽  
Nonlinear Elliptic

We study in the framework of Orlicz Sobolev spacesW01LM(Ω), the existence of entropic solutions to the nonlinear elliptic problems:−div a(x,u,∇u)+div Φ(u)=f in Ω, for the case where the second member of the equationf∈L 1(Ω), andΦ∈(C0(ℝ))N.

Sobolev Spaces

2018 ◽  
Author(s):  
D. E. Edmunds ◽  
W. D. Evans
Keyword(s):  
Sobolev Spaces ◽  
Embedding Theorems ◽  
Approximation Numbers ◽  
Selection Of

This chapter presents a selection of some of the most important results in the theory of Sobolev spacesn. Special emphasis is placed on embedding theorems and the question as to whether an embedding map is compact or not. Some results concerning the k-set contraction nature of certain embedding maps are given, for both bounded and unbounded space domains: also the approximation numbers of embedding maps are estimated and these estimates used to classify the embeddings.

A Comprehensive Investigation of Ensembles of Gaussian-Based and Gradient-Based Transformed Reliability Analyses: When and How to Use Them

2016 ◽  
Author(s):  
Po Ting Lin ◽  
Wei-Hao Lu ◽  
Shu-Ping Lin
Keyword(s):  
Design Optimization ◽  
Design Space ◽  
Optimization Problems ◽  
Nonlinear Problems ◽  
Kernel Functions ◽  
Sensitivity Analyses ◽  
Normal Space ◽  
Highly Nonlinear ◽  
Standard Normal ◽  
Gradient Based

In the past few years, researchers have begun to investigate the existence of arbitrary uncertainties in the design optimization problems. Most traditional reliability-based design optimization (RBDO) methods transform the design space to the standard normal space for reliability analysis but may not work well when the random variables are arbitrarily distributed. It is because that the transformation to the standard normal space cannot be determined or the distribution type is unknown. The methods of Ensemble of Gaussian-based Reliability Analyses (EoGRA) and Ensemble of Gradient-based Transformed Reliability Analyses (EGTRA) have been developed to estimate the joint probability density function using the ensemble of kernel functions. EoGRA performs a series of Gaussian-based kernel reliability analyses and merged them together to compute the reliability of the design point. EGTRA transforms the design space to the single-variate design space toward the constraint gradient, where the kernel reliability analyses become much less costly. In this paper, a series of comprehensive investigations were performed to study the similarities and differences between EoGRA and EGTRA. The results showed that EGTRA performs accurate and effective reliability analyses for both linear and nonlinear problems. When the constraints are highly nonlinear, EGTRA may have little problem but still can be effective in terms of starting from deterministic optimal points. On the other hands, the sensitivity analyses of EoGRA may be ineffective when the random distribution is completely inside the feasible space or infeasible space. However, EoGRA can find acceptable design points when starting from deterministic optimal points. Moreover, EoGRA is capable of delivering estimated failure probability of each constraint during the optimization processes, which may be convenient for some applications.

scholarly journals Real-time automatic uncertainty estimation of coseismic single rectangular fault model using GNSS data

2021 ◽  
Vol 73 (1) ◽  
Author(s):  
Keitaro Ohno ◽  
Yusaku Ohta ◽  
Satoshi Kawamoto ◽  
Satoshi Abe ◽  
Ryota Hino ◽  
...  
Keyword(s):  
Real Time ◽  
Disaster Response ◽  
Order Approximation ◽  
Nonlinear Problems ◽  
Large Earthquake ◽  
Fault Model ◽  
Deformation Monitoring ◽  
Model Parameters ◽  
The Sea Of Japan ◽  
Plate Interface

AbstractRapid estimation of the coseismic fault model for medium-to-large-sized earthquakes is key for disaster response. To estimate the coseismic fault model for large earthquakes, the Geospatial Information Authority of Japan and Tohoku University have jointly developed a real-time GEONET analysis system for rapid deformation monitoring (REGARD). REGARD can estimate the single rectangular fault model and slip distribution along the assumed plate interface. The single rectangular fault model is useful as a first-order approximation of a medium-to-large earthquake. However, in its estimation, it is difficult to obtain accurate results for model parameters due to the strong effect of initial values. To solve this problem, this study proposes a new method to estimate the coseismic fault model and model uncertainties in real time based on the Bayesian inversion approach using the Markov Chain Monte Carlo (MCMC) method. The MCMC approach is computationally expensive and hyperparameters should be defined in advance via trial and error. The sampling efficiency was improved using a parallel tempering method, and an automatic definition method for hyperparameters was developed for real-time use. The calculation time was within 30 s for 1 × 106 samples using a typical single LINUX server, which can implement real-time analysis, similar to REGARD. The reliability of the developed method was evaluated using data from recent earthquakes (2016 Kumamoto and 2019 Yamagata-Oki earthquakes). Simulations of the earthquakes in the Sea of Japan were also conducted exhaustively. The results showed an advantage over the maximum likelihood approach with a priori information, which has initial value dependence in nonlinear problems. In terms of application to data with a small signal-to-noise ratio, the results suggest the possibility of using several conjugate fault models. There is a tradeoff between the fault area and slip amount, especially for offshore earthquakes, which means that quantification of the uncertainty enables us to evaluate the reliability of the fault model estimation results in real time.

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