Stochastic collocation for uncertainty quantification of systems described by neutral delayed differential equations

2017 ◽  
Author(s):  
F. Ferranti ◽  
D. Romano ◽  
G. Antonini ◽  
L. De Camillis
Keyword(s):  
Differential Equations ◽  
Uncertainty Quantification ◽  
Stochastic Collocation ◽  
Delayed Differential Equations

scholarly journals Uncertainty Quantification in Small-Timescale Model-Based Fatigue Crack Growth Analysis Using a Stochastic Collocation Method

Metals ◽  
2020 ◽  
Vol 10 (5) ◽  
pp. 646
Author(s):  
Hesheng Tang ◽  
Xueyuan Guo ◽  
Songtao Xue
Keyword(s):  
Fatigue Crack ◽  
Fatigue Crack Growth ◽  
Uncertainty Quantification ◽  
Crack Growth ◽  
Collocation Method ◽  
Life Prediction ◽  
Remaining Useful Life ◽  
Stochastic Collocation ◽  
Stochastic Collocation Method ◽  
Model Based

Due to the uncertainties originating from the underlying physical model, material properties and the measurement data in fatigue crack growth (FCG) processing, the prediction of fatigue crack growth lifetime is still challenging. The objective of this paper was to investigate a methodology for uncertainty quantification in FCG analysis and probabilistic remaining useful life prediction. A small-timescale growth model for the fracture mechanics-based analysis and predicting crack-growth lifetime is studied. A stochastic collocation method is used to alleviate the computational difficulties in the uncertainty quantification in the small-timescale model-based FCG analysis, which is derived from tensor products based on the solution of deterministic FCG problems on sparse grids of collocation point sets in random space. The proposed method is applied to the prediction of fatigue crack growth lifetime of Al7075-T6 alloy plates and verified by fatigue crack-growth experiments. The results show that the proposed method has the advantage of computational efficiency in uncertainty quantification of remaining life prediction of FCG.

scholarly journals COLLOCATING VOLATILITY: A COMPETITIVE ALTERNATIVE TO STOCHASTIC LOCAL VOLATILITY MODELS

2020 ◽  
Vol 23 (06) ◽  
pp. 2050038
Author(s):  
ANTHONIE W. VAN DER STOEP ◽  
LECH A. GRZELAK ◽  
CORNELIS W. OOSTERLEE
Keyword(s):  
Differential Equations ◽  
Scientific Computing ◽  
Collocation Methods ◽  
Heston Model ◽  
Stochastic Collocation ◽  
Exotic Options ◽  
Siam Journal ◽  
Local Volatility ◽  
Volatility Models ◽  
Stochastic Local Volatility

We discuss a competitive alternative to stochastic local volatility models, namely the Collocating Volatility (CV) framework, introduced in [L. A. Grzelak (2019) The CLV framework — A fresh look at efficient pricing with smile, International Journal of Computer Mathematics 96 (11), 2209–2228]. The CV framework consists of two elements, a “kernel process” that can be efficiently evaluated and a local volatility function. The latter, based on stochastic collocation — e.g. [I. Babuška, F. Nobile & R. Tempone (2007) A stochastic collocation method for elliptic partial differential equations with random input Data, SIAM Journal on Numerical Analysis 45 (3), 1005–1034; B. Ganapathysubramanian & N. Zabaras (2007) Sparse grid collocation schemes for stochastic natural convection problems, Journal of Computational Physics 225 (1), 652–685; J. A. S. Witteveen & G. Iaccarino (2012) Simplex stochastic collocation with random sampling and extrapolation for nonhypercube probability spaces, SIAM Journal on Scientific Computing 34 (2), A814–A838; D. Xiu & J. S. Hesthaven (2005) High-order collocation methods for differential equations with random inputs, SIAM Journal on Scientific Computing 27 (3), 1118–1139] — connects the kernel process to the market and allows the CV framework to be perfectly calibrated to European-type options. In this paper, we consider three different kernel process choices: the Ornstein–Uhlenbeck (OU) and Cox–Ingersoll–Ross (CIR) processes and the Heston model. The kernel process controls the forward smile and allows for an accurate and efficient calibration to exotic options, while the perfect calibration to liquid market quotes is preserved. We confirm this by numerical experiments, in which we calibrate the OU-CV, CIR-CV and Heston-CV frameworks to FX barrier options.

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