On the convergence of the quasi-regression method: polynomial chaos and regularity

2017 ◽  
Vol 54 (2) ◽  
pp. 424-443
Author(s):  
Je Guk Kim
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Method ◽  
Rate Of Convergence ◽  
Polynomial Chaos ◽  
Polynomial Chaos Expansion ◽  
Regression Method ◽  
Basis Functions ◽  
Chaos Expansion ◽  
Value Functions ◽  
Weaker Conditions

Abstract We present an analysis of convergence of a quasi-regression Monte Carlo method proposed by Glasserman and Yu (2004). We show that the method surely converges to the true price of an American option even under multiple underlyings via polynomial chaos expansion and weaker conditions than those used in Glasserman and Yu (2004). Further, we show the number of simulation paths grows exponentially in the number of basis functions to obtain convergence in implementing the method. Finally, we propose a rate of convergence considering regularity of value functions.

scholarly journals Stochastic Modeling of Electrical Field in Potato Tuber using Polynomial Chaos Expansion

2019 ◽  
Vol 29 ◽  
pp. 01008
Author(s):  
Bartosz Sawicki ◽  
Artur Krupa
Keyword(s):  
Monte Carlo ◽  
Potato Tuber ◽  
Polynomial Chaos ◽  
Ohmic Heating ◽  
Stochastic Approach ◽  
Polynomial Chaos Expansion ◽  
Chaos Expansion ◽  
Computational Power ◽  
Electrical Field ◽  
Stochastic Parameters

The paper deals with numerical modeling of objects with a natural origin. The stochastic approach based on description using random variables allows processing such challenges. The Monte-Carlo methods are known a tool for simulations containing stochastic parameters however, they require significant computational power to obtain stable results. Authors compare Monte- Carlo with more advanced Polynomial Chaos Expansion (PCE) method. Both statistical tools have been applied for simulation of the electric field used in ohmic heating of potato tuber probes. Results indicate that PCE is remarkably faster, however, it simplifies some probabilistic features of the solution.

Forward Kinematic Analysis of Non-Deterministic Articulated Multibody Systems With Kinematically Closed-Loops in Polynomial Chaos Expansion Scheme

2015 ◽  
Author(s):  
Sahand Sabet ◽  
Mohammad Poursina
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Method ◽  
Kinematic Analysis ◽  
Multibody Systems ◽  
Polynomial Chaos ◽  
Uncertain Parameters ◽  
Chaos Expansion ◽  
Closed Loops ◽  
Base Functions ◽  
Forward Kinematic

This paper presents the method of polynomial chaos expansion (PCE) for the forward kinematic analysis of nondeterministic multibody systems with kinematically closed-loops. The PCE provides an efficient mathematical framework to introduce uncertainty to the system. This is accomplished by compactly projecting each stochastic response output and random input onto the space of appropriate independent orthogonal polynomial base functions. This paper presents the detailed formulation of the kinematics of a constrained multibody system at the position, velocity, and acceleration levels in the PCE scheme. This analysis is performed by projecting the governing kinematic constraint equations of the system onto the space of appropriate polynomial base functions. Furthermore, forward kinematic analysis is conducted at the position, velocity, and acceleration levels for a non-deterministic four-bar mechanism with single and multiple uncertain parameters in the length of linkages of the system. Time efficiency and accuracy of the intrusive PCE approach are compared with the traditionally used Monte Carlo method. The results demonstrate the drastic increase in the computational time of Monte Carlo method when analyzing complex systems with a large number of uncertain parameters while the intrusive PCE provides better accuracy with much less computation complexity.

UNCERTAINTY QUANTIFICATION IN STOCHASTIC SYSTEMS USING POLYNOMIAL CHAOS EXPANSION

2010 ◽  
Vol 02 (02) ◽  
pp. 305-353 ◽  
Author(s):  
K. SEPAHVAND ◽  
S. MARBURG ◽  
H.-J. HARDTKE
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Simulations ◽  
Uncertainty Quantification ◽  
Stochastic Systems ◽  
Polynomial Chaos ◽  
Random Variables ◽  
Polynomial Chaos Expansion ◽  
Uncertain Parameters ◽  
Chaos Expansion ◽  
Generalized Polynomial

In recent years, extensive research has been reported about a method which is called the generalized polynomial chaos expansion. In contrast to the sampling methods, e.g., Monte Carlo simulations, polynomial chaos expansion is a nonsampling method which represents the uncertain quantities as an expansion including the decomposition of deterministic coefficients and random orthogonal bases. The generalized polynomial chaos expansion uses more orthogonal polynomials as the expansion bases in various random spaces which are not necessarily Gaussian. A general review of uncertainty quantification methods, the theory, the construction method, and various convergence criteria of the polynomial chaos expansion are presented. We apply it to identify the uncertain parameters with predefined probability density functions. The new concepts of optimal and nonoptimal expansions are defined and it demonstrated how we can develop these expansions for random variables belonging to the various random spaces. The calculation of the polynomial coefficients for uncertain parameters by using various procedures, e.g., Galerkin projection, collocation method, and moment method is presented. A comprehensive error and accuracy analysis of the polynomial chaos method is discussed for various random variables and random processes and results are compared with the exact solution or/and Monte Carlo simulations. The method is employed for the basic stochastic differential equation and, as practical application, to solve the stochastic modal analysis of the microsensor quartz fork. We emphasize the accuracy in results and time efficiency of this nonsampling procedure for uncertainty quantification of stochastic systems in comparison with sampling techniques, e.g., Monte Carlo simulation.

An Efficient Application of Polynomial Chaos Expansion for the Dynamic Analysis of Multibody Systems With Uncertainty

2014 ◽  
Author(s):  
Mohammad Poursina
Keyword(s):  
Computational Complexity ◽  
Maximum Degree ◽  
Polynomial Chaos ◽  
Multibody System ◽  
Equations Of Motion ◽  
Polynomial Chaos Expansion ◽  
Stochastic Equations ◽  
Basis Functions ◽  
Chaos Expansion ◽  
Modal Values

In this paper the mathematical framework of an advanced algorithm is presented to efficiently form and solve the equations of motion of a multibody system involving uncertainty in the system parameters and/or the excitations. The uncertainty is introduced to the system through the application of the polynomial chaos expansion. In this scheme, states of the system, nondeterministic parameters, and the constraint loads are expanded using modal values as well as orthogonal basis functions. Computational complexity of the application of traditional methods to solve the stochastic equations of motion of a multibody system drastically grows as a cubic function of the number of the states of the system, uncertain parameters and the maximum degree of the polynomial chosen for the basis function. The presented method forms the equation of motion of the system without forming the entire mass and Jacobian matrices. In this strategy, the stochastic governing equations of motion of each individual body as well as the one associated with the kinematic constraint at the connecting joint are developed in terms of the basis functions and modal coordinates. Then sweeping the system in two passes assembly and disassembly, one can form and solve the stochastic equations of motion. In the assembly pass the non-deterministic equations of motion of the assemblies are obtained. In the disassembly process, these equations are then recursively solved for the modal values of the spatial accelerations and the constraints loads. In the serial and parallel implementations, computational complexity of the method increases as a linear and logarithmic functions of the number of the states of the system, uncertain variables, and the maximum degree of the basis functions used in the expansion.

scholarly journals Performance of non-intrusive uncertainty quantification in the aeroservoelastic simulation of wind turbines

2019 ◽  
Vol 4 (3) ◽  
pp. 397-406 ◽  
Author(s):  
Pietro Bortolotti ◽  
Helena Canet ◽  
Carlo L. Bottasso ◽  
Jaikumar Loganathan
Keyword(s):  
Monte Carlo ◽  
Wind Turbine ◽  
Uncertainty Quantification ◽  
Polynomial Chaos ◽  
Solution Space ◽  
Monte Carlo Sampling ◽  
Polynomial Chaos Expansion ◽  
Offshore Wind ◽  
Offshore Wind Turbine ◽  
Chaos Expansion

Abstract. The present paper characterizes the performance of non-intrusive uncertainty quantification methods for aeroservoelastic wind turbine analysis. Two different methods are considered, namely non-intrusive polynomial chaos expansion and Kriging. Aleatory uncertainties are associated with the wind inflow characteristics and the blade surface state, on account of soiling and/or erosion, and propagated throughout the aeroservoelastic model of a large conceptual offshore wind turbine. Results are compared with a brute-force extensive Monte Carlo sampling, which is used as benchmark. Both methods require at least 1 order of magnitude less simulations than Monte Carlo, with a slight advantage of Kriging over polynomial chaos expansion. The analysis of the solution space clearly indicates the effects of uncertainties and their couplings, and highlights some possible shortcomings of current mostly deterministic approaches based on safety factors.

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