# Girsanov's transformation based variance reduced Monte Carlo simulation schemes for reliability estimation in nonlinear stochastic dynamics

## Abstract

The study considers the problem of simulation based time variant reliability analysis of nonlinear randomly excited dynamical systems. Attention is focused on importance sampling strategies based on the application of Girsanov's transformation method. Controls which minimize the distance function, as in the first order reliability method (FORM), are shown to minimize a bound on the sampling variance of the estimator for the probability of failure. Two schemes based on the application of calculus of variations for selecting control signals are proposed: the first obtains the control force as the solution of a two-point nonlinear boundary value problem, and, the second explores the application of the Volterra series in characterizing the controls. The relative merits of these schemes, vis-à-vis the method based on ideas from the FORM, are discussed. Illustrative examples, involving archetypal single degree of freedom (dof) nonlinear oscillators, and a multi-degree of freedom nonlinear dynamical system, are presented. The credentials of the proposed procedures are established by comparing the solutions with pertinent results from direct Monte Carlo simulations. - Highlights: • The distance minimizing control forces minimize a bound on the sampling variance. • Establishing Girsanov controls via solution of a two-point boundary value problem. • Girsanov controls viamore »

- Authors:

- Publication Date:

- OSTI Identifier:
- 22622308

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Computational Physics; Journal Volume: 341; Other Information: Copyright (c) 2017 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICAL METHODS AND COMPUTING; BOUNDARY-VALUE PROBLEMS; COMPARATIVE EVALUATIONS; COMPUTERIZED SIMULATION; DEGREES OF FREEDOM; FAILURES; MATHEMATICAL SOLUTIONS; MONTE CARLO METHOD; NONLINEAR PROBLEMS; OSCILLATORS; PROBABILITY; RANDOMNESS; REDUCTION; RELIABILITY; SAMPLING; STOCHASTIC PROCESSES; TRANSFER FUNCTIONS; TRANSFORMATIONS

### Citation Formats

```
Kanjilal, Oindrila, E-mail: oindrila@civil.iisc.ernet.in, and Manohar, C.S., E-mail: manohar@civil.iisc.ernet.in.
```*Girsanov's transformation based variance reduced Monte Carlo simulation schemes for reliability estimation in nonlinear stochastic dynamics*. United States: N. p., 2017.
Web. doi:10.1016/J.JCP.2017.03.047.

```
Kanjilal, Oindrila, E-mail: oindrila@civil.iisc.ernet.in, & Manohar, C.S., E-mail: manohar@civil.iisc.ernet.in.
```*Girsanov's transformation based variance reduced Monte Carlo simulation schemes for reliability estimation in nonlinear stochastic dynamics*. United States. doi:10.1016/J.JCP.2017.03.047.

```
Kanjilal, Oindrila, E-mail: oindrila@civil.iisc.ernet.in, and Manohar, C.S., E-mail: manohar@civil.iisc.ernet.in. Sat .
"Girsanov's transformation based variance reduced Monte Carlo simulation schemes for reliability estimation in nonlinear stochastic dynamics". United States.
doi:10.1016/J.JCP.2017.03.047.
```

```
@article{osti_22622308,
```

title = {Girsanov's transformation based variance reduced Monte Carlo simulation schemes for reliability estimation in nonlinear stochastic dynamics},

author = {Kanjilal, Oindrila, E-mail: oindrila@civil.iisc.ernet.in and Manohar, C.S., E-mail: manohar@civil.iisc.ernet.in},

abstractNote = {The study considers the problem of simulation based time variant reliability analysis of nonlinear randomly excited dynamical systems. Attention is focused on importance sampling strategies based on the application of Girsanov's transformation method. Controls which minimize the distance function, as in the first order reliability method (FORM), are shown to minimize a bound on the sampling variance of the estimator for the probability of failure. Two schemes based on the application of calculus of variations for selecting control signals are proposed: the first obtains the control force as the solution of a two-point nonlinear boundary value problem, and, the second explores the application of the Volterra series in characterizing the controls. The relative merits of these schemes, vis-à-vis the method based on ideas from the FORM, are discussed. Illustrative examples, involving archetypal single degree of freedom (dof) nonlinear oscillators, and a multi-degree of freedom nonlinear dynamical system, are presented. The credentials of the proposed procedures are established by comparing the solutions with pertinent results from direct Monte Carlo simulations. - Highlights: • The distance minimizing control forces minimize a bound on the sampling variance. • Establishing Girsanov controls via solution of a two-point boundary value problem. • Girsanov controls via Volterra's series representation for the transfer functions.},

doi = {10.1016/J.JCP.2017.03.047},

journal = {Journal of Computational Physics},

number = ,

volume = 341,

place = {United States},

year = {Sat Jul 15 00:00:00 EDT 2017},

month = {Sat Jul 15 00:00:00 EDT 2017}

}