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A dynamic bi-orthogonal field equation approach to efficient Bayesian inversion

Tagade P. M., Choi H. L.

International Journal of Applied Mathematics and Computer Science

2017

Vol. 27, no. 2

229--243

This paper proposes a novel computationally efficient stochastic spectral projection based approach to Bayesian inversion of a computer simulator with high dimensional parametric and model structure uncertainty. The proposed method is based on the decomposition of the solution into its mean and a random field using a generic Karhunen–Loève expansion. The random field is represented as a convolution of separable Hilbert spaces in stochastic and spatial dimensions that are spectrally represented using respective orthogonal bases. In particular, the present paper investigates generalized polynomial chaos bases for the stochastic dimension and eigenfunction bases for the spatial dimension. Dynamic orthogonality is used to derive closed-form equations for the time evolution of mean, spatial and the stochastic fields. The resultant system of equations consists of a partial differential equation (PDE) that defines the dynamic evolution of the mean, a set of PDEs to define the time evolution of eigenfunction bases, while a set of ordinary differential equations (ODEs) define dynamics of the stochastic field. This system of dynamic evolution equations efficiently propagates the prior parametric uncertainty to the system response. The resulting bi-orthogonal expansion of the system response is used to reformulate the Bayesian inference for efficient exploration of the posterior distribution. The efficacy of the proposed method is investigated for calibration of a 2D transient diffusion simulator with an uncertain source location and diffusivity. The computational efficiency of the method is demonstrated against a Monte Carlo method and a generalized polynomial chaos approach.

Inverse heat transfer problems: an application to bioheat transfer

Rojczyk M., Orlande H. R. B., Colaço M. J., Szczygieł I., Nowak A. J., Białecki R. A., Ostrowski Z.

Computer Assisted Methods in Engineering and Science

2015

Vol. 22, no. 4

365--383

In this work, we applied the Markov chain Monte Carlo (MCMC) method for the estimation of parameters appearing in the Pennes’ formulation of the bioheat transfer equation. The inverse problem of parameter estimation was solved with the simulated transient temperature measurements. A one-dimensional (1D) test case was used to explore the capabilities of using the MCMC method in bioheat transfer problems, specifically for the detection of skin tumors by using surface temperature measurements. The analysis of the sensitivity coefficients was performed in order to examine linear dependence and low sensitivity of the model parameters. The solution of the direct problem was verified with a commercial code. The results obtained in this work show the ability of using inverse heat transfer analysis for the detection of skin tumors.