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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

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2017

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Vol. 27, no. 2

229--243

EN

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.

2

Kelvin transform for (…)-harmonic functions in regular domains

Michalik K., Ryznar M.

Demonstratio Mathematica

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2012

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Vol. 45, nr 2

361-376

EN

We investigate conditional stable processes in a Lipschitz domain D and conditional stable processes in the image of D under the Kelvin transform. We show that, with a suitable change of time, these processes are equal in distribution. As an application, we show the equivalence of the Hardy spaces and the relative Fatou theorem for D and its image.

3

Almost sure and moment stability of stochastic partial differential equations

Kwiecińska A. A.

Probability and Mathematical Statistics

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2001

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Vol. 21, Fasc. 2

405--415

EN

We study the almost sure and moment stability of a class of stochastic partial differential equations and we present an infinite-dimensional version of a theorem proved for stochastic ordinary differential equations by Arnold, Oeljeklaus and Pardoux. We also investigate how adding a term with white noise influences the stability of a deterministic system. The outcome is quite surprising. It turns out that regardless whether the deterministic system was stable or unstable, after adding a term with sufficiently large noise, it becomes pathwise exponentially stable and unstable in the p-th mean for p >1.

4

Unicité Trajectorielle des Équations Différentielles Stochastiques Avec Temps Local

Ouknine Y.

Probability and Mathematical Statistics

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1999

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Vol. 19, Fasc. 1

55--62

EN

We study the pathwise uniqueness of a one-dimensional stochastic differential equation driven by white noise and involving local time of the unknown process. We introduce a very weak condition on the diffusion term which is sufficient for the pathwise uniqueness if one considers an equation of the form [formula] where v stands for a signed Radon measure on R.