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A Low-Rank Solver for Parameter Estimation and Uncertainty Quantification in Time-Dependent Systems of Partial Differential Equations

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Abstract

In this work we propose a low-rank solver in view of performing parameter estimation and uncertainty quantification in systems of partial differential equations. The solution approximation is sought in a space-parameter separated form. The discretisation in the parameter direction is made evolve in time through a Markov Chain Monte Carlo method. The resulting method is a Bayesian sequential estimation of the parameters. The computational burden is mitigated by the introduction of an efficient interpolator, based on a reduced basis built by exploiting the low-rank solves. The method is tested on four different applications.

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Data Availability

The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

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Acknowledgements

The first author acknowledges support by Inria through the PEPR “SantéNumérique”. The third author acknowledges support from ANR Grant ADAPT 18-CE46-0001.

Funding

The first author was financially supported by Inria through the PEPR Santé Numérique program.

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Authors and Affiliations

  1. Sorbonne Université, Inria and CNRS, UMR 7598 Laboratoire Jacques-Louis Lions, Paris, France

    Sébastien Riffaud, Miguel A. Fernández & Damiano Lombardi

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  1. Sébastien Riffaud
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  2. Miguel A. Fernández
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  3. Damiano Lombardi
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Correspondence to Miguel A. Fernández.

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Appendices

SVD of a Second-Order Tensor

Algorithm 2
figure b

SVD of a second-order tensor

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Truncated Preconditioned GMRES Algorithm

Algorithm 3
figure c

Truncated preconditioned GMRES algorithm

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In order to limit memory footprint, Algorithm 3 can also be restarted using the final iterate as initial guess if the maximum number of iterations is reached without converging.

Brand’s Algorithm

Algorithm 4
figure d

Brand’s algorithm [64]

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Riffaud, S., Fernández, M.A. & Lombardi, D. A Low-Rank Solver for Parameter Estimation and Uncertainty Quantification in Time-Dependent Systems of Partial Differential Equations. J Sci Comput 99, 34 (2024). https://doi.org/10.1007/s10915-024-02488-3

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