Bayesian inverse problems for hyperbolic equations
Numerical analysis of Bayesian inverse problems for hyperbolic partial differential equations is analysed in this report. Inverse problems involve constructing a mathematical model while only given limited information on the solution. Furthermore, these information are usually affected by errors cau...
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2020
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sg-ntu-dr.10356-1390602023-02-28T23:19:22Z Bayesian inverse problems for hyperbolic equations Ng, Jeremy Hoang Viet Ha School of Physical and Mathematical Sciences vhhoang@ntu.edu.sg Science::Mathematics::Applied mathematics::Numerical analysis Numerical analysis of Bayesian inverse problems for hyperbolic partial differential equations is analysed in this report. Inverse problems involve constructing a mathematical model while only given limited information on the solution. Furthermore, these information are usually affected by errors caused by the noisy environment. Using classical methods, the inverse problems are typically ill-posed. To make the problems well-posed, a regularising term has to be chosen. However, by treating the error as a random variable, the Bayesian approach guarantees that the problem is well-posed. Given an observation filled with noise which follows a known probability distribution, we seek to find the posterior measure on the coefficient space. We use Markov Chain Monte Carlo method to sample the posterior expectation of a quantity of interest. The forward hyperbolic equation is solved numerically by the finite element method. We analyse the error estimate for the posterior expectation due to MCMC and finite element approximation. Numerical examples confirm the theoretical result. Most of the results in this report are new. Bachelor of Science in Mathematical Sciences 2020-05-15T04:09:17Z 2020-05-15T04:09:17Z 2020 Final Year Project (FYP) https://hdl.handle.net/10356/139060 en application/pdf Nanyang Technological University |
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Science::Mathematics::Applied mathematics::Numerical analysis Ng, Jeremy Bayesian inverse problems for hyperbolic equations |
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Numerical analysis of Bayesian inverse problems for hyperbolic partial differential equations is analysed in this report. Inverse problems involve constructing a mathematical model while only given limited information on the solution. Furthermore, these information are usually affected by errors caused by the noisy environment. Using classical methods, the inverse problems are typically ill-posed. To make the problems well-posed, a regularising term has to be chosen. However, by treating the error as a random variable, the Bayesian approach guarantees that the problem is well-posed. Given an observation filled with noise which follows a known probability distribution, we seek to find the posterior measure on the coefficient space. We use Markov Chain Monte Carlo method to sample the posterior expectation of a quantity of interest. The forward hyperbolic equation is solved numerically by the finite element method. We analyse the error estimate for the posterior expectation due to MCMC and finite element approximation. Numerical examples confirm the theoretical result. Most of the results in this report are new. |
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Hoang Viet Ha |
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Hoang Viet Ha Ng, Jeremy |
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Final Year Project |
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Ng, Jeremy |
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Ng, Jeremy |
title |
Bayesian inverse problems for hyperbolic equations |
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Bayesian inverse problems for hyperbolic equations |
title_full |
Bayesian inverse problems for hyperbolic equations |
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Bayesian inverse problems for hyperbolic equations |
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Bayesian inverse problems for hyperbolic equations |
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bayesian inverse problems for hyperbolic equations |
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Nanyang Technological University |
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2020 |
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https://hdl.handle.net/10356/139060 |
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