Computational Mathematics and Applications Seminar
Upcoming seminars:
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Thu, 24/05 14:00 |
Dr Elias Jarlebring (KTH Stockholm) |
Computational Mathematics and Applications  |
Gibson Grd floor SR |
| The Arnoldi method for standard eigenvalue problems possesses several attractive properties making it robust, reliable and efficient for many problems. We will present here a new algorithm equivalent to the Arnoldi method, but designed for nonlinear eigenvalue problems corresponding to the problem associated with a matrix depending on a parameter in a nonlinear but analytic way. As a first result we show that the reciprocal eigenvalues of an infinite dimensional operator. We consider the Arnoldi method for this and show that with a particular choice of starting function and a particular choice of scalar product, the structure of the operator can be exploited in a very effective way. The structure of the operator is such that when the Arnoldi method is started with a constant function, the iterates will be polynomials. For a large class of NEPs, we show that we can carry out the infinite dimensional Arnoldi algorithm for the operator in arithmetic based on standard linear algebra operations on vectors and matrices of finite size. This is achieved by representing the polynomials by vector coefficients. The resulting algorithm is by construction such that it is completely equivalent to the standard Arnoldi method and also inherits many of its attractive properties, which are illustrated with examples. | |||
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Thu, 31/05 14:00 |
Dr David Kay (University of Oxford) |
Computational Mathematics and Applications |
Gibson Grd floor SR |
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Thu, 07/06 14:00 |
Dr Chris Farmer (University of Oxford) |
Computational Mathematics and Applications |
Rutherford Appleton Laboratory, nr Didcot |
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Uncertainty quantification can begin by specifying the initial state of a system as a probability measure. Part of the state (the 'parameters') might not evolve, and might not be directly observable. Many inverse problems are generalisations of uncertainty quantification such that one modifies the probability measure to be consistent with measurements, a forward model and the initial measure. The inverse problem, interpreted as computing the posterior probability measure of the states, including the parameters and the variables, from a sequence of noise-corrupted observations, is reviewed in the talk. Bayesian statistics provides a natural framework for a solution but leads to very challenging computational problems, particularly when the dimension of the state space is very large, as when arising from the discretisation of a partial differential equation theory.
In this talk we show how the Bayesian framework leads to a new algorithm - the 'Variational Smoothing Filter' - that unifies the leading techniques in use today. In particular the framework provides an interpretation and generalisation of Tikhonov regularisation, a method of forecast verification and a way of quantifying and managing uncertainty. To deal with the problem that a good initial prior may not be Gaussian, as with a general prior intended to describe, for example a geological structure, a Gaussian mixture prior is used. This has many desirable properties, including ease of sampling to make 'numerical rocks' or 'numerical weather' for visualisation purposes and statistical summaries, and in principle can approximate any probability density. Robustness is sought by combining a variational update with this full mixture representation of the conditional posterior density. |
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Thu, 14/06 14:00 |
Dr Christoph Reisinger (University of Oxford) |
Computational Mathematics and Applications |
Gibson Grd floor SR |
If no seminars are displayed this means no further seminars will be held during the current term. Please check again for our new seminar programme nearer the start of next term.