
A priori and a posteriori error analysis for the Nitsche's method of a reduced Landaude Gennes problem
The equilibrium configurations of a two dimensional planar bistable nema...
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Bayesian Filtering for ODEs with Bounded Derivatives
Recently there has been increasing interest in probabilistic solvers for...
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The Viterbi process, decayconvexity and parallelized maximum aposteriori estimation
The Viterbi process is the limiting maximum aposteriori estimate of the...
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Probabilistic Solutions To Ordinary Differential Equations As NonLinear Bayesian Filtering: A New Perspective
We formulate probabilistic numerical approximations to solutions of ordi...
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LSSVR as a Bayesian RBF network
We show the theoretical equivalence between the Least Squares Support Ve...
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On Bayesian posterior mean estimators in imaging sciences and HamiltonJacobi Partial Differential Equations
Variational and Bayesian methods are two approaches that have been widel...
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Convergence analysis of multilevel spectral deferred corrections
The spectral deferred correction (SDC) method is class of iterative solv...
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Bayesian ODE Solvers: The Maximum A Posteriori Estimate
It has recently been established that the numerical solution of ordinary differential equations can be posed as a nonlinear Bayesian inference problem, which can be approximately solved via Gaussian filtering and smoothing, whenever a Gauss–Markov prior is used. In this paper the class of ν times differentiable linear time invariant Gauss–Markov priors is considered. A taxonomy of Gaussian estimators is established, with the maximum a posteriori estimate at the top of the hierarchy, which can be computed with the iterated extended Kalman smoother. The remaining three classes are termed explicit, semiimplicit, and implicit, which are in similarity with the classical notions corresponding to conditions on the vector field, under which the filter update produces a local maximum a posteriori estimate. The maximum a posteriori estimate corresponds to an optimal interpolant in the reproducing Hilbert space associated with the prior, which in the present case is equivalent to a Sobolev space of smoothness ν+1. Consequently, using methods from scattered data approximation and nonlinear analysis in Sobolev spaces, it is shown that the maximum a posteriori estimate converges to the true solution at a polynomial rate in the filldistance (maximum step size) subject to mild conditions on the vector field. The methodology developed provides a novel and more natural approach to study the convergence of these estimators than classical methods of convergence analysis. The methods and theoretical results are demonstrated in numerical examples.
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