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- Agapiou, Sergios, et al.
(författare)
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Posterior consistency of the Bayesian approach to linear ill-posed inverse problems
- 2012
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Ingår i: arXiv:1203.5753v2 [math.ST]. ; :1203.5753, s. 1-30
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- We consider a Bayesian nonparametric approach to a family of linear inverse problems in a separable Hilbert space setting, with Gaussian prior and noise distribution. A method of identifying the posterior distribution using its precision operator is presented. Work- ing with the unbounded precision operator enables us to use partial differential equations (PDE) methodology to study posterior con- sistency in a frequentist sense, and in particular to obtain rates of contraction of the posterior distribution to a Dirac measure centered on the true solution. We show how these rates may be optimized by a choice of the scale parameter in the prior covariance operator. Our methods assume a relatively weak relation between the prior covariance operator, the forward operator and the noise covariance operator; more precisely, we assume that appropriate powers of these operators induce equivalent norms. We compare our results to known minimax rates of convergence in the case where the forward operator and the prior and noise covariances are all simultaneously diagonaliz- able, and confirm that the PDE method provides the same rates for a wide range of parameters. An elliptic PDE inverse problem is used to illustrate the power of the general theory.
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| 3. |
- Almgren, Torgny, 1962, et al.
(författare)
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The opportunistic replacement problem: analysis and case studies
- 2011
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- We consider an optimization model for determining optimal opportunistic maintenance (that is, component replacement) schedules when data is deterministic. This problem generalizes that of Dickman, Epstein, and Wilamowsky [21] and is a natural starting point for the modelling of replacement schedules when component lives are non-deterministic. We show that this basic opportunistic replacement problem is NP-hard. We show that the convex hull of the set of feasible replacement schedules is full-dimensional, and that all the necessary inequalities also are facet-inducing. We show that when maintenance occasions are fixed, the remaining problem can be stated as a linear program; when maintenance costs are monotone with time, the latter is solvable through a greedy procedure. Results from a series of case studies performed in the areas of aircraft engine and wind turbine maintenance are also reported. These illustrate the advantages of utilizing opportunistic maintenance activities based on a complete optimization model, as compared to simpler policies.
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| 4. |
- Andersson, Adam, 1979, et al.
(författare)
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Duality in refined Watanabe-Sobolev spaces and weak approximations of SPDE
- 2013
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- In this paper we introduce a new family of refined Watanabe- Sobolev spaces that capture in a fine way integrability in time of the Malliavin derivative. We consider duality in these spaces and derive a Burkholder type inequality in a dual norm. The theory we develop allows us to prove weak convergence with essentially optimal rate for numerical approximations in space and time of semilinear parabolic stochastic evolution equations driven by Gaussian additive noise. In particular, we combine Galerkin finite element methods with a backward Euler scheme in time. The method of proof does not rely on the use of the Kolmogorov equation or the It¯o formula and is therefore in nature non-Markovian. With this method polynomial growth test functions with mild smoothness assumptions are allowed, meaning in particular that we prove convergence of arbitrary moments with essentially optimal rate. Our Gronwall argument also yields weak error estimates which are uniform in time without any additional effort.
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| 5. |
- Andersson, Adam, 1979, et al.
(författare)
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Weak convergence for a spatial approximation of the nonlinear stochastic heat equation
- 2012
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- We find the weak rate of convergence of approximate solutions of the nonlinear stochastic heat equation, when discretized in space by a standard finite element method. Both multiplicative and additive noise is considered under different assumptions. This extends an earlier result of Debussche in which time discretization is considered for the stochastic heat equation perturbed by white noise. It is known that this equation only has a solution in one space dimension. In order to get results for higher dimensions, colored noise is considered here, besides the white noise case where considerably weaker assumptions on the noise term is needed. Integration by parts in the Malliavin sense is used in the proof. The rate of weak convergence is, as expected, essentially twice the rate of strong convergence.
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