Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Robust Optimization of PDEs with Random Coefficients Using a Multilevel
  Monte Carlo Method   This paper addresses optimization problems constrained by partial
differential equations with uncertain coefficients. In particular, the robust
control problem and the average control problem are considered for a tracking
type cost functional with an additional penalty on the variance of the state.
The expressions for the gradient and Hessian corresponding to either problem
contain expected value operators. Due to the large number of uncertainties
considered in our model, we suggest to evaluate these expectations using a
multilevel Monte Carlo (MLMC) method. Under mild assumptions, it is shown that
this results in the gradient and Hessian corresponding to the MLMC estimator of
the original cost functional. Furthermore, we show that the use of certain
correlated samples yields a reduction in the total number of samples required.
Two optimization methods are investigated: the nonlinear conjugate gradient
method and the Newton method. For both, a specific algorithm is provided that
dynamically decides which and how many samples should be taken in each
iteration. The cost of the optimization up to some specified tolerance $\tau$
is shown to be proportional to the cost of a gradient evaluation with requested
root mean square error $\tau$. The algorithms are tested on a model elliptic
diffusion problem with lognormal diffusion coefficient. An additional nonlinear
term is also considered.

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The categories of this paper are: hep-ph
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The categories of this paper are: math-ph
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The categories of this paper are: hep-th
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The categories of this paper are: nucl-ex
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The categories of this paper are: dg-ga
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The categories of this paper are: solv-int
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The categories of this paper are: q-fin
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The categories of this paper are: astro-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: chao-dyn
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The categories of this paper are: adap-org
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The categories of this paper are: ao-sci
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The categories of this paper are: atom-ph
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The categories of this paper are: physics
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The categories of this paper are: plasm-ph
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The categories of this paper are: patt-sol