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We use second-order Taylor's expansions of the reduced objective function w.r.t. uncertain parameters, allowing us to compute the expected value of the quadratic function explicitly. The objective function of the approximated min-max problem separates into a trust-region problem and a semidefinite program. We construct smoothing functions for the optimal value functions defined by these problems. We prove the existence of optimal solutions for the distributionally robust control problem, and the approximated and smoothed problems, and show that a worst-case distribution exists. For the numerical solution of the approximated problem, we develop a homotopy method that computes a sequence of stationary points of smoothed problems while decreasing smoothing parameters to zero. The adjoint approach is used to compute derivatives of the smoothing functions. Numerical results for two nonlinear optimization problems are presented.<\/jats:p>","DOI":"10.1137\/20m134664x","type":"journal-article","created":{"date-parts":[[2022,5,26]],"date-time":"2022-05-26T14:04:02Z","timestamp":1653573842000},"page":"1410-1435","source":"Crossref","is-referenced-by-count":5,"title":["An Approximation Scheme for Distributionally Robust PDE-Constrained Optimization"],"prefix":"10.1137","volume":"60","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3601-3340","authenticated-orcid":true,"given":"Johannes","family":"Milz","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Michael","family":"Ulbrich","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2022,5,26]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1137\/16M106306X"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1007\/s10444-018-9653-1"},{"key":"atypb3","first-page":"303","author":"M. 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Schulz,\n                      Computational Optimization of Systems Governed by Partial Differential Equations\n                      , Comput. Sci. Eng. 8, SIAM, Philadelphia, 2011,https:\/\/doi.org\/10.1137\/1.9781611972054.","DOI":"10.1137\/1.9781611972054"},{"key":"atypb16","doi-asserted-by":"crossref","unstructured":"V. V. Buldygin and Yu. V. Kozachenko,\n                      Metric Characterization of Random Variables and Random Processes\n                      , Transl. Math. 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