Abstract. The numerical solution of optimization problems governed by partial differential equations (PDEs) with random coefficients is computationally challenging because of the large number of deterministic PDE solves required at each optimization iteration. This paper introduces an efficient algorithm for solving such problems based on a combination of adaptive sparse-grid collocation for the discretization of the PDE in the stochastic space and a trust-region framework for optimization and fidelity management of the stochastic discretization. The overall algorithm adapts the collocation points based on the progress of the optimization algorithm and the impact of the random variables on the solution of the optimization problem. It frequently uses few collocation points initially and increases the number of collocation points only as necessary, thereby keeping the number of deterministic PDE solves low while guaranteeing convergence. Currently an error indicator is used to estimate gradient errors due to adaptive stochastic collocation. The algorithm is applied to three examples, and the numerical results demonstrate a significant reduction in the total number of PDE solves required to obtain an optimal solution when compared with a Newton conjugate gradient algorithm applied to a fixed high-fidelity discretization of the optimization problem.

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This paper improves the trust-region algorithm with adaptive sparse grids introduced in [SIAM J. Sci. Comput., 35 (2013), pp. A1847-A1879] for the solution of optimization problems governed by partial differential equations (PDEs) with uncertain coefficients. The previous algorithm used adaptive sparse-grid discretizations to generate models that are applied in a trust-region framework to generate a trial step. The decision whether to accept this trial step as the new iterate, however, required relatively high-fidelity adaptive discretizations of the objective function. In this paper, we extend the algorithm and convergence theory to allow the use of low-fidelity adaptive sparse-grid models in objective function evaluations. This is accomplished by extending conditions on inexact function evaluations used in previous trust-region frameworks. Our algorithm adaptively builds two separate sparse grids: one to generate optimization models for the step computation and one to approximate the objective function. These adapted sparse grids often contain significantly fewer points than the high-fidelity grids, which leads to a dramatic reduction in the computational cost. This is demonstrated numerically using two examples. Moreover, the numerical results indicate that the new algorithm rapidly identifies the stochastic variables that are relevant to obtaining an accurate optimal solution. When the number of such variables is independent of the dimension of the stochastic space, the algorithm exhibits near dimension-independent behavior. Introduction.The solution of large-scale optimization problems in science and engineering must accommodate model uncertainties, such as unknown material properties and boundary conditions. The coupling of traditional optimization methods with uncertainty quantification faces significant computational challenges due to the potentially large number of stochastic variables. To address this issue, we have developed an algorithm that shows promising results for optimization problems governed by partial differential equations (PDEs) with random coefficients [18]. This algorithm uses a trust-region framework to manage models that are based on sparse grids. In [18] we use adaptive sparse grids to compute the optimization step and a fixed high-fidelity sparse grid to determine whether to accept the step. As dis-

Incomplete malaria control efforts have resulted in a worldwide increase in resistance to drugs used to treat the disease. A complex array of mutations underlying antimalarial drug resistance complicates efficient monitoring of parasite populations and limits the success of malaria control efforts in regions of endemicity. To improve the surveillance of Plasmodium falciparum drug resistance, we developed a multiplex ligase detection reaction-fluorescent-microsphere-based assay (LDR-FMA) that identifies single nucleotide polymorphisms (SNPs) in the P. falciparum dhfr (9 alleles), dhps (10 alleles), and pfcrt (3 alleles) genes associated with resistance to Fansidar and chloroquine. We evaluated 1,121 blood samples from study participants in the Wosera region of Papua New Guinea, where malaria is endemic. Results showed that 468 samples were P. falciparum negative and 453 samples were P. falciparum positive by a Plasmodium species assay and all three gene assays (concordance, 82.2%). Plasmodium falciparum strains exhibit resistance to many antimalarial drugs in most regions of the world where malaria is endemic. In some regions, individual strains are resistant to more than one drug. With a very limited arsenal of safe and effective antimalarial drugs, complex genetic factors contributing to drug resistance pose a constant challenge to efforts to control this important human parasite. Moreover, it has been observed that as drug-resistant P. falciparum evolves and spreads within regions of endemicity and resistant strains become predominant, both transmission and the morbidity and mortality attributable to malaria increase (29). With increasing travel around the world, drug-resistant malaria parasites present further challenges in prescribing effective prophylactic treatment for tourists, military personnel, and humanitarian aid workers. Additionally, infected travelers are likely to increase the exchange of parasite strains between regions where different patterns of drug resistance are observed.Molecular genetic studies of P. falciparum have enabled identification of a number of specific mutations in genes linked to resistance to specific antimalarial drugs (34, 35). These include genes encoding the P. falciparum chloroquine resistance transporter (pfcrt) (10), dihydrofolate reductase (dhfr) (7, 24), and dihydropterate synthetase (dhps) (30, 31), which confer resistance to chloroquine, pyrimethamine, and sulfadoxine, respectively. Mutations in the latter two genes confer resistance to the drug combinations Fansidar (pyrimethaminesulfadoxine) and LAPDAP (chlorproguanil-dapsone) (22, 36). Our interest in monitoring these genes for single nucleotide polymorphisms (SNPs) associated with drug resistance was based on a number of technical and field surveillance objectives. Whereas numerous PCR-based approaches have been used to analyze polymorphisms in the P. falciparum dhps, dhfr, and pfcrt genes, most strategies involve post-PCR restriction fragment length polymorphism or DNA probe hybridization methods that are cumbe...

This work introduces a new method to efficiently solve optimization problems constrained by partial differential equations (PDEs) with uncertain coefficients. The method leverages two sources of inexactness that trade accuracy for speed: (1) stochastic collocation based on dimension-adaptive sparse grids (SGs), which approximates the stochastic objective function with a limited number of quadrature nodes, and (2) projection-based reduced-order models (ROMs), which generate efficient approximations to PDE solutions. These two sources of inexactness lead to inexact objective function and gradient evaluations, which are managed by a trust-region method that guarantees global convergence by adaptively refining the sparse grid and reduced-order model until a proposed error indicator drops below a tolerance specified by trust-region convergence theory. A key feature of the proposed method is that the error indicator-which accounts for errors incurred by both the sparse grid and reduced-order model-must be only an asymptotic error bound, i.e., a bound that holds up to an arbitrary constant that need not be computed. This enables the method to be applicable to a wide range of problems, including those where sharp, computable error bounds are not available; this distinguishes the proposed method from previous works. Numerical experiments performed on a model problem from optimal flow control under uncertainty verify global convergence of the method and demonstrate the method's ability to outperform previously proposed alternatives.

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Abstract. In this work, we apply the MG/OPT framework to a multilevel-in-sample-space discretization of optimization problems governed by PDEs with uncertain coefficients. The MG/OPT algorithm is a template for the application of multigrid to deterministic PDE optimization problems. We employ MG/OPT to exploit the hierarchical structure of sparse grids in order to formulate a multilevel stochastic collocation algorithm. The algorithm is provably first-order convergent under standard assumptions on the hierarchy of discretized objective functions as well as on the optimization routines used as pre-and postsmoothers. We present explicit bounds on the total number of PDE solves and an upper bound on the error for one V-cycle of the MG/OPT algorithm applied to a linear quadratic control problem. We provide numerical results that confirm the theoretical bound on the number of PDE solves and show a dramatic reduction in the total number of PDE solves required to solve these optimization problems when compared with standard optimization routines applied to a fixed sparse-grid discretization of the same problem.

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