Abstract. We consider abstract non-negative self-adjoint operators on L 2 (X) which satisfy the finite speed propagation property for the corresponding wave equation. For such operators we introduce a restriction type condition which in the case of the standard Laplace operator is equivalent to (p, 2) restriction estimate of Stein and Tomas. Next we show that in the considered abstract setting our restriction type condition implies sharp spectral multipliers and endpoint estimates for the BochnerRiesz summability. We also observe that this restriction estimate holds for operators satisfying dispersive or Strichartz estimates. We obtain new spectral multiplier results for several second order differential operators and recover some known results. Our examples include Schrödinger operators with inverse square potentials on R n , the harmonic oscillator, elliptic operators on compact manifolds and Schrödinger operators on asymptotically conic manifolds.

In this work we develop a scalable computational framework for the solution of PDE-constrained optimal control under high-dimensional uncertainty. Specifically, we consider a mean-variance formulation of the control objective and employ a Taylor expansion with respect to the uncertain parameter either to directly approximate the control objective or as a control variate for variance reduction. The expressions for the mean and variance of the Taylor approximation are known analytically, although their evaluation requires efficient computation of the trace of the (preconditioned) Hessian of the control objective. We propose to estimate this trace by solving a generalized eigenvalue problem using a randomized algorithm that only requires the action of the Hessian on a small number of random directions. Then, the computational work does not depend on the nominal dimension of the uncertain parameter, but depends only on the effective dimension (i.e., the rank of the preconditioned Hessian), thus ensuring scalability to high-dimensional problems. Moreover, to increase the estimation accuracy of the mean and variance of the control objective by the Taylor approximation, we use it as a control variate for variance reduction, which results in considerable computational savings (several orders of magnitude) compared to a plain Monte Carlo method. In summary, our approach amounts to solving an optimal control constrained by the original PDE and a set of linearized PDEs, which arise from the computation of the gradient and Hessian of the control objective with respect to the uncertain parameter. We use the Lagrangian formalism to derive expressions for the gradient with respect to the control and apply a gradient-based optimization method to solve the problem. We demonstrate the accuracy, efficiency, and scalability of the proposed computational method for two examples with high-dimensional uncertain parameters: subsurface flow in a porous medium modeled as an elliptic PDE, and turbulent jet flow modeled by the Reynolds-averaged Navier-Stokes equations coupled with a nonlinear advection-diffusion equation characterizing model uncertainty. In particular, for the latter more challenging example we show scalability of our algorithm up to one million parameters resulting from discretization of the uncertain parameter field.

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Abstract. We prove weighted restriction type estimates for Grushin operators. These estimates are then used to prove sharp spectral multiplier theorems as well as Bochner-Riesz summability results with sharp exponent.