An efficient duality-based approach for PDE-constrained sparse optimization

In this paper, elliptic optimal control problems involving the $L^1$-control cost ($L^1$-EOCP) is considered. To numerically discretize $L^1$-EOCP, the standard piecewise linear finite element is employed. However, different from the finite dimensional $l^1$-regularization optimization, the resulting discrete $L^1$-norm does not have a decoupled form. A common approach to overcome this difficulty is employing a nodal quadrature formula to approximately discretize the $L^1$-norm. It is clear that this technique will incur an additional error. To avoid the additional error, solving $L^1$-EOCP via its dual, which can be reformulated as a multi-block unconstrained convex composite minimization problem, is considered. Motivated by the success of the accelerated block coordinate descent (ABCD) method for solving large scale convex minimization problems in finite dimensional space, we consider extending this method to $L^1$-EOCP. Hence, an efficient inexact ABCD method is introduced for solving $L^1$-EOCP. The design of this method combines an inexact 2-block majorized ABCD and the recent advances in the inexact symmetric Gauss-Seidel (sGS) technique for solving a multi-block convex composite quadratic programming whose objective contains a nonsmooth term involving only the first block. The proposed algorithm (called sGS-imABCD) is illustrated at two numerical examples. Numerical results not only confirm the finite element error estimates, but also show that our proposed algorithm is more efficient than (a) the ihADMM (inexact heterogeneous alternating direction method of multipliers), (b) the APG (accelerated proximal gradient) method

Error Estimates for Sparse Optimal Control Problems by Piecewise Linear Finite Element Approximation

Optimization problems with $L^1$-control cost functional subject to an elliptic partial differential equation (PDE) are considered. However, different from the finite dimensional $l^1$-regularization optimization, the resulting discretized $L^1$-norm does not have a decoupled form when the standard piecewise linear finite element is employed to discretize the continuous problem. A common approach to overcome this difficulty is employing a nodal quadrature formula to approximately discretize the $L^1$-norm. It is inevitable that this technique will incur an additional error. Different from the traditional approach, a duality-based approach and an accelerated block coordinate descent (ABCD) method is introduced to solve this type of problem via its dual. Based on the discretized dual problem, a new discretized scheme for the $L^1$-norm is presented. Compared new discretized scheme for $L^1$-norm with the nodal quadrature formula, the advantages of our new discretized scheme can be demonstrated in terms of the approximation order. More importantly, finite element error estimates results for the primal problem with the new discretized scheme for the $L^1$-norm are provided, which confirm that this approximation scheme will not change the order of error estimates.Comment: arXiv admin note: substantial text overlap with arXiv:1709.00005, arXiv:1708.0909

Taylor approximation and variance reduction for PDE-constrained optimal control under uncertainty

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, 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. 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

An FE-dABCD algorithm for elliptic optimal control problems with constraints on the gradient of the state and control

In this paper, elliptic control problems with integral constraint on the gradient of the state and box constraints on the control are considered. The optimal conditions of the problem are proved. To numerically solve the problem, we use the 'First discretize, then optimize' approach. Specifically, we discretize both the state and the control by piecewise linear functions. To solve the discretized problem efficiently, we first transform it into a multi-block unconstrained convex optimization problem via its dual, then we extend the inexact majorized accelerating block coordinate descent (imABCD) algorithm to solve it. The entire algorithm framework is called finite element duality-based inexact majorized accelerating block coordinate descent (FE-dABCD) algorithm. Thanks to the inexactness of the FE-dABCD algorithm, each subproblems are allowed to be solved inexactly. For the smooth subproblem, we use the generalized minimal residual (GMRES) method with preconditioner to slove it. For the nonsmooth subproblems, one of them has a closed form solution through introducing appropriate proximal term, another is solved combining semi-smooth Newton (SSN) method. Based on these efficient strategies, we prove that our proposed FE-dABCD algorithm enjoys $O(\frac{1}{k^2})$ iteration complexity. Some numerical experiments are done and the numerical results show the efficiency of the FE-dABCD algorithm.Comment: 24 page

Mean-variance risk-averse optimal control of systems governed by PDEs with random parameter fields using quadratic approximations

We present a method for optimal control of systems governed by partial differential equations (PDEs) with uncertain parameter fields. We consider an objective function that involves the mean and variance of the control objective, leading to a risk-averse optimal control problem. To make the problem tractable, we invoke a quadratic Taylor series approximation of the control objective with respect to the uncertain parameter. This enables deriving explicit expressions for the mean and variance of the control objective in terms of its gradients and Hessians with respect to the uncertain parameter. The risk-averse optimal control problem is then formulated as a PDE-constrained optimization problem with constraints given by the forward and adjoint PDEs defining these gradients and Hessians. The expressions for the mean and variance of the control objective under the quadratic approximation involve the trace of the (preconditioned) Hessian and are thus prohibitive to evaluate. To address this, we employ trace estimators that only require a modest number of Hessian-vector products. We illustrate our approach with two problems: the control of a semilinear elliptic PDE with an uncertain boundary source term, and the control of a linear elliptic PDE with an uncertain coefficient field. For the latter problem, we derive adjoint-based expressions for efficient computation of the gradient of the risk-averse objective with respect to the controls. Our method ensures that the cost of computing the risk-averse objective and its gradient with respect to the control, measured in the number of PDE solves, is independent of the (discretized) parameter and control dimensions, and depends only on the number of random vectors employed in the trace estimation. Finally, we present a comprehensive numerical study of an optimal control problem for fluid flow in a porous medium with uncertain permeability field.Comment: 27 pages. Minor revisions. Accepted for publication in SIAM/ASA Journal on Uncertainty Quantificatio

A difference of convex functions approach for sparse pde optimal control problems with nonconvex costs

We propose a local regularization of elliptic optimal control problems which involves the nonconvex $L^q$ fractional penalizations in the cost function. The proposed \emph{Huber type} regularization allows us to formulate the PDE constrained optimization formulation as a DC programming problem (difference of convex functions) that is useful to obtain necessary optimality conditions and tackle its numerical solution by applying the well known DC algorithm used in nonconvex optimization problems. By this procedure we approximate the original problem in terms of a consistent family of parameterized problems for which there are efficient numerical methods available. Finally, we present numerical experiments to illustrate our theory with different configurations associated to the parameters of the problem

A practical factorization of a Schur complement for PDE-constrained Distributed Optimal Control

A distributed optimal control problem with the constraint of a linear elliptic partial differential equation is considered. A necessary optimality condition for this problem forms a saddle point system, the efficient and accurate solution of which is crucial. A new factorization of the Schur complement for such a system is proposed and its characteristics discussed. The factorization introduces two complex factors that are complex conjugate to each other. The proposed solution methodology involves the application of a parallel linear domain decomposition solver---FETI-DPH---for the solution of the subproblems with the complex factors. Numerical properties of FETI-DPH in this context are demonstrated, including numerical and parallel scalability and regularization dependence. The new factorization can be used to solve Schur complement systems arising in both range-space and full-space formulations. In both cases, numerical results indicate that the complex factorization is promising

Adaptive finite element methods for sparse PDE-constrained optimization

We propose and analyze reliable and efficient a posteriori error estimators for an optimal control problem that involves a nondifferentiable cost functional, the Poisson problem as state equation and control constraints. To approximate the solution to the state and adjoint equations we consider a piecewise linear finite element method whereas three different strategies are used to approximate the control variable: piecewise constant discretization, piecewise linear discretization and the so-called variational discretization approach. For the first two aforementioned solution techniques we devise an error estimator that can be decomposed as the sum of four contributions: two contributions that account for the discretization of the control variable and the associated subgradient, and two contributions related to the discretization of the state and adjoint equations. The error estimator for the variational discretization approach is decomposed only in two contributions that are related to the discretization of the state and adjoint equations. On the basis of the devised a posteriori error estimators, we design simple adaptive strategies that yield optimal rates of convergence for the numerical examples that we perform

Mesh Independence of a Majorized ABCD Method for Sparse PDE-constrained Optimization Problems

A majorized accelerated block coordinate descent (mABCD) method in Hilbert space is analyzed to solve a sparse PDE-constrained optimization problem via its dual. The finite element approximation method is investigated. The attractive $O(1/k^2)$ iteration complexity of {the mABCD} method for the dual objective function values can be achieved. Based on the convergence result, we prove the robustness with respect to the mesh size $h$ for the mABCD method by establishing that asymptotically the infinite dimensional ABCD method and finite dimensional discretizations have the same convergence property, and the number of iterations of mABCD method remains almost constant as the discretization is refined.Comment: arXiv admin note: substantial text overlap with arXiv:1709.00005, arXiv:1708.09094, arXiv:1709.0953

Data-driven approximations of dynamical systems operators for control

The Koopman and Perron Frobenius transport operators are fundamentally changing how we approach dynamical systems, providing linear representations for even strongly nonlinear dynamics. Although there is tremendous potential benefit of such a linear representation for estimation and control, transport operators are infinite-dimensional, making them difficult to work with numerically. Obtaining low-dimensional matrix approximations of these operators is paramount for applications, and the dynamic mode decomposition has quickly become a standard numerical algorithm to approximate the Koopman operator. Related methods have seen rapid development, due to a combination of an increasing abundance of data and the extensibility of DMD based on its simple framing in terms of linear algebra. In this chapter, we review key innovations in the data-driven characterization of transport operators for control, providing a high-level and unified perspective. We emphasize important recent developments around sparsity and control, and discuss emerging methods in big data and machine learning.Comment: 37 pages, 4 figure