PDE-constrained Optimization Research Articles

Multiobjective PDE-constrained optimization using the reduced-basis method

In this paper the reduced basis (RB) method is applied to solve quadratic multiobjective optimal control problems governed by linear parametrized variational equations. These problems often arise in applications, where the quality of the system behavior has to be measured by more than one criterium. The weighted sum method is exploited for defining scalar-valued linear-quadratic optimal control problems built by introducing additional optimization parameters. The optimal controls corresponding to specific choices of the optimization parameters are efficiently computed by the RB method. The accuracy is guaranteed by an a-posteriori error estimate. An effective sensitivity analysis allows to further reduce the computational times for identifying a suitable and representative set of optimal controls.

On economic-technological optimization of high-voltage electric cables

In this paper, mathematical modelling of high voltage cables for power transmission line design is presented. The Finite Volume Method (FVM) is used to approximate the developed mathematical model (a system of nonlinear multi-physic differential equations) and OpenFOAM (Open source Field Operation And Manipulation) tool is used to implement the obtained parallel finite volume schemes. In order to optimize the design of power lines with respect to technological parameters, different cases of nonstationary load dynamics are investigated and the influence of system nonlinearity and external day, month and years periodical boundary conditions and the source function regimes are simulated. The main aim of this paper is to include into the mathematical model also economic requirements and to optimize sizes of cables with respect to both technologic and economic requirements. Numerical algorithms targeted to solve PDE-constrained optimization problems are developed. Results of computational experiments are presented.

Towards adjoint-based inversion of time-dependent mantle convection with non-linear viscosity

We develop and study an adjoint-based inversion method for the simultaneous recovery of initial temperature conditions and viscosity parameters in time-dependent mantle convection from the current mantle temperature and historic plate motion. Based on a realistic rheological model with temperature-dependent and strain-rate-dependent viscosity, we formulate the inversion as a PDE-constrained optimization problem. The objective functional includes the misfit of surface velocity (plate motion) history, the misfit of the current mantle temperature, and a regularization for the uncertain initial condition. The gradient of this functional with respect to the initial temperature and the uncertain viscosity parameters is computed by solving the adjoint of the mantle convection equations. This gradient is used in a pre-conditioned quasi-Newton minimization algorithm. We study the prospects and limitations of the inversion, as well as the computational performance of the method using two synthetic problems, a sinking cylinder and a realistic subduction model. The subduction model is characterized by the migration of a ridge toward a trench whereby both plate motions and subduction evolve. The results demonstrate: (1) for known viscosity parameters, the initial temperature can be well recovered, as in previous initial condition-only inversions where the effective viscosity was given; (2) for known initial temperature, viscosity parameters can be recovered accurately, despite the existence of trade-offs due to ill-conditioning; (3) for the joint inversion of initial condition and viscosity parameters, initial condition and effective viscosity can be reasonably recovered, but the high dimension of the parameter space and the resulting ill-posedness may limit recovery of viscosity parameters.

Adaptive eigenspace method for inverse scattering problems in the frequency domain

A nonlinear optimization method is proposed for the solution of inverse scattering problems in the frequency domain, when the scattered field is governed by the Helmholtz equation. The time-harmonic inverse medium problem is formulated as a PDE-constrained optimization problem and solved by an inexact truncated Newton-type iteration. Instead of a grid-based discrete representation, the unknown wave speed is projected to a particular finite-dimensional basis of eigenfunctions, which is iteratively adapted during the optimization. Truncating the adaptive eigenspace (AE) basis at a (small and slowly increasing) finite number of eigenfunctions effectively introduces regularization into the inversion and thus avoids the need for standard Tikhonov-type regularization. Both analytical and numerical evidence underpins the accuracy of the AE representation. Numerical experiments demonstrate the efficiency and robustness to missing or noisy data of the resulting adaptive eigenspace inversion method.

A Certified Trust Region Reduced Basis Approach to PDE-Constrained Optimization

Parameter optimization problems constrained by partial differential equations (PDEs) appear in many science and engineering applications. Solving these optimization problems may require a prohibitively large number of computationally expensive PDE solves, especially if the dimension of the design space is large. It is therefore advantageous to replace expensive high-dimensional PDE solvers (e.g., finite element) with lower-dimensional surrogate models. In this paper, the reduced basis (RB) model reduction method is used in conjunction with a trust region optimization framework to accelerate PDE-constrained parameter optimization. Novel a posteriori error bounds on the RB cost and cost gradient for quadratic cost functionals (e.g., least squares) are presented and used to guarantee convergence to the optimum of the high-fidelity model. The proposed certified RB trust region approach uses high-fidelity solves to update the RB model only if the approximation is no longer sufficiently accurate, reducing the number of full-fidelity solves required. We consider problems governed by elliptic and parabolic PDEs and present numerical results for a thermal fin model problem in which we are able to reduce the number of full solves necessary for the optimization by up to 86%.

A Measure Approximation for Distributionally Robust PDE-Constrained Optimization Problems

In numerous applications, scientists and engineers acquire varied forms of data that partially characterize the inputs to an underlying physical system. This data is then used to inform decisions such as controls and designs. Consequently, it is critical that the resulting control or design is robust to the inherent uncertainties associated with the unknown probabilistic characterization of the model inputs. In this paper, we consider optimal control and design problems constrained by partial differential equations with uncertain inputs. We do not assume a known probabilistic model for the inputs, but rather we formulate the problem as a distributionally robust optimization problem where the outer minimization problem determines the control or design, while the inner maximization problem determines the worst-case probability measure that matches desired characteristics of the data. We analyze the inner maximization problem in the space of measures and introduce a novel measure approximation technique, bas...

Optimal Control of Non-Smooth Hyperbolic Evolution Maxwell Equations in Type-II Superconductivity

We analyze the optimal control of an electromagnetic process in type-II superconductivity. The PDE-constrained optimization problem is to find an optimal applied current density, which steers the electromagnetic fields to the desired ones in the presence of a type-II superconductor. The governing PDE system for the electromagnetic fields consists of hyperbolic evolution Maxwell equations with a nonlinear and nonsmooth constitutive law for the electric field and the current density based on the Bean critical-state model. Through the use of the Maxwell theory, the semigroup theory, Helmholtz decomposition, and results on maximal monotone operators, we develop a mathematical theory including an existence analysis and first-order necessary optimality conditions for the nonsmooth PDE-constrained optimization problem.

POD-based error control for reduced-order bicriterial PDE-constrained optimization

In the present paper, a bicriterial optimal control problem governed by an abstract evolution problem and bilateral control constraints is considered. To compute Pareto optimal points and the Pareto front numerically, the (Euclidean) reference point method is applied, where many scalar constrained optimization problems have to be solved. For this reason a reduced-order approach based on proper orthogonal decomposition (POD) is utilized. An a-posteriori error analysis ensures a desired accuracy for the Pareto optimal points and for the Pareto front computed by the POD method. Numerical experiments for evolution problems with convection-diffusion illustrate the efficiency of the presented approach.

Low-Rank Solution to an Optimization Problem Constrained by the Navier--Stokes Equations

The numerical solution of PDE-constrained optimization problems subject to the nonstationary Navier--Stokes equation is a challenging task. While space-time approaches often show favorable convergence properties, they often suffer from storage problems. Here we propose to approximate the solution to the optimization problem in a low-rank form, which is similar to the model order reduction (MOR) approach. However, in contrast to classical MOR schemes we do not compress the full solution at the end of the algorithm but start our algorithm with low-rank data and maintain this form throughout the iteration. Numerical experiments indicate that this approach reduces the computational costs by two orders of magnitude.

Robust preconditioners for PDE-constrained optimization with limited observations

Regularization robust preconditioners for PDE-constrained optimization problems have been successfully developed. These methods, however, typically assume observation data and control throughout the entire domain of the state equation. For many inverse problems, this is an unrealistic assumption. In this paper we propose and analyze preconditioners for PDE-constrained optimization problems with limited observation data, e.g. observations are only available at the boundary of the solution domain. Our methods are robust with respect to both the regularization parameter and the mesh size. That is, the condition number of the preconditioned optimality system is uniformly bounded, independently of the size of these two parameters. The method does, however, require extra regularity. We first consider a prototypical elliptic control problem and thereafter more general PDE-constrained optimization problems. Our theoretical findings are illuminated by several numerical results.

Trajectory piecewise quadratic reduced-order model for subsurface flow, with application to PDE-constrained optimization

A new reduced-order model based on trajectory piecewise quadratic (TPWQ) approximations and proper orthogonal decomposition (POD) is introduced and applied for subsurface oil–water flow simulation. The method extends existing techniques based on trajectory piecewise linear (TPWL) approximations by incorporating second-derivative terms into the reduced-order treatment. Both the linear and quadratic reduced-order methods, referred to as POD-TPWL and POD-TPWQ, entail the representation of new solutions as expansions around previously simulated high-fidelity (full-order) training solutions, along with POD-based projection into a low-dimensional space. POD-TPWQ entails significantly more offline preprocessing than POD-TPWL as it requires generating and projecting several third-order (Hessian-type) terms. The POD-TPWQ method is implemented for two-dimensional systems. Extensive numerical results demonstrate that it provides consistently better accuracy than POD-TPWL, with speedups of about two orders of magnitude relative to high-fidelity simulations for the problems considered. We demonstrate that POD-TPWQ can be used as an error estimator for POD-TPWL, which motivates the development of a trust-region-based optimization framework. This procedure uses POD-TPWL for fast function evaluations and a POD-TPWQ error estimator to determine when retraining, which entails a high-fidelity simulation, is required. Optimization results for an oil–water problem demonstrate the substantial speedups that can be achieved relative to optimizations based on high-fidelity simulation.

A note on preconditioners for complex linear systems arising from PDE-constrained optimization problems

In this note, the block-diagonal preconditioner proposed and the block triangular proposed in Krendl et al. (2013) and Pearson and Wathen (2012), respectively, are further studied and optimized. Two improved preconditioners are proposed for solving a class of complex linear systems arising from optimal control with time-periodic parabolic equation. Theoretical analyses show that the eigenvalues of the improved block-diagonal and block triangular preconditioned matrices are located in [ − 1 , − 2 / 2 ) ∪ ( 2 / 2 , 1 ] and ( 1 / 2 , 1 ] , respectively. Numerical experiments illustrate the feasibility and effectiveness of these preconditioners for Krylov subspace methods.

Fast iterative solvers for large matrix systems arising from time-dependent Stokes control problems

In this manuscript we consider the development of fast iterative solvers for Stokes control problems, an important class of PDE-constrained optimization problems. In particular we wish to develop effective preconditioners for the matrix systems arising from finite element discretizations of time-dependent variants of such problems. To do this we consider a suitable rearrangement of the matrix systems, and exploit the saddle point structure of many of the relevant sub-matrices involved – we may then use this to construct representations of these sub-matrices based on good approximations of their (1,1)-block and Schur complement. We test our recommended iterative methods on a distributed control problem with Dirichlet boundary conditions, and on a time-periodic problem.

Comparison of preconditioned Krylov subspace iteration methods for PDE-constrained optimization problems

The governing dynamics of fluid flow is stated as a system of partial differential equations referred to as the Navier-Stokes system. In industrial and scientific applications, fluid flow control becomes an optimization problem where the governing partial differential equations of the fluid flow are stated as constraints. When discretized, the optimal control of the Navier-Stokes equations leads to large sparse saddle point systems in two levels. In this paper, we consider distributed optimal control for the Stokes system and test the particular case when the arising linear system can be compressed after eliminating the control function. In that case, a system arises in a form which enables the application of an efficient block matrix preconditioner that previously has been applied to solve complex-valued systems in real arithmetic. Under certain conditions, the condition number of the so preconditioned matrix is bounded by 2. The numerical and computational efficiency of the method in terms of number of iterations and execution time is favorably compared with other published methods.

Stability of Saddle Points Via Explicit Coderivatives of Pointwise Subdifferentials

We derive stability criteria for saddle points of a class of nonsmooth optimization problems in Hilbert spaces arising in PDE-constrained optimization, using metric regularity of infinite-dimensional set-valued mappings. A main ingredient is an explicit pointwise characterization of the regular coderivative of the subdifferential of convex integral functionals. This is applied to several stability properties for parameter identification problems for an elliptic partial differential equation with non-differentiable data fitting terms.

Comparison of preconditioned Krylov subspace iteration methods for PDE-constrained optimization problems

Comparison of preconditioned Krylov subspace iteration methods for PDE-constrained optimization problems : Poisson and convection–diffusion control

The split Bregman algorithm applied to PDE-constrained optimization problems with total variation regularization

We derive an efficient solution method for ill-posed PDE-constrained optimization problems with total variation regularization. This regularization technique allows discontinuous solutions, which is desirable in many applications. Our approach is to adapt the split Bregman technique to handle such PDE-constrained optimization problems. This leads to an iterative scheme where we must solve a linear saddle point problem in each iteration. We prove that the spectra of the corresponding saddle point operators are almost contained in three bounded intervals, not containing zero, with a very limited number of isolated eigenvalues. Krylov subspace methods handle such operators very well and thus provide an efficient algorithm. In fact, we can guarantee that the number of iterations needed cannot grow faster than $$O([\ln (\alpha ^{-1})]^2)$$O([ln(ź-1)]2) as $$\alpha \rightarrow 0$$źź0, where $$\alpha $$ź is a small regularization parameter. Moreover, in our numerical experiments we demonstrate that one can expect iteration numbers of order $$O(\ln (\alpha ^{-1}))$$O(ln(ź-1)).

Risk-Averse PDE-Constrained Optimization Using the Conditional Value-At-Risk

Uncertainty is inevitable when solving science and engineering application problems. In the face of uncertainty, it is essential to determine robust and risk-averse solutions. In this work, we consider a class of PDE-constrained optimization problems in which the PDE coefficients and inputs may be uncertain. We introduce two approximations for minimizing the conditional value-at-risk (CVaR) for such PDE-constrained optimization problems. These approximations are based on the primal and dual formulations of CVaR. For the primal problem, we introduce a smooth approximation of CVaR in order to utilize derivative-based optimization algorithms and to take advantage of the convergence properties of quadrature-based discretizations. For this smoothed CVaR, we prove differentiability as well as consistency of our approximation. For the dual problem, we regularize the inner maximization problem, rigorously derive optimality conditions, and demonstrate the consistency of our approximation. Furthermore, we propose a...

PDE-Constrained Optimization with Local Control and Boundary Observations: Robust Preconditioners

We consider PDE-constrained optimization problems with control functions defined on a subregion of the domain of the state equation. The main purpose of this paper is to define and analyze robust preconditioners for KKT systems associated with such optimization tasks, that is, preconditioners that lead to iteration bounds, for the MINRES scheme, that are independent of the regularization parameter $\alpha$ and the mesh size $h$. Our analysis addresses elliptic control problems, subject to Tikhonov regularization, and covers cases with boundary observations only and locally defined control functions. A number of numerical experiments are presented. We also explain how the results can be adapted to an inverse problem arising in connection with electrocardiography.

A Posteriori Error Estimate for a PDE-Constrained Optimization Formulation for the Flow in DFNs

Flows in fractured media have been modeled with many different approaches in order to get reliable and efficient simulations for many critical applications. The common issues to be tackled are the wide range of scales involved in the phenomenon, the complexity of the domain, and the huge computational cost. In this paper we introduce residual-based “a posteriori” error estimates for a formulation of the flow in the discrete fracture networks based on a constrained optimization approach (see Berrone, Pieraccini, and Scialò [SIAM J. Sci. Comput., 35 (2013), pp. B487--B510], [SIAM J. Sci. Comput., 35 (2013), pp. A908--A935], [J. Comput. Phys., 256 (2014), pp. 838--853]), suitable to overcome all the difficulties related to a good quality mesh generation with conformity requirement.