Efficient Solution of Large-Scale Algebraic Riccati Equations Associated with Index-2 DAEs via the Inexact Low-Rank Newton-ADI Method

This paper extends the algorithm of Benner, Heinkenschloss, Saak, and Weichelt: An inexact low-rank Newton-ADI method for large-scale algebraic Riccati equations, Applied Numerical Mathematics Vol.~108 (2016), pp.~125--142, doi:10.1016/j.apnum.2016.05.006 to Riccati equations associated with Hessenberg index-2 Differential Algebratic Equation (DAE) systems. Such DAE systems arise, e.g., from semi-discretized, linearized (around steady state) Navier-Stokes equations. The solution of the associated Riccati equation is important, e.g., to compute feedback laws that stabilize the Navier-Stokes equations. Challenges in the numerical solution of the Riccati equation arise from the large-scale of the underlying systems and the algebraic constraint in the DAE system. These challenges are met by a careful extension of the inexact low-rank Newton-ADI method to the case of DAE systems. A main ingredient in the extension to the DAE case is the projection onto the manifold described by the algebraic constraints. In the algorithm, the equations are never explicitly projected, but the projection is only applied as needed. Numerical experience indicates that the algorithmic choices for the control of inexactness and line-search can help avoid subproblems with matrices that are only marginally stable. The performance of the algorithm is illustrated on a large-scale Riccati equation associated with the stabilization of Navier-Stokes flow around a cylinder.Comment: 21 pages, 2 figures, 4 table

A fast and accurate domain-decomposition nonlinear manifold reduced order model

This paper integrates nonlinear-manifold reduced order models (NM-ROMs) with domain decomposition (DD). NM-ROMs approximate the FOM state in a nonlinear-manifold by training a shallow, sparse autoencoder using FOM snapshot data. These NM-ROMs can be advantageous over linear-subspace ROMs (LS-ROMs) for problems with slowly decaying Kolmogorov $n$-width. However, the number of NM-ROM parameters that need to trained scales with the size of the FOM. Moreover, for "extreme-scale" problems, the storage of high-dimensional FOM snapshots alone can make ROM training expensive. To alleviate the training cost, this paper applies DD to the FOM, computes NM-ROMs on each subdomain, and couples them to obtain a global NM-ROM. This approach has several advantages: Subdomain NM-ROMs can be trained in parallel, each involve fewer parameters to be trained than global NM-ROMs, require smaller subdomain FOM dimensional training data, and training of subdomain NM-ROMs can tailor them to subdomain-specific features of the FOM. The shallow, sparse architecture of the autoencoder used in each subdomain NM-ROM allows application of hyper-reduction (HR), reducing the complexity caused by nonlinearity and yielding computational speedup of the NM-ROM. This paper provides the first application of NM-ROM (with HR) to a DD problem. In particular, it details an algebraic DD formulation of the FOM, trains a NM-ROM with HR for each subdomain, and develops a sequential quadratic programming (SQP) solver to evaluate the coupled global NM-ROM. Theoretical convergence results for the SQP method and a priori and a posteriori error estimates for the DD NM-ROM with HR are provided. The proposed DD NM-ROM with HR approach is numerically compared to a DD LS-ROM with HR on 2D steady-state Burgers' equation, showing an order of magnitude improvement in accuracy of the proposed DD NM-ROM over the DD LS-ROM

Local Error Analysis of Discontinuous Galerkin Methods for Advection-Dominated Elliptic Linear-Quadratic Optimal Control Problems

This paper analyzes the local properties of the symmetric interior penalty upwind discontinuous Galerkin (SIPG) method for the numerical solution of optimal control problems governed by linear reaction-advection-diffusion equations with distributed controls. The theoretical and numerical results presented in this paper show that for advection-dominated problems the convergence properties of the SIPG discretization can be superior to the convergence properties of stabilized finite element discretizations such as the streamline upwind Petrov Galerkin (SUPG) method. For example, we show that for a small diffusion parameter the SIPG method is optimal in the interior of the domain. This is in sharp contrast to SUPG discretizations, for which it is known that the existence of boundary layers can pollute the numerical solution of optimal control problems everywhere even into domains where the solution is smooth and, as a consequence, in general reduces the convergence rates to only first order. In order to prove the nice convergence properties of the SIPG discretization for optimal control problems, we first improve local error estimates of the SIPG discretization for single advection-dominated equations by showing that the size of the numerical boundary layer is controlled not by the mesh size but rather by the size of the diffusion parameter. As a result, for small diffusion, the boundary layers are too “weak” to pollute the SIPG solution into domains of smoothness in optimal control problems. This favorable property of the SIPG method is due to the weak treatment of boundary conditions, which is natural for discontinuous Galerkin methods, while for SUPG methods strong imposition of boundary conditions is more conventional. The importance of the weak treatment of boundary conditions for the solution of advection dominated optimal control problems with distributed controls is also supported by our numerical results

Analysis of Inexact Trust-Region Interior-Point SQP Algorithms

. In this paper we analyze inexact trust--region interior--point (TRIP) sequential quadra-- tic programming (SQP) algorithms for the solution of optimization problems with nonlinear equality constraints and simple bound constraints on some of the variables. Such problems arise in many engineering applications, in particular in optimal control problems with bounds on the control. The nonlinear constraints often come from the discretization of partial differential equations. In such cases the calculation of derivative information and the solution of linearized equations is expensive. Often, the solution of linear systems and derivatives are computed inexactly yielding nonzero residuals. This paper analyzes the effect of the inexactness onto the convergence of TRIP SQP and gives practical rules to control the size of the residuals of these inexact calculations. It is shown that if the size of the residuals is of the order of both the size of the constraints and the trust--region radius, t..

Analysis of Inexact Trust-Region SQP Algorithms

In this paper we extend the design of a class of composite–step trust–region SQP methods and their global convergence analysis to allow inexact problem information. The inexact problem information can result from iterative linear systems solves within the trust–region SQP method or from approximations of first–order derivatives. Accuracy requirements in our trust–region SQP methods are adjusted based on feasibility and optimality of the iterates. Our accuracy requirements are stated in general terms, but we show how they can be enforced using information that is already available in matrix–free implementations of SQP methods. In the absence of inexactness our global convergence theory is equal to that of Dennis, El–Alem, Maciel (SIAM J. Optim., 7 (1997), pp. 177–207). If all iterates are feasible, i.e., if all iterates satisfy the equality constraints, then our results are related to the known convergence analyses for trust–region methods with inexact gradient information for unconstrained optimizatio

Shape Optimization of Shell Structure Acoustics

This paper provides a rigorous framework for the numerical solution of shape optimization problems in shell structure acoustics using a reference-domain approach. The structure is modeled with Naghdi shell equations, fully coupled to boundary integral equations on a minimally regular surface, permitting the formulation of three-dimensional radiation and scattering problems on a two-dimensional set of reference coordinates. We prove well-posedness of this model, and Fréchet differentiability of the state with respect to the surface shape. For a class of shape optimization problems we prove existence of optimal solutions under slightly stronger surface regularity assumptions. Finally, adjoint equations are used to efficiently compute derivatives of the radiated field with respect to large numbers of shape parameters, which allows consideration of a rich space of shapes and, thus, of a broad range of design problems. A numerical example is presented to illustrate the applicability of our theoretical results

Troltzsch: Analysis of the Lagrange-SQP-Newton method for the control of a Phase field equation

This paper investigates the local convergence of the Lagrange–SQP–Newton method applied to an optimal control problem governed by a phase field equation with distributed control. The phase field equation is a system of two semilinear parabolic differential equations. Stability analysis of optimization problems and regularity results for parabolic differential equations are used to proof convergence of the controls with respect to the L 2 (Q) norm and with respect to the L 1 (Q) norm. Key words Sequential quadratic programming method, Lagrange–SQP–Newton method, optimal control, phase field equation, control constraints. AMS subject classifications 49M37, 49K20

Formulation and Analysis of a Sequential Quadratic Programming Method for the Optimal Dirichlet Boundary Control of Navier-Stokes Flow

The optimal boundary control of Navier--Stokes flow is formulated as a constrained optimization problem and a sequential quadratic programming (SQP) approach is studied for its solution. Since SQP methods treat states and controls as independent variables and do not insist on satisfying the constraints during the iterations, care must be taken to avoid a possible incompatibility of Dirichlet boundary conditions and incompressibility constraint. In this paper, compatibility is enforced by choosing appropriate function spaces. The resulting optimization problem is analyzed. Differentiability of the constraints and surjectivity of linearized constraints are verified and adjoints are computed. An SQP method is applied to the optimization problem and compared with other approaches