Adaptive computation of smallest eigenvalues of self‐adjoint elliptic partial differential equations

We consider a new adaptive finite element (AFEM) algorithm for self‐adjoint elliptic PDE eigenvalue problems. In contrast to other approaches we incorporate the inexact solutions of the resulting finite‐dimensional algebraic eigenvalue problems into the adaptation process. In this way we can balance the costs of the adaptive refinement of the mesh with the costs for the iterative eigenvalue method. We present error estimates that incorporate the discretization errors, approximation errors in the eigenvalue solver and roundoff errors, and use these for the adaptation process. We show that it is also possible to restrict to very few iterations of a Krylov subspace solver for the eigenvalue problem on coarse meshes. Several examples are presented to show that this new approach achieves much better complexity than the previous AFEM approaches which assume that the algebraic eigenvalue problem is solved to full accuracy. Copyright © 2010 John Wiley & Sons, Ltd.     

A Priori Error Estimates for Least-Squares Mixed Finite Element Approximation of Elliptic Optimal Control Problems

In this paper, a constrained distributed optimal control problem governed by a first-order elliptic system is considered. Least-squares mixed finite element methods, which are not subject to the Ladyzhenkaya-Babuska-Brezzi consistency condition, are used for solving the elliptic system with two unknown state variables. By adopting the Lagrange multiplier approach, continuous and discrete optimality systems including a primal state equation, an adjoint state equation, and a variational inequality for the optimal control are derived, respectively. Both the discrete state equation and discrete adjoint state equation yield a symmetric and positive definite linear algebraic system. Thus, the popular solvers such as preconditioned conjugate gradient (PCG) and algebraic multi-grid (AMG) can be used for rapid solution. Optimal a priori error estimates are obtained, respectively, for the control function in $L^2(Ω)$-norm, for the original state and adjoint state in $H^1(Ω)$-norm, and for the flux state and adjoint flux state in $H$(div; $Ω$)-norm. Finally, we use one numerical example to validate the theoretical findings.     

Adaptive solution of elliptic PDE-eigenvalue problems.

In this paper we introduce a new adaptive algorithm (AFEMLA) for elliptic PDE-eigenvalue problems. In contrast to other approaches the algebraic eigenvalue problem does not have to be solved to full accuracy. We incorporate the iterative solution of the resulting finite dimensional algebraic eigenvalue problems in the adaptation process in order to balance the cost with the costs for the iterative eigenvalue method. We present error estimates that incorporate the discretization errors, approximation errors in the eigenvalue solver and roundoff errors. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

State Trajectory Compression in Optimal Control

In optimal control problems with nonlinear time-dependent 3D PDEs, the computation of the reduced gradient by adjoint methods requires one solve of the state equation forward in time, and one backward solve of the adjoint equation. Since the state enters into the adjoint equation, the storage of a 4D discretization is necessary. We propose a lossy compression algorithm using a cheap predictor for the state data, with additional entropy coding of prediction errors. Analytical and numerical results indicate that compression factors around 30 can be obtained without exceeding the FE discretization error. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Adjoint equation and Lyapunov regularity for linear stochastic differential algebraic equations of index 1

We introduce a concept of adjoint equation and Lyapunov regularity of a stochastic differential algebraic Equation (SDAE) of index 1. The notion of adjoint SDAE is introduced in a similar way as in the deterministic differential algebraic equation case. We prove a multiplicative ergodic theorem for the adjoint SDAE and the adjoint Lyapunov spectrum. Employing the notion of adjoint equation and Lyapunov spectrum of an SDAE, we are able to define Lyapunov regularity of SDAEs. Some properties and an example of a metal oxide semiconductor field-effect transistor ring oscillator under thermal noise are discussed.     

Lie algebras with an algebraic adjoint representation revisited

A well-known theorem due to E. Zelmanov proves that PI-Lie algebras with an algebraic adjoint representation over a field of characteristic zero are locally finite-dimensional. In particular, a Lie algebra (over a field of characteristic zero) whose adjoint representation is algebraic of bounded degree is locally finite-dimensional. In this paper it is proved that a prime nondegenerate PI-Lie algebra with an algebraic adjoint representation over a field of characteristic zero is simple and finite-dimensional over its centroid, which is an algebraic field extension of the base field. We also give a new and shorter proof of the local finiteness of Lie algebras with an algebraic adjoint representation of bounded degree.     

An adaptive finite element method for the sparse optimal control of fractional diffusion

We propose and analyze an a posteriori error estimator for a partial differential equation (PDE)-constrained optimization problem involving a nondifferentiable cost functional, fractional diffusion, and control-constraints. We realize fractional diffusion as the Dirichlet-to-Neumann map for a nonuniformly PDE and propose an equivalent optimal control problem with a local state equation. For such an equivalent problem, we design an a posteriori error estimator which can be defined as the sum of four contributions: two contributions related to the approximation of the state and adjoint equations and two contributions that account for the discretization of the control variable and its associated subgradient. The contributions related to the discretization of the state and adjoint equations rely on anisotropic error estimators in weighted Sobolev spaces. We prove that the proposed a posteriori error estimator is locally efficient and, under suitable assumptions, reliable. We design an adaptive scheme that yields, for the examples that we perform, optimal experimental rates of convergence.     

Residual-based a posteriori error control for Space-time finite element approximations of parabolic optimal control problems

In this paper we investigate a space-time finite element approximation of parabolic optimal control problems. The first order optimality conditions are transformed into an elliptic equation of fourth order in space and second order in time involving only the state or the adjoint state in the space-time domain. We derive a priori and a posteriori error estimates for the time discretization of the state and the adjoint state. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Explicit One-Step P-Stable Methods for Second Order Periodic Initial Value Problems

In this paper, we present an explicit one-step method for solving periodic initial value problems of second order ordinary differential equations. The method is P-stable, and of first algebraic order and high phase-lag order. To improve the algebraic order, we give a composition second order scheme with the proposed method and its adjoint. We report some numerical results to illustrate the efficiency of our methods.     

Towards Riccati-Feedback Control of Complex Flows with Moving Interfaces

Problems featuring moving interfaces appear in many applications. They can model solidification and melting of pure materials, crystal growth and other multi-phase problems. The control of the moving interface enables to, for example, influence production processes and, thus, the product material quality. We consider the two-phase Stefan problem that models a solid and a liquid phase separated by the moving interface. In the liquid phase, the heat distribution is characterized by a convection-diffusion equation. The fluid flow in the liquid phase is described by the Navier–Stokes equations which introduces a differential algebraic structure to the system. The interface movement is coupled with the temperature through the Stefan condition, which adds additional algebraic constraints. Our formulation uses a sharp interface representation and we define a quadratic tracking-type cost functional as a target of a control input. We compute an open loop optimal control for the Stefan problem using an adjoint system. For a feedback representation, we linearize the system about the trajectory defined by the open loop control. This results in a linear-quadratic regulator problem, for which we formulate the differential Riccati equation with time varying coefficients. This Riccati equation defines the corresponding feedback gain. Further, we present the feedback formulation that takes into account the structure and the differential algebraic components of the problem. Also, we discuss how the complexities that come, for example, with mesh movements, can be handled in a feedback setting. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Algebraic multigrid preconditioning for iterative eigensolvers

The objective is a comparative study of iterative solvers for eigenproblems arising from elliptic and self–adjoint partial differential operators. Typically only a few of the smallest eigenvalues of these problems are to be computed. We discuss various gradient based preconditioned eigensolvers which make use of algebraic multigrid preconditioning. We present some algorithms together with numerical results. Performance characteristics are derived by comparison with the solutions of standard problems. We show that known advantages of algebraic multigrid preconditioning (e.g. for boundary–value problems with large jumps in the coefficients) transfer to AMG–preconditioned eigensolvers. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Goal-oriented anisotropic grid adaptation

Venditti and Darmofal have introduced a grid adaptation strategy for estimating and reducing simulation errors in functional outputs of partial differential equations. The procedure is based on an adjoint formulation in which the estimated error in the functional can be directly related to the local residual errors of both the primal and adjoint solutions. In this note, we propose an extension of this method to the anisotropic case. The strategy proposed for grid adaptation is also compared with the anisotropic Hessian approach, based on the minimization of interpolation error. To cite this article: G. Rogé, L. Martin, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Adjoint concepts for the optimal control of Burgers equation

Adjoint techniques are a common tool in the numerical treatment of optimal control problems. They are used for efficient evaluations of the gradient of the objective in gradient-based optimization algorithms. Different adjoint techniques for the optimal control of Burgers equation with Neumann boundary control are studied. The methods differ in the point in the numerical algorithm at which the adjoints are incorporated. Discretization methods for the continuous adjoint are discussed and compared with methods applying the application of the discrete adjoint. At the example of the implicit Euler method and the Crank Nicolson method it is shown that a discretization for the adjoint problem that is adjoint to the discretized optimal control problem avoids additional errors in gradient-based optimization algorithms. The approach of discrete adjoints coincides with that of automatic differentiation tools (AD) which provide exact gradient calculations on the discrete level.     

A posteriori error estimates for optimal control problems constrained by convection-diffusion equations

We propose a characteristic finite element discretization of evolutionary type convection-diffusion optimal control problems. Nondivergence-free velocity fields and bilateral inequality control constraints are handled. Then some residual type a posteriori error estimates are analyzed for the approximations of the control, the state, and the adjoint state. Based on the derived error estimators, we use them as error indicators in developing efficient multi-set adaptive meshes characteristic finite element algorithm for such optimal control problems. Finally, one numerical example is given to check the feasibility and validity of multi-set adaptive meshes refinements.     

Adaptive Surface Reconstruction Based on Tensor Product Algebraic Splines

Surface reconstruction from unorganized data points is a challenging problem in Computer Aided Design and Geometric Modeling. In this paper, we extend the mathematical model proposed by Juttler and Felis (Adv. Comput. Math., 17 (2002), pp. 135-152) based on tensor product algebraic spline surfaces from fixed meshes to adaptive meshes. We start with a tensor product algebraic B-spline surface defined on an initial mesh to fit the given data based on an optimization approach. By measuring the fitting errors over each cell of the mesh, we recursively insert new knots in cells over which the errors are larger than some given threshold, and construct a new algebraic spline surface to better fit the given data locally. The algorithm terminates when the error over each cell is less than the threshold. We provide some examples to demonstrate our algorithm and compare it with Jiittler's method. Examples suggest that our method is effective and is able to produce reconstruction surfaces of high quality.AMS subject classifications: 65D17     

Adaptive Symmetric Interior Penalty Galerkin (SIPG) method for optimal control of convection diffusion equations with control constraints

In this paper, we study a posteriori error estimates of the upwind symmetric interior penalty Galerkin (SIPG) method for the control constrained optimal control problems governed by linear diffusion–convection–reaction partial differential equations. Residual based error estimators are used for the state, the adjoint and the control. An adaptive mesh refinement indicated by a posteriori error estimates is applied. Numerical examples are presented for convection dominated problems to illustrate the theoretical findings and the effectiveness of the adaptivity.     

Optimal Control of Differential–Algebraic Equations of Higher Index,Part 2: Necessary Optimality Conditions

This paper deals with optimal control problems described by higher index DAEs. We introduce a class of problems which can be transformed to index one control problems. For these problems we show in the accompanying paper that, if the solutions to the adjoint equations are well–defined, then the first-order approximations to the functionals defining the problem can be expressed in terms of the adjoint variables. In this paper we show that the solutions to the adjoint equations are essentially bounded measurable functions. Then, based on the first order approximations, we derive the necessary optimality conditions for the considered class of control problems. These conditions do not require the transformation of the DAEs to index-one system; however, higher-index DAEs and their associated adjoint equations have to be solved.     

Automatic step size and order control in one-step collocation methods with higher derivatives

On the basis of symmetric E-methods with higher derivatives having the convergence order four, six, or eight, implicit extrapolation schemes are constructed for the numerical solution of ordinary differential equations. The combined step size and order control used in these schemes implements an automatic global error control in the extrapolation E-methods, which makes it possible to solve differential problems in automatic mode up to the accuracy specified by the user (without taking into account round-off errors). The theory of adjoint and symmetric methods presented in this paper is an extension of the results that are well known for the conventional Runge-Kutta schemes to methods involving higher derivatives. Since the implicit extrapolation based on multi-stage Runge-Kutta methods can be very time consuming, special emphasis is made on the efficiency of calculations. All the theoretical conclusions of this paper are confirmed by the numerical results obtained for test problems.     

Polynomial cost for solving IVP for high-index DAE

We show that the cost of solving initial value problems for high-index differential algebraic equations is polynomial in the number of digits of accuracy requested. The algorithm analyzed is built on a Taylor series method developed by Pryce for solving a general class of differential algebraic equations. The problem may be fully implicit, of arbitrarily high fixed index and contain derivatives of any order. We give estimates of the residual which are needed to design practical error control algorithms for differential algebraic equations. We show that adaptive meshes are always more efficient than non-adaptive meshes. Finally, we construct sufficiently smooth interpolants of the discrete solution. AMS subject classification (2000) 34A09, 65L80, 68Q25