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Dautray, Robert; Lions, J. L.

doi:10.1007/978-3-642-58004-8_2

Cited by 41 publications

(72 citation statements)

References 2 publications

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“…[DL,Chapter XVIII,Sect. 3]) that the parametric operator family {B(σ) : σ ∈ S} defined by the bilinear form b(σ; ·, ·) in (4.9) is a family of isomorphisms from X to Y .…”

Section: Space-time Variational Formulations Of Parabolic State Equatmentioning

confidence: 99%

Analytic Regularity and GPC Approximation for Control Problems Constrained by Linear Parametric Elliptic and Parabolic PDEs

Kunoth¹,

Schwab²

2013

SIAM J. Control Optim.

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Abstract. This paper deals with linear-quadratic optimal control problems constrained by a parametric or stochastic elliptic or parabolic PDE. We address the (difficult) case that the state equation depends on a countable number of parameters i.e., on σ j with j ∈ N, and that the PDE operator may depend non-affinely on the parameters. We consider tracking-type functionals and distributed as well as boundary controls. Building on recent results in [CDS1, CDS2], we show that the state and the control are analytic as functions depending on these parameters σ j . We establish sparsity of generalized polynomial chaos (gpc) expansions of both, state and control, in terms of the stochastic coordinate sequence σ = (σ j ) j≥1 of the random inputs, and prove convergence rates of best N -term truncations of these expansions. Such truncations are the key for subsequent computations since they do not assume that the stochastic input data has a finite expansion. In the follow-up paper [KS], we explain two methods how such best N -term truncations can practically be computed, by greedy-type algorithms as in [SG, G], or by multilevel Monte-Carlo methods as in [KSS]. The sparsity result allows in conjunction with adaptive wavelet Galerkin schemes for sparse, adaptive tensor discretizations of control problems constrained by linear elliptic and parabolic PDEs developed in [DK, GK, K], see [KS].

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“…[DL,Chapter XVIII,Sect. 3]) that the parametric operator family {B(σ) : σ ∈ S} defined by the bilinear form b(σ; ·, ·) in (4.9) is a family of isomorphisms from X to Y .…”

Section: Space-time Variational Formulations Of Parabolic State Equatmentioning

confidence: 99%

Analytic Regularity and GPC Approximation for Control Problems Constrained by Linear Parametric Elliptic and Parabolic PDEs

Kunoth¹,

Schwab²

2013

SIAM J. Control Optim.

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“…However, for transient solutions, we use the second-order, strong stability preserving, Runge-Kutta (SSP-RK2) method [14,39]. As a reminder to the reader, if we write the semi-discrete system (3.5) in the abstract form 6) then the SSP-RK2 method takes the form …”

Section: Description Of the Schemementioning

confidence: 99%

“…We implement the plane-source problem assuming a purely scattering material with σ t = σ a = 1.0, with no source (S = 0) and initial condition 6) where F floor = 5.0 × 10 −5 . Again a small, positive baseline value is used to maintain stability of the scheme, which is otherwise compromised by small errors in the solution of the optimization problem (2.11).…”

Section: Plane Sourcementioning

confidence: 99%

High-order entropy-based closures for linear transport in slab geometry

Hauck

2011

Communications in Mathematical Sciences

107

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Abstract. We compute high-order entropy-based (M N ) models for a linear transport equation on a one-dimensional slab geometry. We simulate two test problems from the literature: the twobeam instability and the plane-source problem. In the former case, we compute solutions for systems up to order N = 6; in the latter, up to N = 15. The most notable outcome of these results is the existence of shocks in the steady-state profiles of the two-beam instability for all odd values of N .Key words. Particle transport, maximum entropy, moment closures.AMS subject classifications. 82C70, 82D75, 94A17. IntroductionIn transport and kinetic theory, moment models are used to reduce the size of the state space required for a kinetic description while still maintaining basic features of a kinetic model. They do so by replacing the velocity component of phase space by a finite number of velocity moments. Moment models are commonly derived using an approximate reconstruction of the kinetic description from these moments. The reconstruction prescribes a closure, i.e. a recipe for expressing the moment model as a closed system of the retained moments. Entropy-based methods specify this reconstruction as the solution to a constrained, convex optimization problem. In many situations the cost functional for the optimization problem is directly related to the kinetic entropy of the system. In other cases it simply enforces physically relevant features. The benefit of the entropy approach is a reduced model which retains fundamental properties from the kinetic formalism not found in traditional moment models-properties such as hyperbolicity, entropy dissipation, and positivity. The main disadvantage of the entropy approach is that, unlike traditional moment models, the entropy-based kinetic reconstruction can rarely be expressed as an analytic function of the given moments. Thus the optimization problem must be solved numerically via the associated dual problem. This can increase computational costs significantly.Recent advances in both analysis and implementation of entropy-based methods have been made in several application areas. For gas dynamics, the formal properties of entropy-based models were elucidated in [26]. However, it is also known [17,20,21,38] that the defining optimization problem in this case is ill-posed. As a result, alternative approaches are currently being pursued which regularize the problem in some suitable fashion; see [15] and references therein. For charge transport in semiconductors, the issue of ill-posedness also exists for the so-called parabolic band approximation [29, p.69]. However, for experimental dispersion relations or for more realistic approximations, like the Kane dispersion relation [3, p.3], the optimization

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“…Therefore, by the extraction theorem of Rellich, we can find a subsequence, say fy n g, of fy n g and find It is verified by the standard manupulation as in Dautray and Lions [8] that the limit z satisfies the initial conditions zð0Þ ¼ y 0 and z 0 ð0Þ ¼ y 1 . Now we use the condition (B2).…”

Section: Temam [15; P 274])mentioning

confidence: 89%

“…The introduction of this V 2 admits us to choose an appropriate space of di¤erential operators according to the types of damping e¤ects, so that our treatment provides a convenient and unified way of obtaining the solutions to the nonlinear control system (1.1), (1.2). The basic results for the free system with Bv ¼ 0, y 0 A V , y 1 A H are established in Ha and Nakagiri [9] due to the variational formulation of Dautray-Lions [8].…”

Section: Introductionmentioning

confidence: 99%

Optimal Control Problems for Nonlinear Hyperbolic Distributed Parameter Systems with Damping Terms

Nakagiri

2004

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Cited by 41 publications

References 2 publications

Analytic Regularity and GPC Approximation for Control Problems Constrained by Linear Parametric Elliptic and Parabolic PDEs

Analytic Regularity and GPC Approximation for Control Problems Constrained by Linear Parametric Elliptic and Parabolic PDEs

High-order entropy-based closures for linear transport in slab geometry

Optimal Control Problems for Nonlinear Hyperbolic Distributed Parameter Systems with Damping Terms

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