A Sparse Grid Stochastic Collocation Upwind Finite Volume Element Method for the Constrained Optimal Control Problem Governed by Random Convection Diffusion Equations

Ge, Liang; Wang, Lianhai; Chang, Yanzhen

doi:10.1007/s10915-018-0713-y

Cited by 3 publications

(4 citation statements)

References 29 publications

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“…We have Ψ(u * ), Ψ(u * N ) ∈ R. Choosing u = u * N in the first and u = u * in the second estimate in (15), and adding the resulting inequalities yields (14). Lemma 6.…”

Section: Exponential Tail and Mean-squared Error Boundsmentioning

confidence: 99%

“…where α > 0, H and U are real Hilbert spaces, and K(ξ) : U → H is a bounded, linear operator and h(ξ) ∈ H for each ξ ∈ Ξ. The control problems governed by affine-linear PDEs with random inputs considered, for example, in [15], [16], [18], [25], [29], [34], and [35] can be formulated as instances of (6). In many of these works, the operator K(ξ) is compact for each ξ ∈ Ξ, the expectation function…”

Section: Introductionmentioning

confidence: 99%

“…Further approaches to discretize the expected value in (6) are, for example, stochastic collocation [15], [28], quasi-Monte Carlo sampling [21], and low-rank tensor approximations [14]. A solution method for ( 1) is (robust) stochastic approximation.…”

Section: Introductionmentioning

confidence: 99%

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Sample Average Approximations of Strongly Convex Stochastic Programs in Hilbert Spaces

Milz

2021

Preprint

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We analyze the tail behavior of solutions to sample average approximations (SAAs) of stochastic programs posed in Hilbert spaces. We require that the integrand be strongly convex with the same convexity parameter for each realization. Combined with a standard condition from the literature on stochastic programming, we establish non-asymptotic exponential tail bounds for the distance between the SAA solutions and the stochastic program's solution, without assuming compactness of the feasible set. Our assumptions are verified on a class of infinite-dimensional optimization problems governed by affine-linear partial differential equations with random inputs.

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“…We have Ψ(u * ), Ψ(u * N ) ∈ R. Choosing u = u * N in the first and u = u * in the second estimate in (15), and adding the resulting inequalities yields (14). Lemma 6.…”

Section: Exponential Tail and Mean-squared Error Boundsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Sample Average Approximations of Strongly Convex Stochastic Programs in Hilbert Spaces

Milz

2021

Preprint

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“…We also assume that the following condition holds [8]: b 2 1 + b 2 2 ≤ 4(1 − γ)ac for some γ ∈ (0, 1). In recent years, numerous numerical methods have been widely applied to various optimal control problems governed by partial differential equations; see, e.g., [5,21,22] for standard finite element methods, [3,16,17] for mixed finite element methods, [11,18] for finite volume methods, and [6,9] for spectral methods. Chen and Liu [2] first used Raviart-Thomas mixed finite element method to solve a class of elliptic optimal control problems, in which objective functional contains the gradient of the state variable.…”

Section: H Chenmentioning

confidence: 99%

A new two-grid $P_0^2$-$P_1$ mixed finite element algorithm for general elliptic optimal control problems

2022

J. Nonlinear Funct. Anal.

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In this paper, a new two-grid mixed finite element scheme for distributed optimal control governed by general elliptic equations is presented. P 2 0 -P 1 mixed finite elements and piecewise constant functions are used for spatial discretization. Convergence of the proposed two-grid algorithm is discussed. In the two-grid scheme, the solution of the elliptic optimal control problem on a fine grid is reduced to the solution of the elliptic optimal control problem on a much coarser grid and the solution of a symmetric linear algebraic system on the fine grid and the resulting solution still maintains an asymptotically optimal accuracy.

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Sample average approximations of strongly convex stochastic programs in Hilbert spaces

Milz

2022

Optim Lett

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We analyze the tail behavior of solutions to sample average approximations (SAAs) of stochastic programs posed in Hilbert spaces. We require that the integrand be strongly convex with the same convexity parameter for each realization. Combined with a standard condition from the literature on stochastic programming, we establish non-asymptotic exponential tail bounds for the distance between the SAA solutions and the stochastic program’s solution, without assuming compactness of the feasible set. Our assumptions are verified on a class of infinite-dimensional optimization problems governed by affine-linear partial differential equations with random inputs. We present numerical results illustrating our theoretical findings.

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A Sparse Grid Stochastic Collocation Upwind Finite Volume Element Method for the Constrained Optimal Control Problem Governed by Random Convection Diffusion Equations

Cited by 3 publications

References 29 publications

Sample Average Approximations of Strongly Convex Stochastic Programs in Hilbert Spaces

Sample Average Approximations of Strongly Convex Stochastic Programs in Hilbert Spaces

A new two-grid $P_0^2$-$P_1$ mixed finite element algorithm for general elliptic optimal control problems

Sample average approximations of strongly convex stochastic programs in Hilbert spaces

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