Abstract
We verify strong rates of convergence for a time-implicit, finite-element based space-time discretization of the backward stochastic heat equation, and the forward-backward stochastic heat equation from stochastic optimal control. The fully discrete version of the forward-backward stochastic heat equation is then used within a gradient descent algorithm to approximately solve the linear-quadratic control problem for the stochastic heat equation driven by additive noise. This work is thus giving a theoretical foundation for the computational findings in Dunst and Prohl, SIAM J. Sci. Comput. 38 (2016) A2725–A2755.
Highlights
Let D ⊂ Rd be a bounded domain with C2 boundary, T > 0, and a function X ≡ {X(t); t ∈ [0, T ]} ∈ C([0, T ]; H10 ∩ H2) be given
We consider problem SLQ as a prototype example of a stochastic optimal control problem involving a stochastic PDE, for which corresponding numerical analyses so far are rare in the existing literature; see e.g. [15, 35]. This is in contrast to the deterministic counterpart problem LQ which involves a linear PDE, where optimal rates of convergence are available for space-time discretization of related optimality conditions, which may be used as part of a gradient descent algorithm with step size control [21] to approximate the minimizing tuple (X∗, U ∗), which here consists of deterministic state and control functions
To obtain a convergence rate for the space-time discretization of (1.2), we assume that X0 ∈ H10 ∩ H2, that U ∈ L2F Ω; L2(0, T ; H10), and σ ∈ L∞ F 0, T ; L2(Ω; H10 ∩ H2)
Summary
Let D ⊂ Rd be a bounded domain with C2 boundary, T > 0, and a (deterministic) function X ≡ {X(t); t ∈ [0, T ]} ∈ C([0, T ]; H10 ∩ H2) be given. [15, 35] This is in contrast to the deterministic counterpart problem LQ which involves a linear PDE, where optimal rates of convergence are available for (finite-element based) space-time discretization of related optimality conditions We use Malliavin calculus and variational analysis as technical tools to exploit inherent stability properties of timeimplicit finite-element based discretizations to validate strong error estimates for (the space-time discretization of) both problems, where both discretization parameters enter additively, and no coupling terms arise. The second goal in this work is addressed, where strong error estimates for a space-time discretization (4.15)–(4.16) of the coupled forward-backward SPDE (1.2)-(1.3) (FBSPDE for short) are shown: its derivation starts
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