Robust Optimization Approaches for PDE-Constrained Problems under Uncertainty Stefan Ulbrich Department of Mathematics TU Darmstadt, Germany Joint work with Philip Kolvenbach, Oliver Lass, and Adrian Sichau and in parts with Alessandro Alla, Michael Hinze, Sebastian Sch¨ ops and Herbert De Gersem Supported by DFG within SFB 805 and by BMBF within SIMUROM/PASIROM Nonlinear Optimization Optimization and Inversion under Uncertainty, RICAM, Linz, November 13, 2019 Optimization and Inversion under Uncertainty, RICAM, Linz, 2019 November 13, 2019 | S. Ulbrich | 1
Outline ◮ Robust formulation of PDE-constrained optimization with uncertain data ◮ (First and) second order approximation of the robust counterpart ◮ Equivalent reformulations for second order approximation using optimality or duality theory ◮ Nonsmooth reduced formulation ◮ Update strategy for the expansion point ◮ Invoking reduced order models with error estimation ◮ Application to shape optimization of synchronous motors and for the elastodynamic wave equation ◮ Conclusion and outlook Optimization and Inversion under Uncertainty, RICAM, Linz, 2019 November 13, 2019 | S. Ulbrich | 2
PDE-Constrained Optimization under Uncertainty Uncertain PDE-Constrained Optimization Problem min h 0 ( y , x ; p ) y ∈ Y , x ∈ X s.t. h i ( y , x ; p ) ≤ 0, i ∈ I , (P) C ( y , x ; p ) = 0. ◮ Typically nonconvex, design variables x , state y , uncertain parameters p ◮ h 0 , h i : Y × X × R n p → R , C : Y × X × R n p → Z sufficiently smooth ◮ C ( y , x ; p ) = 0 has a unique solution y = y ( x ; p ) for all relevant x , p ◮ ∂ y C ∈ L ( Y , Z ) is invertible Optimization and Inversion under Uncertainty, RICAM, Linz, 2019 November 13, 2019 | S. Ulbrich | 3
PDE-Constrained Optimization under Uncertainty Uncertain PDE-Constrained Optimization Problem min h 0 ( y , x ; p ) y ∈ Y , x ∈ X s.t. h i ( y , x ; p ) ≤ 0, i ∈ I , (P) C ( y , x ; p ) = 0. ◮ Typically nonconvex, design variables x , state y , uncertain parameters p ◮ h 0 , h i : Y × X × R n p → R , C : Y × X × R n p → Z sufficiently smooth ◮ C ( y , x ; p ) = 0 has a unique solution y = y ( x ; p ) for all relevant x , p ◮ ∂ y C ∈ L ( Y , Z ) is invertible Uncertainty to be considered: ◮ Parameter p is uncertain with p ∈ U p = { p ∈ R n p . . � p − ¯ p � B p ≤ 1 } � v � B . . = ( v T Bv ) 1 / 2 for a symmetric positive definite matrix B ◮ p can also be coefficients in an expansion, e.g. Karhunen-Lo´ eve expansion ◮ Constraint-wise uncertainties also possible ◮ Also possible: Design x uncertain, x ∈ U x = { x ∈ X = R n x . . � x − ¯ x � B x ≤ 1 } Optimization and Inversion under Uncertainty, RICAM, Linz, 2019 November 13, 2019 | S. Ulbrich | 3
Robust Optimization - Basic Idea Uncertain Optimization Problem ˆ min h 0 ( x ; p ) x (Pr) ˆ s.t. h i ( x ; p ) ≤ 0, i ∈ I . Assumption: Parameter p is uncertain. We only know that p ∈ U p . Optimization and Inversion under Uncertainty, RICAM, Linz, 2019 November 13, 2019 | S. Ulbrich | 4
Robust Optimization - Basic Idea Uncertain Optimization Problem ˆ min h 0 ( x ; p ) x (Pr) ˆ s.t. h i ( x ; p ) ≤ 0, i ∈ I . Assumption: Parameter p is uncertain. We only know that p ∈ U p . Consider the “Robust Counterpart” of (Pr): ˆ min max h 0 ( x ; p ) x p ∈U p ˆ s.t. h i ( x ; p ) ≤ 0 ∀ p ∈ U p , i ∈ I . [e.g. Ben-Tal, Bertsimas, El Ghaoui, Nemirovski, Nesterov, Zowe,. . . ] Optimization and Inversion under Uncertainty, RICAM, Linz, 2019 November 13, 2019 | S. Ulbrich | 4
Robust Optimization - Basic Idea Uncertain Optimization Problem ˆ min h 0 ( x ; p ) x (Pr) ˆ s.t. h i ( x ; p ) ≤ 0, i ∈ I . Assumption: Parameter p is uncertain. We only know that p ∈ U p . Consider the “Robust Counterpart” of (Pr): ˆ ˆ min max h 0 ( x ; p ) min max h 0 ( x ; p ) x p ∈U p x p ∈U p ⇐ ⇒ ˆ ˆ s.t. h i ( x ; p ) ≤ 0 ∀ p ∈ U p , i ∈ I . s.t. max h i ( x ; p ) ≤ 0, i ∈ I . p ∈U p [e.g. Ben-Tal, Bertsimas, El Ghaoui, Nemirovski, Nesterov, Zowe,. . . ] Optimization and Inversion under Uncertainty, RICAM, Linz, 2019 November 13, 2019 | S. Ulbrich | 4
Relation to Probabilistic Constraints ˆ min max h 0 ( x ; p ) x p ∈U p ˆ s.t. max h i ( x ; p ) ≤ 0, i ∈ I . p ∈U p If U p is confidence region for the random variable p of probability α then the solution x satisfies the constraints with probability ≥ α . Alternative approach: Probabilistic constraints, e.g. [Pr´ ekopa 95, Henrion, R¨ omisch 10, Van Ackooij, Henrion 14, Chen, Ghattas et al. 18]. Optimization and Inversion under Uncertainty, RICAM, Linz, 2019 November 13, 2019 | S. Ulbrich | 5
PDE-Constrained Optimization under Uncertainty Uncertain PDE-Constrained Optimization Problem min h 0 ( y , x ; p ) y ∈ Y , x ∈ X s.t. h i ( y , x ; p ) ≤ 0, i ∈ I , (P) C ( y , x ; p ) = 0. ◮ Typically nonconvex, design variables x , state y , uncertain parameters p ◮ h 0 , h i : Y × X × R n p → R , C : Y × X × R n p → Z sufficiently smooth ◮ C ( y , x ; p ) = 0 has a unique solution y = y ( x ; p ) for all relevant x , p ◮ ∂ y C ∈ L ( Y , Z ) is invertible Uncertainty to be considered ◮ Parameter p is uncertain with p ∈ U p = { p ∈ R n p . . � p − ¯ p � B p ≤ 1 } √ � v � B . v T Bv for a symmetric positive definite matrix B . = ◮ Also possible: Design x uncertain, x ∈ U x = { x ∈ X = R n x . . � x − ¯ x � B x ≤ 1 } Optimization and Inversion under Uncertainty, RICAM, Linz, 2019 November 13, 2019 | S. Ulbrich | 6
Robust Formulation of (P) Worst-case values of objective function and inequality constraints: i ( x ) . h wc . = y ∈ Y , s ∈ R np h i ( y , x ; ¯ max p + s ) s.t. C ( y , x ; ¯ p + s ) = 0, � s � B p ≤ 1. Reduced formulation: i ( x ) . h wc s ∈ R np ˆ . = max p + s ) s.t. � s � B p ≤ 1, h i ( x ; ¯ p + s ) := h i ( y ( x ; ¯ p + s ), x ; ¯ where C ( y ( x ; ¯ p + s ), x ; ¯ p + s ) = 0 . Optimization and Inversion under Uncertainty, RICAM, Linz, 2019 November 13, 2019 | S. Ulbrich | 7
Robust Formulation of (P) Worst-case values of objective function and inequality constraints: i ( x ) . h wc . = y ∈ Y , s ∈ R np h i ( y , x ; ¯ max p + s ) s.t. C ( y , x ; ¯ p + s ) = 0, � s � B p ≤ 1. Reduced formulation: i ( x ) . h wc s ∈ R np ˆ . = max p + s ) s.t. � s � B p ≤ 1, h i ( x ; ¯ p + s ) := h i ( y ( x ; ¯ p + s ), x ; ¯ where C ( y ( x ; ¯ p + s ), x ; ¯ p + s ) = 0 . Robust Counterpart of (P) h wc min 0 ( x ) x ∈ X (R) h wc s.t. i ( x ) ≤ 0, i ∈ I . Optimization and Inversion under Uncertainty, RICAM, Linz, 2019 November 13, 2019 | S. Ulbrich | 7
Robust Formulation of (P) Worst-case values of objective function and inequality constraints: i ( x ) . h wc . = y ∈ Y , s ∈ R np h i ( y , x ; ¯ max p + s ) s.t. C ( y , x ; ¯ p + s ) = 0, � s � B p ≤ 1. Reduced formulation: i ( x ) . h wc s ∈ R np ˆ . = max p + s ) s.t. � s � B p ≤ 1, h i ( x ; ¯ p + s ) := h i ( y ( x ; ¯ p + s ), x ; ¯ where C ( y ( x ; ¯ p + s ), x ; ¯ p + s ) = 0 . Robust Counterpart of (P) h wc min 0 ( x ) x ∈ X (R) h wc s.t. i ( x ) ≤ 0, i ∈ I . In the nonconvex case (R) is in general computationally intractable! Optimization and Inversion under Uncertainty, RICAM, Linz, 2019 November 13, 2019 | S. Ulbrich | 7
Approximation of Robust Formulation of (P) Robust Counterpart of (P) h wc ˆ min 0 ( x ) := max h 0 ( x ; p ) x ∈ X p ∈U p (R) h wc ˆ s.t. i ( x ) := max h i ( x ; p ) ≤ 0, i ∈ I . p ∈U p In the nonconvex case (R) is in general computationally intractable! Possible approaches: Approximate h wc by ˜ h wc such that ˜ h wc and ∇ ˜ h wc can be computed efficiently h wc can be characterized conveniently by differentiable constraints. or ˜ ◮ Linearize ˆ h i ( x ; p ) w.r.t. p [Diehl, Bock, Kostina 06; Zhang 07] ◮ In this talk: Approximate ˆ h i ( x ; p ) by second order Taylor expansion w.r.t. p [Sichau 13; Lass, SU 17; Alla, Hinze, Lass, Kolvenbach, SU 19; Kolvenbach, Lass, SU 18; cf. also Houska, Diehl 12; Alexanderian, Petra, Stadler, Ghattas 16; Chen, Villa, Ghattas 18; Milz, Ulbrich 19] Optimization and Inversion under Uncertainty, RICAM, Linz, 2019 November 13, 2019 | S. Ulbrich | 8
Approximation of Robust Formulation of (P) Approximated Robust Counterpart of (P) h appr h wc ˜ ˆ min 0 ( x ) := max ( x ; p ) 0 x ∈ X p ∈U p (RA) h wc h appr ˜ ˆ s.t. i ( x ) := max ( x ; p ) ≤ 0, i ∈ I . i p ∈U p In the nonconvex case (R) is in general computationally intractable! Possible approaches: Approximate h wc by ˜ h wc such that ˜ h wc and ∇ ˜ h wc can be computed efficiently h wc can be characterized conveniently by differentiable constraints. or ˜ ◮ Linearize ˆ h i ( x ; p ) w.r.t. p [Diehl, Bock, Kostina 06, Zhang 07] ◮ In this talk: Approximate ˆ h i ( x ; p ) by second order Taylor expansion w.r.t. p [Sichau 13; Lass, SU 17; Alla, Hinze, Lass, Kolvenbach, SU 19; Kolvenbach, Lass, SU 18; cf. also Houska, Diehl 12; Alexanderian, Petra, Stadler, Ghattas 16; Chen, Villa, Ghattas 18; Milz, Ulbrich 19] Optimization and Inversion under Uncertainty, RICAM, Linz, 2019 November 13, 2019 | S. Ulbrich | 9
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