Multiobjective Parameter Optimization of Elliptic PDEs using the Reduced Basis Method S. Banholzer, B. Gebken, M. Dellnitz, S. Peitz, S. Volkwein University of Konstanz / Paderborn University Workshop: New trends in PDE constrained optimization RICAM Linz, 17th October 2019 S. Banholzer, B. Gebken, M. Dellnitz, S. Peitz, S. Volkwein (University of Konstanz / Paderborn University Workshop: New trends in PDE constrained optimization RICAM Linz) Multiobjective Parameter Optimization of Elliptic PDEs using the Reduced Basis Method 17th October 2019 1 / 18
Problem Formulation Consider the multiobjective parameter optimization problem J 1 ( u; y ) . . . s.t. (MPOP) min e ( y; u ) = 0 ; u;y J k ( u; y ) where J i : U � Y ! R , i = 1 ; :::; k , are the objective functions , - U = R n is the parameter space , Y is the state space , - e is a PDE called the state equation , which is uniquely solvable for every u 2 U . - The parameter-to-state mapping is given by S : U ! Y . We write J i ( u ) = J i ( u; S ( u )) . - S. Banholzer, B. Gebken, M. Dellnitz, S. Peitz, S. Volkwein (University of Konstanz / Paderborn University Workshop: New trends in PDE constrained optimization RICAM Linz) Multiobjective Parameter Optimization of Elliptic PDEs using the Reduced Basis Method 17th October 2019 2 / 18
Problem Formulation Consider the multiobjective parameter optimization problem J 1 ( u ) J 1 ( u; S ( u )) . . . . = . . ; (MPOP) min u J k ( u ) J k ( u; S ( u )) where J i : U � Y ! R , i = 1 ; :::; k , are the objective functions , - U = R n is the parameter space , Y is the state space , - e is a PDE called the state equation , which is uniquely solvable for every u 2 U . - The parameter-to-state mapping is given by S : U ! Y . We write J i ( u ) = J i ( u; S ( u )) . - S. Banholzer, B. Gebken, M. Dellnitz, S. Peitz, S. Volkwein (University of Konstanz / Paderborn University Workshop: New trends in PDE constrained optimization RICAM Linz) Multiobjective Parameter Optimization of Elliptic PDEs using the Reduced Basis Method 17th October 2019 2 / 18
How can we solve (MPOP) (numerically)? - How can approximation errors induced by using model order reduction be - handled? Problem Formulation Consider the multiobjective parameter optimization problem J 1 ( u ) J 1 ( u; S ( u )) . . . . = . . ; (MPOP) min u J k ( u ) J k ( u; S ( u )) where J i : U � Y ! R , i = 1 ; :::; k , are the objective functions , - U = R n is the parameter space , Y is the state space , - e is a PDE called the state equation , which is uniquely solvable for every u 2 U . - The parameter-to-state mapping is given by S : U ! Y . We write J i ( u ) = J i ( u; S ( u )) . - Questions: What is a meaningful optimality concept? (Problem: No total order on R k ) - S. Banholzer, B. Gebken, M. Dellnitz, S. Peitz, S. Volkwein (University of Konstanz / Paderborn University Workshop: New trends in PDE constrained optimization RICAM Linz) Multiobjective Parameter Optimization of Elliptic PDEs using the Reduced Basis Method 17th October 2019 2 / 18
How can approximation errors induced by using model order reduction be - handled? Problem Formulation Consider the multiobjective parameter optimization problem J 1 ( u ) J 1 ( u; S ( u )) . . . . = . . ; (MPOP) min u J k ( u ) J k ( u; S ( u )) where J i : U � Y ! R , i = 1 ; :::; k , are the objective functions , - U = R n is the parameter space , Y is the state space , - e is a PDE called the state equation , which is uniquely solvable for every u 2 U . - The parameter-to-state mapping is given by S : U ! Y . We write J i ( u ) = J i ( u; S ( u )) . - Questions: What is a meaningful optimality concept? (Problem: No total order on R k ) - How can we solve (MPOP) (numerically)? - S. Banholzer, B. Gebken, M. Dellnitz, S. Peitz, S. Volkwein (University of Konstanz / Paderborn University Workshop: New trends in PDE constrained optimization RICAM Linz) Multiobjective Parameter Optimization of Elliptic PDEs using the Reduced Basis Method 17th October 2019 2 / 18
Problem Formulation Consider the multiobjective parameter optimization problem J r J 1 ( u; S r ( u )) 1 ( u ) . . . . = . . ; (MPOP) min u J r J k ( u; S r ( u )) k ( u ) where J i : U � Y ! R , i = 1 ; :::; k , are the objective functions , - U = R n is the parameter space , Y is the state space , - e is a PDE called the state equation , which is uniquely solvable for every u 2 U . - The parameter-to-state mapping is given by S : U ! Y . We write J i ( u ) = J i ( u; S ( u )) . - Questions: What is a meaningful optimality concept? (Problem: No total order on R k ) - How can we solve (MPOP) (numerically)? - How can approximation errors induced by using model order reduction be - handled? S. Banholzer, B. Gebken, M. Dellnitz, S. Peitz, S. Volkwein (University of Konstanz / Paderborn University Workshop: New trends in PDE constrained optimization RICAM Linz) Multiobjective Parameter Optimization of Elliptic PDEs using the Reduced Basis Method 17th October 2019 2 / 18
Multiobjective Parameter Optimization of PDEs Pareto Optimality Definition: Pareto optimality A parameter � u 2 U is called Pareto optimal for (MPOP), if there is no other para- meter u 2 U with J i ( u ) � J i (� 8 i 2 f 1 ; :::; k g ; u ) for at least one l 2 f 1 ; :::; k g : J l ( u ) < J l (� u ) ( k u � ( � 1 ; � 1) > k 2 ) Example: J : R 2 ! R 2 ; u 7! 2 : k u � (1 ; 1) > k 2 2 S. Banholzer, B. Gebken, M. Dellnitz, S. Peitz, S. Volkwein (University of Konstanz / Paderborn University Workshop: New trends in PDE constrained optimization RICAM Linz) Multiobjective Parameter Optimization of Elliptic PDEs using the Reduced Basis Method 17th October 2019 3 / 18
Multiobjective Parameter Optimization of PDEs Pareto Optimality Definition: Pareto optimality A parameter � u 2 U is called Pareto optimal for (MPOP), if there is no other para- meter u 2 U with J i ( u ) � J i (� 8 i 2 f 1 ; :::; k g ; u ) for at least one l 2 f 1 ; :::; k g : J l ( u ) < J l (� u ) Goal: Find the Pareto set of (MPOP), i.e., the set of all Pareto optimal parameters of (MPOP). S. Banholzer, B. Gebken, M. Dellnitz, S. Peitz, S. Volkwein (University of Konstanz / Paderborn University Workshop: New trends in PDE constrained optimization RICAM Linz) Multiobjective Parameter Optimization of Elliptic PDEs using the Reduced Basis Method 17th October 2019 3 / 18
Definition: Pareto criticality If u and � satisfy (KKT), then u is Pareto critical with KKT vec- tor � . The set P c of all Pareto critical points is the Pareto critical set . Multiobjective Parameter Optimization of PDEs Optimality conditions Let J be differentiable. Theorem: If u is Pareto optimal, then there is some � 2 ( R � 0 ) k with ∑ k i =1 � i = 1 such that k ∑ (KKT) � i r J i ( u ) = DJ ( u ) > � = 0 : i =1 S. Banholzer, B. Gebken, M. Dellnitz, S. Peitz, S. Volkwein (University of Konstanz / Paderborn University Workshop: New trends in PDE constrained optimization RICAM Linz) Multiobjective Parameter Optimization of Elliptic PDEs using the Reduced Basis Method 17th October 2019 4 / 18
Multiobjective Parameter Optimization of PDEs Optimality conditions Let J be differentiable. Theorem: If u is Pareto optimal, then there is some � 2 ( R � 0 ) k with ∑ k i =1 � i = 1 such that k ∑ (KKT) � i r J i ( u ) = DJ ( u ) > � = 0 : i =1 Definition: Pareto criticality If u and � satisfy (KKT), then u is Pareto critical with KKT vec- tor � . The set P c of all Pareto critical points is the Pareto critical set . S. Banholzer, B. Gebken, M. Dellnitz, S. Peitz, S. Volkwein (University of Konstanz / Paderborn University Workshop: New trends in PDE constrained optimization RICAM Linz) Multiobjective Parameter Optimization of Elliptic PDEs using the Reduced Basis Method 17th October 2019 4 / 18
Continuation Method with Exact Gradients Continuation Method via Boxes We use a set-oriented continuation method to compute the Pareto critical set. (see [Hillermeier, 2001]) Divide the variable space into boxes with radius r - B ( r ) := f [ � r; r ] n + (2 i 1 r; : : : ; 2 i n r ) T j ( i 1 ; : : : ; i n ) 2 Z n g : Compute B c ( r ) := f B 2 B ( r ) j B \ P c 6 = ;g . - � 2 It holds B \ P c 6 = ; ( � � 2 = 0 . - � DJ ( u ) T � ) min u 2 B;� 2 � k If a box B 2 B ( r ) with B \ P c 6 = ; is found, use the tangent space of P c to get - candidates for neighbouring boxes containing Pareto critical points. S. Banholzer, B. Gebken, M. Dellnitz, S. Peitz, S. Volkwein (University of Konstanz / Paderborn University Workshop: New trends in PDE constrained optimization RICAM Linz) Multiobjective Parameter Optimization of Elliptic PDEs using the Reduced Basis Method 17th October 2019 5 / 18
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