Multiobjective Parameter Optimization of Elliptic PDEs using the - - PowerPoint PPT Presentation

▶
multiobjective parameter optimization of elliptic pdes
SMART_READER_LITE
LIVE PREVIEW

Multiobjective Parameter Optimization of Elliptic PDEs using the - - PowerPoint PPT Presentation

Nov 16, 2022 42 likes •657 views

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

slide-1
SLIDE 1

Reduced Basis Method

  • f Elliptic PDEs using the

Multiobjective Parameter Optimization

  • 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

slide-2
SLIDE 2

Problem Formulation

Consider the multiobjective parameter optimization problem min

u;y

  J1(u; y) . . . Jk(u; y)   s.t. e(y; u) = 0; (MPOP) where

  • Ji : U Y ! R, i = 1; :::; k, are the objective functions,
  • U = Rn 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 Ji(u) = Ji(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

slide-3
SLIDE 3

Problem Formulation

Consider the multiobjective parameter optimization problem min

u

  J1(u) . . . Jk(u)   =   J1(u; S(u)) . . . Jk(u; S(u))   ; (MPOP) where

  • Ji : U Y ! R, i = 1; :::; k, are the objective functions,
  • U = Rn 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 Ji(u) = Ji(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

slide-4
SLIDE 4

Problem Formulation

Consider the multiobjective parameter optimization problem min

u

  J1(u) . . . Jk(u)   =   J1(u; S(u)) . . . Jk(u; S(u))   ; (MPOP) where

  • Ji : U Y ! R, i = 1; :::; k, are the objective functions,
  • U = Rn 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 Ji(u) = Ji(u; S(u)).

Questions:

  • What is a meaningful optimality concept? (Problem: No total order on Rk)
  • 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

slide-5
SLIDE 5

Problem Formulation

Consider the multiobjective parameter optimization problem min

u

  J1(u) . . . Jk(u)   =   J1(u; S(u)) . . . Jk(u; S(u))   ; (MPOP) where

  • Ji : U Y ! R, i = 1; :::; k, are the objective functions,
  • U = Rn 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 Ji(u) = Ji(u; S(u)).

Questions:

  • What is a meaningful optimality concept? (Problem: No total order on Rk)
  • 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

slide-6
SLIDE 6

Problem Formulation

Consider the multiobjective parameter optimization problem min

u

  Jr

1(u)

. . . Jr

k(u)

  =   J1(u; Sr(u)) . . . Jk(u; Sr(u))   ; (MPOP) where

  • Ji : U Y ! R, i = 1; :::; k, are the objective functions,
  • U = Rn 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 Ji(u) = Ji(u; S(u)).

Questions:

  • What is a meaningful optimality concept? (Problem: No total order on Rk)
  • 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

slide-7
SLIDE 7

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 Ji(u) Ji( u) 8i 2 f1; :::; kg; Jl(u) < Jl( u) for at least one l 2 f1; :::; kg:

Example: J : R2 ! R2; u 7! (ku (1; 1)>k2

2

ku (1; 1)>k2

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

slide-8
SLIDE 8

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 Ji(u) Ji( u) 8i 2 f1; :::; kg; Jl(u) < Jl( u) for at least one l 2 f1; :::; kg:

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

slide-9
SLIDE 9

Multiobjective Parameter Optimization of PDEs

Optimality conditions Let J be differentiable.

Theorem:

If u is Pareto optimal, then there is some 2 (R0)k with ∑k

i=1 i = 1 such that k

∑

i=1

irJi(u) = DJ(u)> = 0: (KKT)

Definition: Pareto criticality

If u and satisfy (KKT), then u is Pareto critical with KKT vec- tor . The set Pc 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

slide-10
SLIDE 10

Multiobjective Parameter Optimization of PDEs

Optimality conditions Let J be differentiable.

Theorem:

If u is Pareto optimal, then there is some 2 (R0)k with ∑k

i=1 i = 1 such that k

∑

i=1

irJi(u) = DJ(u)> = 0: (KKT)

Definition: Pareto criticality

If u and satisfy (KKT), then u is Pareto critical with KKT vec- tor . The set Pc 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

slide-11
SLIDE 11

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 + (2i1r; : : : ; 2inr)T j (i1; : : : ; in) 2 Zng:

  • Compute Bc(r) := fB 2 B(r) j B \ Pc 6= ;g.
  • It holds B \ Pc 6= ; (

) minu2B;2k

  • DJ(u)T
  • 2

2 = 0.

  • If a box B 2 B(r) with B \ Pc 6= ; is found, use the tangent space of Pc 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

slide-12
SLIDE 12

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 + (2i1r; : : : ; 2inr)T j (i1; : : : ; in) 2 Zng:

  • Compute Bc(r) := fB 2 B(r) j B \ Pc 6= ;g.
  • It holds B \ Pc 6= ; (

) minu2B;2k

  • DJ(u)T
  • 2

2 = 0.

  • If a box B 2 B(r) with B \ Pc 6= ; is found, use the tangent space of Pc 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

slide-13
SLIDE 13

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 + (2i1r; : : : ; 2inr)T j (i1; : : : ; in) 2 Zng:

  • Compute Bc(r) := fB 2 B(r) j B \ Pc 6= ;g.
  • It holds B \ Pc 6= ; (

) minu2B;2k

  • DJ(u)T
  • 2

2 = 0.

  • If a box B 2 B(r) with B \ Pc 6= ; is found, use the tangent space of Pc 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

slide-14
SLIDE 14

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 + (2i1r; : : : ; 2inr)T j (i1; : : : ; in) 2 Zng:

  • Compute Bc(r) := fB 2 B(r) j B \ Pc 6= ;g.
  • It holds B \ Pc 6= ; (

) minu2B;2k

  • DJ(u)T
  • 2

2 = 0.

  • If a box B 2 B(r) with B \ Pc 6= ; is found, use the tangent space of Pc 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

slide-15
SLIDE 15

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 + (2i1r; : : : ; 2inr)T j (i1; : : : ; in) 2 Zng:

  • Compute Bc(r) := fB 2 B(r) j B \ Pc 6= ;g.
  • It holds B \ Pc 6= ; (

) minu2B;2k

  • DJ(u)T
  • 2

2 = 0.

  • If a box B 2 B(r) with B \ Pc 6= ; is found, use the tangent space of Pc 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

slide-16
SLIDE 16

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 + (2i1r; : : : ; 2inr)T j (i1; : : : ; in) 2 Zng:

  • Compute Bc(r) := fB 2 B(r) j B \ Pc 6= ;g.
  • It holds B \ Pc 6= ; (

) minu2B;2k

  • DJ(u)T
  • 2

2 = 0.

  • If a box B 2 B(r) with B \ Pc 6= ; is found, use the tangent space of Pc 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

slide-17
SLIDE 17

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 + (2i1r; : : : ; 2inr)T j (i1; : : : ; in) 2 Zng:

  • Compute Bc(r) := fB 2 B(r) j B \ Pc 6= ;g.
  • It holds B \ Pc 6= ; (

) minu2B;2k

  • DJ(u)T
  • 2

2 = 0.

  • If a box B 2 B(r) with B \ Pc 6= ; is found, use the tangent space of Pc 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

slide-18
SLIDE 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 + (2i1r; : : : ; 2inr)T j (i1; : : : ; in) 2 Zng:

  • Compute Bc(r) := fB 2 B(r) j B \ Pc 6= ;g.
  • It holds B \ Pc 6= ; (

) minu2B;2k

  • DJ(u)T
  • 2

2 = 0.

  • If a box B 2 B(r) with B \ Pc 6= ; is found, use the tangent space of Pc 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

slide-19
SLIDE 19

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 + (2i1r; : : : ; 2inr)T j (i1; : : : ; in) 2 Zng:

  • Compute Bc(r) := fB 2 B(r) j B \ Pc 6= ;g.
  • It holds B \ Pc 6= ; (

) minu2B;2k

  • DJ(u)T
  • 2

2 = 0.

  • If a box B 2 B(r) with B \ Pc 6= ; is found, use the tangent space of Pc 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

slide-20
SLIDE 20

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 + (2i1r; : : : ; 2inr)T j (i1; : : : ; in) 2 Zng:

  • Compute Bc(r) := fB 2 B(r) j B \ Pc 6= ;g.
  • It holds B \ Pc 6= ; (

) minu2B;2k

  • DJ(u)T
  • 2

2 = 0.

  • If a box B 2 B(r) with B \ Pc 6= ; is found, use the tangent space of Pc 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

slide-21
SLIDE 21

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 + (2i1r; : : : ; 2inr)T j (i1; : : : ; in) 2 Zng:

  • Compute Bc(r) := fB 2 B(r) j B \ Pc 6= ;g.
  • It holds B \ Pc 6= ; (

) minu2B;2k

  • DJ(u)T
  • 2

2 = 0.

  • If a box B 2 B(r) with B \ Pc 6= ; is found, use the tangent space of Pc 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

slide-22
SLIDE 22

Continuation Method with Inexact Gradients

Using inexact gradients

Assumption:

Let Jr : U ! Rk be the inexact objective function satisfying sup

u2U

krJi(u) rJr

i (u)k2 "i; i 2 f1; :::; kg

with error bounds "1; :::; "k 0. Question: How can we approximatively compute Pc only using these information?

Lemma:

Let u 2 Pc with KKT vector 2 k. Then it holds kDJr(u)>k2 >":

  • 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 6 / 18

slide-23
SLIDE 23

Continuation Method with Inexact Gradients

Using inexact gradients

Assumption:

Let Jr : U ! Rk be the inexact objective function satisfying sup

u2U

krJi(u) rJr

i (u)k2 "i; i 2 f1; :::; kg

with error bounds "1; :::; "k 0. Question: How can we approximatively compute Pc only using these information?

Lemma:

Let u 2 Pc with KKT vector 2 k. Then it holds kDJr(u)>k2 >":

  • 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 6 / 18

slide-24
SLIDE 24

Continuation Method with Inexact Gradients

Using inexact gradients

Assumption:

Let Jr : U ! Rk be the inexact objective function satisfying sup

u2U

krJi(u) rJr

i (u)k2 "i; i 2 f1; :::; kg

with error bounds "1; :::; "k 0. Question: How can we approximatively compute Pc only using these information?

Lemma:

Let u 2 Pc with KKT vector 2 k. Then it holds kDJr(u)>k2 >":

  • 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 6 / 18

slide-25
SLIDE 25

Continuation Method with Inexact Gradients

Using inexact gradients

Corollary:

Let P r := fu 2 Rn j min

2k(kDJr(u)>k2 2 (>")2) 0g:

Then P r

c P r and Pc P r. Furthermore, P r is tight.

(a)

" = (0:2; 0:05)>

(b)

" = (0; 0:2)>

  • 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 7 / 18

slide-26
SLIDE 26

Continuation Method with Inexact Gradients

Using inexact gradients

Corollary:

Let P r := fu 2 Rn j min

2k(kDJr(u)>k2 2 (>")2) 0g:

Then P r

c P r and Pc P r. Furthermore, P r is tight.

  • 1
  • 0.5

0.5 1

  • 1
  • 0.5

0.5 1

(a) " = (0:2; 0:05)>

  • 1
  • 0.5

0.5 1

  • 1
  • 0.5

0.5 1

(b) " = (0; 0:2)>

  • 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 7 / 18

slide-27
SLIDE 27

Continuation Method with Inexact Gradients

Using inexact gradients

Corollary:

Let P r := fu 2 Rn j min

2k(kDJr(u)>k2 2 (>")2) 0g:

Then P r

c P r and Pc P r. Furthermore, P r is tight.

  • 1
  • 0.5

0.5 1

  • 1
  • 0.5

0.5 1

(a) " = (0:2; 0:05)>

  • 1
  • 0.5

0.5 1

  • 1
  • 0.5

0.5 1

(b) " = (0; 0:2)>

  • 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 7 / 18

slide-28
SLIDE 28

Continuation Method with Inexact Gradients

Computing P r

We want to compute a box covering of P r := fu 2 Rn j min

2k(kDJr(u)>k2 2 (>")2) 0g;

i.e., fB 2 B(r) j B \ P r 6= ;g.

Lemma/Problem:

P r contains a nonempty, open subset of Rn. Two options:

  • Strategy 1: Compute P r directly.
  • Strategy 2: Only compute @P r.
  • 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 8 / 18

slide-29
SLIDE 29

Continuation Method with Inexact Gradients

Computing P r

We want to compute a box covering of P r := fu 2 Rn j min

2k(kDJr(u)>k2 2 (>")2) 0g;

i.e., fB 2 B(r) j B \ P r 6= ;g.

Lemma/Problem:

P r contains a nonempty, open subset of Rn. Two options:

  • Strategy 1: Compute P r directly.
  • Strategy 2: Only compute @P r.
  • 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 8 / 18

slide-30
SLIDE 30

Continuation Method with Inexact Gradients

Computing P r

We want to compute a box covering of P r := fu 2 Rn j min

2k(kDJr(u)>k2 2 (>")2) 0g;

i.e., fB 2 B(r) j B \ P r 6= ;g.

Lemma/Problem:

P r contains a nonempty, open subset of Rn.

  • 1
  • 0.5

0.5 1

  • 1
  • 0.5

0.5 1

Two options:

  • Strategy 1: Compute P r directly.
  • Strategy 2: Only compute @P r.
  • 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 8 / 18

slide-31
SLIDE 31

Continuation Method with Inexact Gradients

Computing P r Let B 2 B(r) and P r := fu 2 Rn j min

2k(kDJr(u)>k2 2 (>")2) 0g:

Subproblem for Strategy 1 (Computing P r):

B \ P r 6= ; ( ) min

u2B;2k(kDJr(u)>k2 2 (>")2) 0

Subproblem for Strategy 2 (Computing @P r):

B \ @P r 6= ; ( ) min

u2B '(u)2 = 0

where ' : Rn ! R; u 7! min

2k(kDJr(u)>k2 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 9 / 18

slide-32
SLIDE 32

Continuation Method with Inexact Gradients

Computing P r Let B 2 B(r) and P r := fu 2 Rn j min

2k(kDJr(u)>k2 2 (>")2) 0g:

Subproblem for Strategy 1 (Computing P r):

B \ P r 6= ; ( ) min

u2B;2k(kDJr(u)>k2 2 (>")2) 0

Subproblem for Strategy 2 (Computing @P r):

B \ @P r 6= ; ( ) min

u2B '(u)2 = 0

where ' : Rn ! R; u 7! min

2k(kDJr(u)>k2 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 9 / 18

slide-33
SLIDE 33

Continuation Method with Inexact Gradients

Computing P r Let B 2 B(r) and P r := fu 2 Rn j min

2k(kDJr(u)>k2 2 (>")2) 0g:

Subproblem for Strategy 1 (Computing P r):

B \ P r 6= ; ( ) min

u2B;2k(kDJr(u)>k2 2 (>")2) 0

Subproblem for Strategy 2 (Computing @P r):

B \ @P r 6= ; ( ) min

u2B '(u)2 = 0

where ' : Rn ! R; u 7! min

2k(kDJr(u)>k2 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 9 / 18

slide-34
SLIDE 34

Multiobjective Parameter Optimization of Elliptic PDEs using the RB method

Multiobjective Parameter Optimization of an Elliptic PDE

Given a domain R2, we consider the problem min

y;u J(y; u) :=

    

1 2ky y 1k2 L2() 1 2ky y 2k2 L2() 1 2ky y 3k2 L2() 1 2kuk2 R2

     s:t: y(x) + cb(x) ry(x) + 0:5y(x) = f (x) for x 2 ; @y @(x) = 0 for x 2 @; ua u ub: with u = (; c) 2 [0:5; 3] [1; 1], i.e., we optimize the diffusivity in and the strength and orientation c of the advection field b.

  • Denote by S the solution operator of the state equation.
  • We choose y 1 = S((0:7; 0:8)), y 2 = S((2; 0:5)), y 3 = S((3; 0:5)).
  • The solution operators of the adjoint equations (w.r.t. the cost functions

J1; J2; J3) are given by A1; A2; A3.

  • 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 10 / 18

slide-35
SLIDE 35

Multiobjective Parameter Optimization of Elliptic PDEs using the RB method

Multiobjective Parameter Optimization of an Elliptic PDE

Given a domain R2, we consider the problem min

y;u J(y; u) :=

    

1 2ky y 1k2 L2() 1 2ky y 2k2 L2() 1 2ky y 3k2 L2() 1 2kuk2 R2

     s:t: y(x) + cb(x) ry(x) + 0:5y(x) = f (x) for x 2 ; @y @(x) = 0 for x 2 @; ua u ub: with u = (; c) 2 [0:5; 3] [1; 1], i.e., we optimize the diffusivity in and the strength and orientation c of the advection field b.

  • Denote by S the solution operator of the state equation.
  • We choose y 1 = S((0:7; 0:8)), y 2 = S((2; 0:5)), y 3 = S((3; 0:5)).
  • The solution operators of the adjoint equations (w.r.t. the cost functions

J1; J2; J3) are given by A1; A2; A3.

  • 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 10 / 18

slide-36
SLIDE 36

Multiobjective Parameter Optimization of Elliptic PDEs using the RB method

Multiobjective Parameter Optimization of an Elliptic PDE

Given a domain R2, we consider the problem min

u J(u) :=

    

1 2kS(u) S((0:7; 0:8))k2 L2() 1 2kS(u) S((2; 0:5))k2 L2() 1 2kS(u) S((3; 0:5))k2 L2() 1 2kuk2 R2

     s:t: ua u ub: with u = (; c) 2 [0:5; 3] [1; 1], i.e., we optimize the diffusivity in and the strength and orientation c of the advection field b.

  • Denote by S the solution operator of the state equation.
  • We choose y 1 = S((0:7; 0:8)), y 2 = S((2; 0:5)), y 3 = S((3; 0:5)).
  • The solution operators of the adjoint equations (w.r.t. the cost functions

J1; J2; J3) are given by A1; A2; A3.

  • 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 10 / 18

slide-37
SLIDE 37

Multiobjective Parameter Optimization of Elliptic PDEs using the RB method

Reduced Basis Method

For solving (MPOP) by the continuation method the problem min

u2B;2k

  • DJ(u)T
  • 2

2

has to be solved for numerous boxes B.

!

Many solves of the state and adjoint equations are required.

!

Use the Reduced Basis method to lower the computational effort.

  • 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 11 / 18

slide-38
SLIDE 38

Multiobjective Parameter Optimization of Elliptic PDEs using the RB method

Reduced Basis Method

For solving (MPOP) by the continuation method the problem min

u2B;2k

  • DJ(u)T
  • 2

2

has to be solved for numerous boxes B.

!

Many solves of the state and adjoint equations are required.

!

Use the Reduced Basis method to lower the computational effort.

  • 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 11 / 18

slide-39
SLIDE 39

Multiobjective Parameter Optimization of Elliptic PDEs using the RB method

Reduced Basis Method

Weak formulation of the state equation: Find y 2 V := H1() such that a(u; y; ') = b(') for all ' 2 V:

  • Denote the solution operator of the RB state equation by Sr.
  • Let Ar

1; Ar 2; Ar 3 be the solution operators of the RB adjoint equations.

  • Define the RB objective function Jr(u) := J(Sr(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 12 / 18

slide-40
SLIDE 40

Multiobjective Parameter Optimization of Elliptic PDEs using the RB method

Reduced Basis Method

Weak formulation of the state equation: Find y 2 V := H1() such that a(u; y; ') = b(') for all ' 2 V: Idea of the RB method: Replace the infinite-dimensional space V in the weak formulation by a low-dimensional subspace V r V , which is typically spanned by snapshots, i.e., solutions to the state (and adjoint) equation to different parameter values.

  • Denote the solution operator of the RB state equation by Sr.
  • Let Ar

1; Ar 2; Ar 3 be the solution operators of the RB adjoint equations.

  • Define the RB objective function Jr(u) := J(Sr(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 12 / 18

slide-41
SLIDE 41

Multiobjective Parameter Optimization of Elliptic PDEs using the RB method

Reduced Basis Method

FE discretization of the state equation: Find y N 2 V N H1() such that a(u; y; ') = b(') for all ' 2 V N: Idea of the RB method: Replace the FE space V N in the FE discretization by a low-dimensional subspace V r V N, which is typically spanned by snapshots, i.e., solutions to the state (and adjoint) equation to different parameter values.

  • Denote the solution operator of the RB state equation by Sr.
  • Let Ar

1; Ar 2; Ar 3 be the solution operators of the RB adjoint equations.

  • Define the RB objective function Jr(u) := J(Sr(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 12 / 18

slide-42
SLIDE 42

Multiobjective Parameter Optimization of Elliptic PDEs using the RB method

Reduced Basis Method

FE discretization of the state equation: Find y N 2 V N H1() such that a(u; y; ') = b(') for all ' 2 V N: Idea of the RB method: Replace the FE space V N in the FE discretization by a low-dimensional subspace V r V N, which is typically spanned by snapshots, i.e., solutions to the state (and adjoint) equation to different parameter values. ! RB state equation: Find y r 2 V r such that a(u; y r; ') = b(') for all ' 2 V r:

  • Denote the solution operator of the RB state equation by Sr.
  • Let Ar

1; Ar 2; Ar 3 be the solution operators of the RB adjoint equations.

  • Define the RB objective function Jr(u) := J(Sr(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 12 / 18

slide-43
SLIDE 43

Multiobjective Parameter Optimization of Elliptic PDEs using the RB method

Reduced Basis Method

FE discretization of the state equation: Find y N 2 V N H1() such that a(u; y; ') = b(') for all ' 2 V N: Idea of the RB method: Replace the FE space V N in the FE discretization by a low-dimensional subspace V r V N, which is typically spanned by snapshots, i.e., solutions to the state (and adjoint) equation to different parameter values. ! RB state equation: Find y r 2 V r such that a(u; y r; ') = b(') for all ' 2 V r:

  • Denote the solution operator of the RB state equation by Sr.
  • Let Ar

1; Ar 2; Ar 3 be the solution operators of the RB adjoint equations.

  • Define the RB objective function Jr(u) := J(Sr(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 12 / 18

slide-44
SLIDE 44

Multiobjective Parameter Optimization of Elliptic PDEs using the RB method

Constructing the Reduced Basis

To apply the strategies for the continuation method with inexact gradients, the RB has to be constructed such that the inequalities sup

u2U

krJi(u) rJr

i (u)k2 "i;

for all i 2 f1; :::; kg hold for given error thresholds "1; : : : ; "4 0. ! Use a greedy algorithm. Data: Parameter set P [0:5; 3] [1; 1], greedy tolerances "1; : : : ; "k > 0. Choose u 2 P, compute S(u),A1(u); A2(u); A3(u); Set V r = spanfS(u); A1(u); A2(u); A3(u)g and compute the reduced basis by orthonormalization; while maxu2P maxi2f1;2;3;4g > "i do Choose ( u; i) = arg maxu2P; i2f1;2;3;4g rJi(u) rJr

i (u) 2;

Compute S( u) and Ai( u); Set V r = span fV r [ fS( u); Ai( u)gg and compute the reduced basis by

  • rthonormalization
  • 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 13 / 18

slide-45
SLIDE 45

Multiobjective Parameter Optimization of Elliptic PDEs using the RB method

Constructing the Reduced Basis

To apply the strategies for the continuation method with inexact gradients, the RB has to be constructed such that the inequalities sup

u2U

krJi(u) rJr

i (u)k2 "i;

for all i 2 f1; :::; kg hold for given error thresholds "1; : : : ; "4 0. ! Use a greedy algorithm. Data: Parameter set P [0:5; 3] [1; 1], greedy tolerances "1; : : : ; "k > 0. Choose u 2 P, compute S(u),A1(u); A2(u); A3(u); Set V r = spanfS(u); A1(u); A2(u); A3(u)g and compute the reduced basis by orthonormalization; while maxu2P maxi2f1;2;3;4g

  • rJi(u) rJr

i (u)

  • 2 > "i do

Choose ( u; i) = arg maxu2P; i2f1;2;3;4g

  • rJi(u) rJr

i (u)

  • 2;

Compute S( u) and Ai( u); Set V r = span fV r [ fS( u); Ai( u)gg and compute the reduced basis by

  • rthonormalization
  • 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 13 / 18

slide-46
SLIDE 46

Multiobjective Parameter Optimization of Elliptic PDEs using the RB method

Constructing the Reduced Basis

To apply the strategies for the continuation method with inexact gradients, the RB has to be constructed such that the inequalities sup

u2U

krJi(u) rJr

i (u)k2 "i;

for all i 2 f1; :::; kg hold for given error thresholds "1; : : : ; "4 0. ! Use a greedy algorithm. Data: Parameter set P [0:5; 3] [1; 1], greedy tolerances "1; : : : ; "k > 0. Choose u 2 P, compute S(u),A1(u); A2(u); A3(u); Set V r = spanfS(u); A1(u); A2(u); A3(u)g and compute the reduced basis by orthonormalization; while maxu2P maxi2f1;2;3;4g

  • rJi(u) rJr

i (u)

  • 2 > "i do

Choose ( u; i) = arg maxu2P; i2f1;2;3;4g

  • rJi(u) rJr

i (u)

  • 2;

Compute S( u) and Ai( u); Set V r = span fV r [ fS( u); Ai( u)gg and compute the reduced basis by

  • rthonormalization
  • 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 13 / 18

slide-47
SLIDE 47

Multiobjective Parameter Optimization of Elliptic PDEs using the RB method

A-posteriori error estimation

It can be shown j@Ji(u) @Jr

i (u)j = jhrS(u); rAi(u)i hrSr(u); rAr i (u)ij

kr (S(u) Sr(u)) rAr

i (u)k + kr (S(u) Sr(u)) r (Ai(u) Ar i (u))k

+ krSr(u) r (Ai(u) Ar

i (u))k

(1) kr (S(u) Sr(u))k krAr

i (u)k + kr (S(u) Sr(u))k kr (Ai(u) Ar i (u))k

+ krSr(u)k kr (Ai(u) Ar

i (u))k

(2) S(u) krAr

i (u)k + S(u)Ai(u) + krSr(u)k Ai(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 14 / 18

slide-48
SLIDE 48

Multiobjective Parameter Optimization of Elliptic PDEs using the RB method

A-posteriori error estimation

It can be shown j@Ji(u) @Jr

i (u)j = jhrS(u); rAi(u)i hrSr(u); rAr i (u)ij

kr (S(u) Sr(u)) rAr

i (u)k + kr (S(u) Sr(u)) r (Ai(u) Ar i (u))k

+ krSr(u) r (Ai(u) Ar

i (u))k

(1) kr (S(u) Sr(u))k krAr

i (u)k + kr (S(u) Sr(u))k kr (Ai(u) Ar i (u))k

+ krSr(u)k kr (Ai(u) Ar

i (u))k

(2) S(u) krAr

i (u)k + S(u)Ai(u) + krSr(u)k Ai(u):

200 400 600 800 1000 100 101 102 103 104 105 106 Overstimation of gradient errors

Estimate (1) Estimate (2) Estimate (3)

Figure: Overestimation of gradient errors

  • 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 14 / 18

slide-49
SLIDE 49

Multiobjective Parameter Optimization of Elliptic PDEs using the RB method

A-posteriori error estimation

It can be shown j@Ji(u) @Jr

i (u)j = jhrS(u); rAi(u)i hrSr(u); rAr i (u)ij

kr (S(u) Sr(u)) rAr

i (u)k + kr (S(u) Sr(u)) r (Ai(u) Ar i (u))k

+ krSr(u) r (Ai(u) Ar

i (u))k

(1) kr (S(u) Sr(u))k krAr

i (u)k + kr (S(u) Sr(u))k kr (Ai(u) Ar i (u))k

+ krSr(u)k kr (Ai(u) Ar

i (u))k

(2) S(u) krAr

i (u)k + S(u)Ai(u) + krSr(u)k Ai(u):

200 400 600 800 1000 100 101 102 103 104 105 106 Overstimation of gradient errors

Estimate (1) Estimate (2) Estimate (3)

Figure: Overestimation of gradient errors

  • 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 14 / 18

slide-50
SLIDE 50

Multiobjective Parameter Optimization of Elliptic PDEs using the RB method

A-posteriori error estimation

It can be shown j@Ji(u) @Jr

i (u)j = jhrS(u); rAi(u)i hrSr(u); rAr i (u)ij

kr (S(u) Sr(u)) rAr

i (u)k + kr (S(u) Sr(u)) r (Ai(u) Ar i (u))k

+ krSr(u) r (Ai(u) Ar

i (u))k

(1) kr (S(u) Sr(u))k krAr

i (u)k + kr (S(u) Sr(u))k kr (Ai(u) Ar i (u))k

+ krSr(u)k kr (Ai(u) Ar

i (u))k

(2) S(u) krAr

i (u)k + S(u)Ai(u) + krSr(u)k Ai(u):

(3)

200 400 600 800 1000 100 101 102 103 104 105 106 Overstimation of gradient errors

Estimate (1) Estimate (2) Estimate (3)

Figure: Overestimation of gradient errors

  • 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 14 / 18

slide-51
SLIDE 51

Multiobjective Parameter Optimization of Elliptic PDEs using the RB method

A-posteriori error estimation

It can be shown j@Ji(u) @Jr

i (u)j = jhrS(u); rAi(u)i hrSr(u); rAr i (u)ij

kr (S(u) Sr(u)) rAr

i (u)k + kr (S(u) Sr(u)) r (Ai(u) Ar i (u))k

+ krSr(u) r (Ai(u) Ar

i (u))k

(1) kr (S(u) Sr(u))k krAr

i (u)k + kr (S(u) Sr(u))k kr (Ai(u) Ar i (u))k

+ krSr(u)k kr (Ai(u) Ar

i (u))k

(2) S(u) krAr

i (u)k + S(u)Ai(u) + krSr(u)k Ai(u):

(3)

200 400 600 800 1000 100 101 102 103 104 105 106 Overstimation of gradient errors

Estimate (1) Estimate (2) Estimate (3)

Figure: Overestimation of gradient errors

! Huge overestimation of the gradient error (mainly caused by using the Cauchy-Schwarz inequality in (2)). ! Computation of exact errors necessary (strong greedy algorithm, see [Haasdonk, Salomon, Wohlmuth, 2013]).

  • 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 14 / 18

slide-52
SLIDE 52

Multiobjective Parameter Optimization of Elliptic PDEs using the RB method

A-posteriori error estimation

It can be shown j@Ji(u) @Jr

i (u)j = jhrS(u); rAi(u)i hrSr(u); rAr i (u)ij

kr (S(u) Sr(u)) rAr

i (u)k + kr (S(u) Sr(u)) r (Ai(u) Ar i (u))k

+ krSr(u) r (Ai(u) Ar

i (u))k

(1) kr (S(u) Sr(u))k krAr

i (u)k + kr (S(u) Sr(u))k kr (Ai(u) Ar i (u))k

+ krSr(u)k kr (Ai(u) Ar

i (u))k

(2) S(u) krAr

i (u)k + S(u)Ai(u) + krSr(u)k Ai(u):

(3)

200 400 600 800 1000 100 101 102 103 104 105 106 Overstimation of gradient errors

Estimate (1) Estimate (2) Estimate (3)

Figure: Overestimation of gradient errors

! Huge overestimation of the gradient error (mainly caused by using the Cauchy-Schwarz inequality in (2)). ! Computation of exact errors necessary (strong greedy algorithm, see [Haasdonk, Salomon, Wohlmuth, 2013]).

  • 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 14 / 18

slide-53
SLIDE 53

Multiobjective Parameter Optimization of Elliptic PDEs using the RB method

Numerical Results – Comparison of Methods For " = (0:03; 0:03; 0:01; 0:01)>:

(a) Strategy 1 (b) Strategy 2

Algorithm # Boxes # Subproblems Runtime (in seconds) Exact cont. 15916 18746 17501s Strategy 1 21750 24515 1426s Strategy 2 899 1252 276s

  • 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 15 / 18

slide-54
SLIDE 54

Multiobjective Parameter Optimization of Elliptic PDEs using the RB method

Numerical Results – Individual Error Bounds Varying the error bounds ": " = (0:1; 0:1; 0:1; 0:01)

  • 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 16 / 18

slide-55
SLIDE 55

Multiobjective Parameter Optimization of Elliptic PDEs using the RB method

Numerical Results – Individual Error Bounds Varying the error bounds ": " = (0:0885; 0:0885; 0:0885; 0:01)

  • 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 16 / 18

slide-56
SLIDE 56

Multiobjective Parameter Optimization of Elliptic PDEs using the RB method

Numerical Results – Individual Error Bounds Varying the error bounds ": " = (0:03; 0:03; 0:03; 0:01)

  • 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 16 / 18

slide-57
SLIDE 57

Multiobjective Parameter Optimization of Elliptic PDEs using the RB method

Numerical Results – Individual Error Bounds Varying the error bounds ": " = (0:03; 0:03; 0:01; 0:01)

  • 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 16 / 18

slide-58
SLIDE 58

Multiobjective Parameter Optimization of Elliptic PDEs using the RB method

Numerical Results – Individual Error Bounds Varying the error bounds ": " = (0:03; 0:01; 0:01; 0:01)

  • 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 16 / 18

slide-59
SLIDE 59

Conclusion

Conclusion

  • Solving MPOPs by a set-oriented continuation method via boxes.
  • Development of efficient strategies to deal with inexactness in the gradients.
  • Application of the RB method via a greedy algorithm ! reduction of the

computational time by a factor of up to 60.

  • Individual error bounds allow a precise control of the tightness of the

covering.

Outlook:

  • In certain situations the subproblems for computing @P r are quite hard to

solve ! development of more sophisticated solvers

  • Development of a more efficient a-posteriori error estimator for the error in

the gradients.

  • 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 17 / 18

slide-60
SLIDE 60

Conclusion

Conclusion

  • Solving MPOPs by a set-oriented continuation method via boxes.
  • Development of efficient strategies to deal with inexactness in the gradients.
  • Application of the RB method via a greedy algorithm ! reduction of the

computational time by a factor of up to 60.

  • Individual error bounds allow a precise control of the tightness of the

covering.

Outlook:

  • In certain situations the subproblems for computing @P r are quite hard to

solve ! development of more sophisticated solvers

  • Development of a more efficient a-posteriori error estimator for the error in

the gradients.

  • 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 17 / 18

slide-61
SLIDE 61

Conclusion

Literature

  • S. B., B. Gebken, M. Dellnitz, S. Peitz & S. Volkwein. “ROM-based multiobjective
  • ptimization of elliptic PDEs via numerical continuation”. Preprint, 2019.

https://arxiv.org/abs/1906.09075.

  • B. Gebken, S. Peitz & M. Dellnitz. “On the hierarchical structure of Pareto critical sets”.

Journal of Global Optimization, 2019.

  • C. Hillermeier, “Nonlinear Multiobjective Optimization: A Generalized Homotopy

Approach”, Birkhäuser Basel, 2001.

  • O. Schütze, A. Dell’Aere & M. Dellnitz, “On Continuation Methods for the Numerical

Treatment of Multi-Objective Optimization Problems”. Practical Approaches to Multi-Objective Optimization, 2005.

  • 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 18 / 18