Solutions of Polynomial Systems Derived from the ... - ISSAC 2009
Solutions of Polynomial Systems Derived from the Steady Cavity Flow Problem Martin Mevissen1
Kosuke Yokoyama2 1 Tokyo
Nobuki Takayama2
Institute of Technology
2 Kobe
University
ISSAC 2009, Seoul, Korea
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
1 / 26
The steady cavity flow problem 0 = 0 =
2 ∂2ψ + ∂∂yω2 + ω ∂x 2 ∂ψ ∂ω ∂ψ ∂ω 1 ∂x ∂y − ∂y ∂x + R
A
∀(x, y) ∈ [0, 1]2 , !
∂x 2
!
B
v1 =
∂2ω
∂ψ , ∂y
+
∂2ω ∂y 2
"
∀(x, y) ∈ [0, 1]2 .
D
C
v2 = −
∂ψ ∂x
Parameter R denotes the Reynolds number. Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
2 / 26
The steady cavity flow problem 0 = 0 =
2 ∂2ψ + ∂∂yω2 + ω ∂x 2 ∂ψ ∂ω ∂ψ ∂ω 1 ∂x ∂y − ∂y ∂x + R
A
∀(x, y) ∈ [0, 1]2 , !
∂x 2
!
B
v1 =
∂2ω
∂ψ , ∂y
+
∂2ω ∂y 2
"
∀(x, y) ∈ [0, 1]2 .
D
C
v2 = −
∂ψ ∂x
Parameter R denotes the Reynolds number. Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
2 / 26
The steady cavity flow problem 0 = 0 =
2 ∂2ψ + ∂∂yω2 + ω ∂x 2 ∂ψ ∂ω ∂ψ ∂ω 1 ∂x ∂y − ∂y ∂x + R
A
∀(x, y) ∈ [0, 1]2 , !
∂x 2
!
B
v1 =
∂2ω
∂ψ , ∂y
+
∂2ω ∂y 2
"
∀(x, y) ∈ [0, 1]2 .
D
C
v2 = −
∂ψ ∂x
Parameter R denotes the Reynolds number. Discretize the steady cavity flow problem via a finite difference scheme! Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
2 / 26
Discretization yields Discrete steady cavity flow problem DSCF(R,N) −4ψi,j + ψi+1,j + ψi−1,j + ψi,j+1 + ψi,j−1 +h2 ωi,j −4ωi,j + ωi+1,j + ωi−1,j + ωi,j+1 + ωi,j−1 + R4 (ψi+1,j − ψi−1,j )(ωi,j+1 − ωi,j−1) − R4 (ψi,j+1 − ψi,j−1 )(ωi+1,j − ωi−1,j ) ψ1,j = ψN,j = ψi,1 = ψi,N ψ ωN,j ω1,j = −2 h2,j2 , ψi,2 ωi,1 = −2 h2 , ωi,N
= 0 (i, j = 2, ..., N-1)
= 0 (i, j = 2, ..., N-1) = 0 (i, j = 1, ..., N) ψ = −2 N−1,j h2 ψ +h = −2 i,N−1 h2
Problem with two parameters R and N and dimension n = 2(N − 2)2 .
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
3 / 26
Discretization yields Discrete steady cavity flow problem DSCF(R,N) −4ψi,j + ψi+1,j + ψi−1,j + ψi,j+1 + ψi,j−1 +h2 ωi,j −4ωi,j + ωi+1,j + ωi−1,j + ωi,j+1 + ωi,j−1 + R4 (ψi+1,j − ψi−1,j )(ωi,j+1 − ωi,j−1) − R4 (ψi,j+1 − ψi,j−1 )(ωi+1,j − ωi−1,j ) ψ1,j = ψN,j = ψi,1 = ψi,N ψ ωN,j ω1,j = −2 h2,j2 , ψi,2 ωi,1 = −2 h2 , ωi,N
= 0 (i, j = 2, ..., N-1)
= 0 (i, j = 2, ..., N-1) = 0 (i, j = 1, ..., N) ψ = −2 N−1,j h2 ψ +h = −2 i,N−1 h2
Problem with two parameters R and N and dimension n = 2(N − 2)2 .
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
3 / 26
Discretization yields Discrete steady cavity flow problem DSCF(R,N) −4ψi,j + ψi+1,j + ψi−1,j + ψi,j+1 + ψi,j−1 +h2 ωi,j −4ωi,j + ωi+1,j + ωi−1,j + ωi,j+1 + ωi,j−1 + R4 (ψi+1,j − ψi−1,j )(ωi,j+1 − ωi,j−1) − R4 (ψi,j+1 − ψi,j−1 )(ωi+1,j − ωi−1,j ) ψ1,j = ψN,j = ψi,1 = ψi,N ψ ωN,j ω1,j = −2 h2,j2 , ψi,2 ωi,1 = −2 h2 , ωi,N
= 0 (i, j = 2, ..., N-1)
= 0 (i, j = 2, ..., N-1) = 0 (i, j = 1, ..., N) ψ = −2 N−1,j h2 ψ +h = −2 i,N−1 h2
Problem with two parameters R and N and dimension n = 2(N − 2)2 .
Conjecture DSCF(R, N) has finitely many complex solutions for any R and N.
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
3 / 26
Discretization yields Discrete steady cavity flow problem DSCF(R,N) −4ψi,j + ψi+1,j + ψi−1,j + ψi,j+1 + ψi,j−1 +h2 ωi,j −4ωi,j + ωi+1,j + ωi−1,j + ωi,j+1 + ωi,j−1 + R4 (ψi+1,j − ψi−1,j )(ωi,j+1 − ωi,j−1) − R4 (ψi,j+1 − ψi,j−1 )(ωi+1,j − ωi−1,j ) ψ1,j = ψN,j = ψi,1 = ψi,N ψ ωN,j ω1,j = −2 h2,j2 , ψi,2 ωi,1 = −2 h2 , ωi,N
= 0 (i, j = 2, ..., N-1)
= 0 (i, j = 2, ..., N-1) = 0 (i, j = 1, ..., N) ψ = −2 N−1,j h2 ψ +h = −2 i,N−1 h2
Problem with two parameters R and N and dimension n = 2(N − 2)2 .
Conjecture DSCF(R, N) has finitely many complex solutions for any R and N.
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
3 / 26
Discretization yields Discrete steady cavity flow problem DSCF(R,N) −4ψi,j + ψi+1,j + ψi−1,j + ψi,j+1 + ψi,j−1 +h2 ωi,j −4ωi,j + ωi+1,j + ωi−1,j + ωi,j+1 + ωi,j−1 + R4 (ψi+1,j − ψi−1,j )(ωi,j+1 − ωi,j−1) − R4 (ψi,j+1 − ψi,j−1 )(ωi+1,j − ωi−1,j ) ψ1,j = ψN,j = ψi,1 = ψi,N ψ ωN,j ω1,j = −2 h2,j2 , ψi,2 ωi,1 = −2 h2 , ωi,N
= 0 (i, j = 2, ..., N-1)
= 0 (i, j = 2, ..., N-1) = 0 (i, j = 1, ..., N) ψ = −2 N−1,j h2 ψ +h = −2 i,N−1 h2
Problem with two parameters R and N and dimension n = 2(N − 2)2 .
Conjecture DSCF(R, N) has finitely many complex solutions for any R and N. Aim: Enumerate all solutions of DSCF(R, N) w.r.t. kinetic energy! Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
3 / 26
Outline
1
The discrete steady cavity flow problem
2
Sparse SDP relaxation method Improving the accuracy of SDPR Gröbner basis method and SDPR Enumeration algorithm Numerical results
3
Relations of Reynolds number R and CF(R, N)
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
4 / 26
Sparse SDP relaxations for POP Idea of our approach: Apply the sparse semidefinite program relaxations (SDPR) to a polynomial optimization problem (POP) derived from DSCF(R, N).
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
5 / 26
Sparse SDP relaxations for POP Idea of our approach: Apply the sparse semidefinite program relaxations (SDPR) to a polynomial optimization problem (POP) derived from DSCF(R, N).
An SDP in standard form (P) min s.t.
$A0 , X % $Ai , X % = bi (i = 1, . . . , k) X !0 (X positive semidefinite) #k (D) max b y i=1 #i i s.t. A0 − ki=1 Ai yi ! 0 # A0 , . . . , Ak , X ∈ Sn , b, y ∈ Rk , $G, H% := i,j Gi,j Hi,j .
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
5 / 26
Sparse SDP relaxations for POP Idea of our approach: Apply the sparse semidefinite program relaxations (SDPR) to a polynomial optimization problem (POP) derived from DSCF(R, N).
An SDP in standard form (P) min s.t.
$A0 , X % $Ai , X % = bi (i = 1, . . . , k) X !0 (X positive semidefinite) #k (D) max b y i=1 #i i s.t. A0 − ki=1 Ai yi ! 0 # A0 , . . . , Ak , X ∈ Sn , b, y ∈ Rk , $G, H% := i,j Gi,j Hi,j .
¯ , y¯ ), min(P) = max(D) holds. If there exists interior feasible (X
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
5 / 26
Sparse SDP relaxations for POP Idea of our approach: Apply the sparse semidefinite program relaxations (SDPR) to a polynomial optimization problem (POP) derived from DSCF(R, N).
An SDP in standard form (P) min s.t.
$A0 , X % $Ai , X % = bi (i = 1, . . . , k) X !0 (X positive semidefinite) #k (D) max b y i=1 #i i s.t. A0 − ki=1 Ai yi ! 0 # A0 , . . . , Ak , X ∈ Sn , b, y ∈ Rk , $G, H% := i,j Gi,j Hi,j .
¯ , y¯ ), min(P) = max(D) holds. If there exists interior feasible (X (SDP) can be solved in poly time [Khachian 1979], efficiently by primal-dual interior point method [Nesterov, Nemirovski 1994].
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
5 / 26
Sparse SDP relaxations for POP POP min F (x) s.t. gj (x) ≥ 0 ∀ j ∈ {1, . . . , k} , hi (x) = 0 ∀ i ∈ {1, . . . , l} .
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
6 / 26
Sparse SDP relaxations for POP POP min F (x) s.t. gj (x) ≥ 0 ∀ j ∈ {1, . . . , k} , hi (x) = 0 ∀ i ∈ {1, . . . , l} . Sparse POP can be approximated by a hierarchy of sparse SDP relaxations SDPR(w) [Lasserre, Waki, Kim, Kojima]. min(SDPR(w )) → min(POP), for w → ∞, holds if compactness conditions for feas(POP) are satisfied.
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
6 / 26
Sparse SDP relaxations for POP POP min F (x) s.t. gj (x) ≥ 0 ∀ j ∈ {1, . . . , k} , hi (x) = 0 ∀ i ∈ {1, . . . , l} . Sparse POP can be approximated by a hierarchy of sparse SDP relaxations SDPR(w) [Lasserre, Waki, Kim, Kojima]. min(SDPR(w )) → min(POP), for w → ∞, holds if compactness conditions for feas(POP) are satisfied. POP derived from PDE satisfy structured sparsity patterns [Mevissen, Kojima, Nie, Takayama].
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
6 / 26
Sparse SDP relaxations for POP POP min F (x) s.t. gj (x) ≥ 0 ∀ j ∈ {1, . . . , k} , hi (x) = 0 ∀ i ∈ {1, . . . , l} . Sparse POP can be approximated by a hierarchy of sparse SDP relaxations SDPR(w) [Lasserre, Waki, Kim, Kojima]. min(SDPR(w )) → min(POP), for w → ∞, holds if compactness conditions for feas(POP) are satisfied. POP derived from PDE satisfy structured sparsity patterns [Mevissen, Kojima, Nie, Takayama]. Take DSCF(R,N) as constraints and choose an objective to obtain a POP! Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
6 / 26
An objective is needed to derive a POP: Kinetic energy of the flow:
$ $
[0,1]2
Discretize kinetic energy: F (ψ, ω) =
Mevissen (Tokyo Tech)
1 4
)
2≤i,j≤N−1
%&
∂ψ ∂y
'2
+
&
∂ψ ∂x
'2 (
dxdy
* +2 * +2 ψi+1,j − ψi−1,j + ψi,j+1 − ψi,j−1
Solutions for Polynomial Systems
ISSAC 2009
7 / 26
An objective is needed to derive a POP: Kinetic energy of the flow:
$ $
[0,1]2
Discretize kinetic energy: F (ψ, ω) =
1 4
)
2≤i,j≤N−1
%&
∂ψ ∂y
'2
+
&
∂ψ ∂x
'2 (
dxdy
* +2 * +2 ψi+1,j − ψi−1,j + ψi,j+1 − ψi,j−1
The cavity flow optimization problem CF(R, N) min F (ψ, ω) s.t. DSCF(R, N)
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
7 / 26
An objective is needed to derive a POP: Kinetic energy of the flow:
$ $
[0,1]2
Discretize kinetic energy: F (ψ, ω) =
1 4
)
2≤i,j≤N−1
%&
∂ψ ∂y
'2
+
&
∂ψ ∂x
'2 (
dxdy
* +2 * +2 ψi+1,j − ψi−1,j + ψi,j+1 − ψi,j−1
The cavity flow optimization problem CF(R, N) min F (ψ, ω) s.t. DSCF(R, N)
Proposition
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
7 / 26
An objective is needed to derive a POP: Kinetic energy of the flow:
$ $
[0,1]2
Discretize kinetic energy: F (ψ, ω) =
1 4
)
2≤i,j≤N−1
%&
∂ψ ∂y
'2
+
&
∂ψ ∂x
'2 (
dxdy
* +2 * +2 ψi+1,j − ψi−1,j + ψi,j+1 − ψi,j−1
The cavity flow optimization problem CF(R, N) min F (ψ, ω) s.t. DSCF(R, N)
Proposition a) CF(0, N) is a convex quadratic for any N.
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
7 / 26
An objective is needed to derive a POP: Kinetic energy of the flow:
$ $
[0,1]2
Discretize kinetic energy: F (ψ, ω) =
1 4
)
2≤i,j≤N−1
%&
∂ψ ∂y
'2
+
&
∂ψ ∂x
'2 (
dxdy
* +2 * +2 ψi+1,j − ψi−1,j + ψi,j+1 − ψi,j−1
The cavity flow optimization problem CF(R, N) min F (ψ, ω) s.t. DSCF(R, N)
Proposition a) CF(0, N) is a convex quadratic for any N. b) CF(R, N) is non-convex quadratic for any N, if R )= 0. Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
7 / 26
Sparsity pattern It can be shown that CF(R, N) satisfies a some structured sparsity pattern.
Figure: Chordal correlative sparsity pattern matrix for CF (R, 20)
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
8 / 26
Sparsity pattern It can be shown that CF(R, N) satisfies a some structured sparsity pattern.
Figure: Chordal correlative sparsity pattern matrix for CF (R, 20)
Therefore, the sparse SDP relaxations [Waki et al.] are efficient to solve CF(R, N)! Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
8 / 26
Sparsity pattern It can be shown that CF(R, N) satisfies a some structured sparsity pattern.
Figure: Chordal correlative sparsity pattern matrix for CF (R, 20)
Therefore, the sparse SDP relaxations [Waki et al.] are efficient to solve CF(R, N)! We will apply SDPR(w ) with order w ∈ {1, 2}. Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
8 / 26
Outline
1
The discrete steady cavity flow problem
2
Sparse SDP relaxation method Improving the accuracy of SDPR Gröbner basis method and SDPR Enumeration algorithm Numerical results
3
Relations of Reynolds number R and CF(R, N)
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
9 / 26
How to improve the accuracy of SDPR Problem SDPR(1) and SDPR(2) may not yield accurate approximation to the global optimizers of CF(R, N).
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
10 / 26
How to improve the accuracy of SDPR Problem SDPR(1) and SDPR(2) may not yield accurate approximation to the global optimizers of CF(R, N).
Technique 1 Impose tight lower and upper bounds for all ψi and ωi , ψ ω ω lbdψ i ≤ ψi ≤ ubdi and lbdi ≤ ωi ≤ ubdi
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
∀ 1 ≤ i ≤ N 2.
ISSAC 2009
10 / 26
How to improve the accuracy of SDPR Problem SDPR(1) and SDPR(2) may not yield accurate approximation to the global optimizers of CF(R, N).
Technique 1 Impose tight lower and upper bounds for all ψi and ωi , ψ ω ω lbdψ i ≤ ψi ≤ ubdi and lbdi ≤ ωi ≤ ubdi
∀ 1 ≤ i ≤ N 2.
Technique 2 Apply locally convergent optimization techniques like Newton’s method for nonlinear systems or sequential quadratic programming (SQP) starting from the SDPR(w ) solution.
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
10 / 26
Combining these techniques yields
The SDPR method
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
11 / 26
Combining these techniques yields
The SDPR method 1
Choose the two parameters R and N.
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
11 / 26
Combining these techniques yields
The SDPR method 1
Choose the two parameters R and N.
2
˜ ω ˜ := (ψ, Apply SDPR(w ) to CF(R, N) and obtain solution u ˜ ).
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
11 / 26
Combining these techniques yields
The SDPR method 1
Choose the two parameters R and N.
2
˜ ω ˜ := (ψ, Apply SDPR(w ) to CF(R, N) and obtain solution u ˜ ).
3
Apply sequential quadratic programming (SQP) to CF(R, N) or ˜, Newton’s method to DSCF(R, N), each of them starting from u and obtain u := (ψ, ω).
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
11 / 26
Outline
1
The discrete steady cavity flow problem
2
Sparse SDP relaxation method Improving the accuracy of SDPR Gröbner basis method and SDPR Enumeration algorithm Numerical results
3
Relations of Reynolds number R and CF(R, N)
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
12 / 26
Gröbner basis method Use Gröbner basis method to tune SDPR method and validate its numerical results!
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
13 / 26
Gröbner basis method Use Gröbner basis method to tune SDPR method and validate its numerical results!
Apply rational univariate representation (RUR) to DSCF(R, 5): Find all complex solutions of a system with 18 variables, 9 appear as linear and 9 appear as quadratic variables. Then, all real solutions can be enumerated w.r.t. their discretized kinetic energy.
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
13 / 26
Gröbner basis method Use Gröbner basis method to tune SDPR method and validate its numerical results!
Apply rational univariate representation (RUR) to DSCF(R, 5): Find all complex solutions of a system with 18 variables, 9 appear as linear and 9 appear as quadratic variables. Then, all real solutions can be enumerated w.r.t. their discretized kinetic energy.
Claim: Results obtained by RUR for N = 5 can be used for tuning SDPR method for N > 5, too!
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
13 / 26
Outline
1
The discrete steady cavity flow problem
2
Sparse SDP relaxation method Improving the accuracy of SDPR Gröbner basis method and SDPR Enumeration algorithm Numerical results
3
Relations of Reynolds number R and CF(R, N)
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
14 / 26
Algorithm to enumerate all solutions of DSCF(R, N) Iterate the 5 steps to approximate the k smallest energy solutions:
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
15 / 26
Algorithm to enumerate all solutions of DSCF(R, N) Iterate the 5 steps to approximate the k smallest energy solutions: 1
Given u (k −1) , the approximation to the (k − 1)th smallest energy solution obtained by solving SDPRk −1 (w ).
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
15 / 26
Algorithm to enumerate all solutions of DSCF(R, N) Iterate the 5 steps to approximate the k smallest energy solutions: 1
2
Given u (k −1) , the approximation to the (k − 1)th smallest energy solution obtained by solving SDPRk −1 (w ). , Choose $k1 , $k2 > 0 and integers b1k , b2k ∈ 1, . . . , (N − 2)2 .
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
15 / 26
Algorithm to enumerate all solutions of DSCF(R, N) Iterate the 5 steps to approximate the k smallest energy solutions: 1
2 3
Given u (k −1) , the approximation to the (k − 1)th smallest energy solution obtained by solving SDPRk −1 (w ). , Choose $k1 , $k2 > 0 and integers b1k , b2k ∈ 1, . . . , (N − 2)2 . Add the following quadratic constraints to SDPRk −1 (w ) and denote the resulting (tighter) SDP relaxation as SDPRk (w ). (k −1) 2 )
(uj − uj
(uj+(N−2)2 −
Mevissen (Tokyo Tech)
≥ $k1 (k −1) uj+(N−2)2 )2
∀1 ≤ j ≤ b1k , ≥ $k2 ∀1 ≤ j ≤ b2k .
Solutions for Polynomial Systems
ISSAC 2009
(1)
15 / 26
Algorithm to enumerate all solutions of DSCF(R, N) Iterate the 5 steps to approximate the k smallest energy solutions: 1
2 3
Given u (k −1) , the approximation to the (k − 1)th smallest energy solution obtained by solving SDPRk −1 (w ). , Choose $k1 , $k2 > 0 and integers b1k , b2k ∈ 1, . . . , (N − 2)2 . Add the following quadratic constraints to SDPRk −1 (w ) and denote the resulting (tighter) SDP relaxation as SDPRk (w ). (k −1) 2 )
(uj − uj
(uj+(N−2)2 − 4
≥ $k1 (k −1) uj+(N−2)2 )2
∀1 ≤ j ≤ b1k , ≥ $k2 ∀1 ≤ j ≤ b2k .
(1)
˜ (k ) . Solve SDPRk (w ) and obtain a first approximation u
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
15 / 26
Algorithm to enumerate all solutions of DSCF(R, N) Iterate the 5 steps to approximate the k smallest energy solutions: 1
2 3
Given u (k −1) , the approximation to the (k − 1)th smallest energy solution obtained by solving SDPRk −1 (w ). , Choose $k1 , $k2 > 0 and integers b1k , b2k ∈ 1, . . . , (N − 2)2 . Add the following quadratic constraints to SDPRk −1 (w ) and denote the resulting (tighter) SDP relaxation as SDPRk (w ). (k −1) 2 )
(uj − uj
(uj+(N−2)2 − 4 5
≥ $k1 (k −1) uj+(N−2)2 )2
∀1 ≤ j ≤ b1k , ≥ $k2 ∀1 ≤ j ≤ b2k .
(1)
˜ (k ) . Solve SDPRk (w ) and obtain a first approximation u ˜ (k ) as starting point. Obtain Apply Newton’s method or SQP with u u (k ) as an approximation to u (k )$ .
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
15 / 26
Algorithm to enumerate all solutions of DSCF(R, N) Iterate the 5 steps to approximate the k smallest energy solutions: 1
2 3
Given u (k −1) , the approximation to the (k − 1)th smallest energy solution obtained by solving SDPRk −1 (w ). , Choose $k1 , $k2 > 0 and integers b1k , b2k ∈ 1, . . . , (N − 2)2 . Add the following quadratic constraints to SDPRk −1 (w ) and denote the resulting (tighter) SDP relaxation as SDPRk (w ). (k −1) 2 )
(uj − uj
(uj+(N−2)2 − 4 5
≥ $k1 (k −1) uj+(N−2)2 )2
∀1 ≤ j ≤ b1k , ≥ $k2 ∀1 ≤ j ≤ b2k .
(1)
˜ (k ) . Solve SDPRk (w ) and obtain a first approximation u ˜ (k ) as starting point. Obtain Apply Newton’s method or SQP with u u (k ) as an approximation to u (k )$ .
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
15 / 26
Algorithm to enumerate all solutions of DSCF(R, N) Iterate the 5 steps to approximate the k smallest energy solutions: 1
2 3
Given u (k −1) , the approximation to the (k − 1)th smallest energy solution obtained by solving SDPRk −1 (w ). , Choose $k1 , $k2 > 0 and integers b1k , b2k ∈ 1, . . . , (N − 2)2 . Add the following quadratic constraints to SDPRk −1 (w ) and denote the resulting (tighter) SDP relaxation as SDPRk (w ). (k −1) 2 )
(uj − uj
(uj+(N−2)2 − 4 5
≥ $k1 (k −1) uj+(N−2)2 )2
∀1 ≤ j ≤ b1k , ≥ $k2 ∀1 ≤ j ≤ b2k .
(1)
˜ (k ) . Solve SDPRk (w ) and obtain a first approximation u ˜ (k ) as starting point. Obtain Apply Newton’s method or SQP with u u (k ) as an approximation to u (k )$ .
Constraints (1) have shown to be superior to alternatives. Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
15 / 26
Convergence result for the enumeration algorithm Proposition 2 Let R and N be fixed, (u (1) , . . . , u (k −1) ) be the output of the first (k − 1) iterations. If this output is a sufficiently close approximation of (u (1)$ , . . . , u (k −1)$ ), and if DSCF(R, N) is finite and distinct in terms of F , i.e. F (u (1)$ ) < F (u (2)$ ) < . . ., then there exist b ∈ {1, . . . , n} and $ ∈ Rb such that u (k ) from the kth iteration satisfies u (k ) (w ) → u (k )$
Mevissen (Tokyo Tech)
when w → ∞.
Solutions for Polynomial Systems
ISSAC 2009
16 / 26
Convergence result for the enumeration algorithm Proposition 2 Let R and N be fixed, (u (1) , . . . , u (k −1) ) be the output of the first (k − 1) iterations. If this output is a sufficiently close approximation of (u (1)$ , . . . , u (k −1)$ ), and if DSCF(R, N) is finite and distinct in terms of F , i.e. F (u (1)$ ) < F (u (2)$ ) < . . ., then there exist b ∈ {1, . . . , n} and $ ∈ Rb such that u (k ) from the kth iteration satisfies u (k ) (w ) → u (k )$
when w → ∞.
Proof: Apply of Lasserre’s theorem to sequence of POPs generated by the algorithm.
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
16 / 26
Convergence result for the enumeration algorithm Proposition 2 Let R and N be fixed, (u (1) , . . . , u (k −1) ) be the output of the first (k − 1) iterations. If this output is a sufficiently close approximation of (u (1)$ , . . . , u (k −1)$ ), and if DSCF(R, N) is finite and distinct in terms of F , i.e. F (u (1)$ ) < F (u (2)$ ) < . . ., then there exist b ∈ {1, . . . , n} and $ ∈ Rb such that u (k ) from the kth iteration satisfies u (k ) (w ) → u (k )$
when w → ∞.
Proof: Apply of Lasserre’s theorem to sequence of POPs generated by the algorithm.
But: In numerical experiments we are restricted to w ∈ {1, 2}.
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
16 / 26
Convergence result for the enumeration algorithm Proposition 2 Let R and N be fixed, (u (1) , . . . , u (k −1) ) be the output of the first (k − 1) iterations. If this output is a sufficiently close approximation of (u (1)$ , . . . , u (k −1)$ ), and if DSCF(R, N) is finite and distinct in terms of F , i.e. F (u (1)$ ) < F (u (2)$ ) < . . ., then there exist b ∈ {1, . . . , n} and $ ∈ Rb such that u (k ) from the kth iteration satisfies u (k ) (w ) → u (k )$
when w → ∞.
Proof: Apply of Lasserre’s theorem to sequence of POPs generated by the algorithm.
But: In numerical experiments we are restricted to w ∈ {1, 2}.
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
16 / 26
Convergence result for the enumeration algorithm Proposition 2 Let R and N be fixed, (u (1) , . . . , u (k −1) ) be the output of the first (k − 1) iterations. If this output is a sufficiently close approximation of (u (1)$ , . . . , u (k −1)$ ), and if DSCF(R, N) is finite and distinct in terms of F , i.e. F (u (1)$ ) < F (u (2)$ ) < . . ., then there exist b ∈ {1, . . . , n} and $ ∈ Rb such that u (k ) from the kth iteration satisfies u (k ) (w ) → u (k )$
when w → ∞.
Proof: Apply of Lasserre’s theorem to sequence of POPs generated by the algorithm.
But: In numerical experiments we are restricted to w ∈ {1, 2}. Use results from RUR for N = 5 to tune $ and b for N ≥ 5. Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
16 / 26
Outline
1
The discrete steady cavity flow problem
2
Sparse SDP relaxation method Improving the accuracy of SDPR Gröbner basis method and SDPR Enumeration algorithm Numerical results
3
Relations of Reynolds number R and CF(R, N)
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
17 / 26
CF(4000,5) k 0 1 2
$k1 1e-3 1e-3
w 1 1 1
b1k 3 3
b2k 0 0
tC 2 5 8
F (u (k ) ) 4.6e-4 6.3e-4 1.0e-3
$sc 2e-10 5e-4 5e-4
1
1
1
0.9
0.9
0.9
0.8
0.8
0.8
0.7
0.7
0.7
0.6
0.6
0.6
0.5
0.5
0.5
0.4
0.4
0.4
0.3
0.3
0.3
0.2
0.2
0.2
0.1
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
solution u (0) u (1) u (2)
0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Can verify by Gröbner basis method [RUR]: u (0) , u (1) and u (2) are indeed the 3 smallest energy solutions! Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
18 / 26
CF(20000,7) k 0 1 1 1 2
$k1 1e-3 1e-6 1e-5 1e-5
w 1 1 1 1 1
b1k 1 5 5 5
b2k 0 0 0 0
tC 2 5 5 9 14
F (u (k ) ) 3.4e-4 3.4e-4 3.4e-4 5.9e-4 5.2e-3
$sc 3e-7 5e-4 6e-6 5e-6 5e-6
1
1
1
0.9
0.9
0.9
0.8
0.8
0.8
0.7
0.7
0.7
0.6
0.6
0.6
0.5
0.5
0.5
0.4
0.4
0.4
0.3
0.3
0.3
0.2
0.2
0.2
0.1
0.1
0
0
0
0.2
0.4
0.6
0.8
u (0) Mevissen (Tokyo Tech)
1
solution u (0) u (0) u (0) u (1) u (2)
0.1
0
0.2
0.4
0.6
0.8
1
u (1) Solutions for Polynomial Systems
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
u (2) ISSAC 2009
19 / 26
CF(20000,7) k 0 1 1 1 2
$k1 1e-3 1e-6 1e-5 1e-5
w 1 1 1 1 1
b1k 1 5 5 5
b2k 0 0 0 0
tC 2 5 5 9 14
F (u (k ) ) 3.4e-4 3.4e-4 3.4e-4 5.9e-4 5.2e-3
$sc 3e-7 5e-4 6e-6 5e-6 5e-6
1
1
1
0.9
0.9
0.9
0.8
0.8
0.8
0.7
0.7
0.7
0.6
0.6
0.6
0.5
0.5
0.5
0.4
0.4
0.4
0.3
0.3
0.3
0.2
0.2
0.2
0.1
0.1
0
0
0
0.2
0.4
0.6
0.8
1
solution u (0) u (0) u (0) u (1) u (2)
0.1
0
0.2
u (0)
0.4
0.6
0.8
1
u (1)
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
u (2)
Right choice for $ and b crucial! Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
19 / 26
CF(40000,7) k 0 1 2 3 0
$k1 5e-6 5e-6 8e-6 -
w 1 1 1 1 2
b1k 5 5 5 -
b2k 0 0 0 -
tC 3 7 11 16 5872
$sc 2e-7 6e-9 3e-6 5e-6 8e-10
F (u (k ) ) 3.4e-4 7.3e-4 5.9e-4 2.3e-4 2.6e-4
1
1
1
0.9
0.9
0.9
0.8
0.8
0.8
0.7
0.7
0.7
0.6
0.6
0.6
0.5
0.5
0.5
0.4
0.4
0.4
0.3
0.3
0.3
0.2
0.2
0.2
0.1
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
u (0) (1)
1
0
solution u (0) (1) u (1) (1) u (2) (1) u (3) (1) u (0) (2)
0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
u (0) (2)
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
u (3) (1)
Enumeration algorithm more efficient in approximating global minimizer than classical SDP relaxation! Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
20 / 26
Outline
1
The discrete steady cavity flow problem
2
Sparse SDP relaxation method Improving the accuracy of SDPR Gröbner basis method and SDPR Enumeration algorithm Numerical results
3
Relations of Reynolds number R and CF(R, N)
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
21 / 26
Minimal kinetic energy solution for increasing R Observe: DSCF(R, N) is more difficult to solve for larger R.
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
22 / 26
Minimal kinetic energy solution for increasing R Observe: DSCF(R, N) is more difficult to solve for larger R. Question: How does u $ behave for increasing R?
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
22 / 26
Minimal kinetic energy solution for increasing R Observe: DSCF(R, N) is more difficult to solve for larger R. Question: How does u $ behave for increasing R?
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
22 / 26
Minimal kinetic energy solution for increasing R Observe: DSCF(R, N) is more difficult to solve for larger R. Question: How does u $ behave for increasing R? Compare the SDPR method results with
Naive homotopy-like continuation method
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
22 / 26
Minimal kinetic energy solution for increasing R Observe: DSCF(R, N) is more difficult to solve for larger R. Question: How does u $ behave for increasing R? Compare the SDPR method results with
Naive homotopy-like continuation method 1
Choose R # , N and a step size ∆R.
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
22 / 26
Minimal kinetic energy solution for increasing R Observe: DSCF(R, N) is more difficult to solve for larger R. Question: How does u $ behave for increasing R? Compare the SDPR method results with
Naive homotopy-like continuation method 1
Choose R # , N and a step size ∆R.
2
Solve DSCF(0, N) and obtain its unique solution u 0 .
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
22 / 26
Minimal kinetic energy solution for increasing R Observe: DSCF(R, N) is more difficult to solve for larger R. Question: How does u $ behave for increasing R? Compare the SDPR method results with
Naive homotopy-like continuation method 1
Choose R # , N and a step size ∆R.
2
Solve DSCF(0, N) and obtain its unique solution u 0 .
3
Increase R k −1 : R k = R k −1 + ∆R
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
22 / 26
Minimal kinetic energy solution for increasing R Observe: DSCF(R, N) is more difficult to solve for larger R. Question: How does u $ behave for increasing R? Compare the SDPR method results with
Naive homotopy-like continuation method 1
Choose R # , N and a step size ∆R.
2
Solve DSCF(0, N) and obtain its unique solution u 0 .
3
Increase R k −1 : R k = R k −1 + ∆R
4
Apply Newton’s method to DSCF(R k , N) starting from u k −1 . Obtain solution u k as an approximation to a solution of DSCF(R k , N).
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
22 / 26
Minimal kinetic energy solution for increasing R Observe: DSCF(R, N) is more difficult to solve for larger R. Question: How does u $ behave for increasing R? Compare the SDPR method results with
Naive homotopy-like continuation method 1
Choose R # , N and a step size ∆R.
2
Solve DSCF(0, N) and obtain its unique solution u 0 .
3
Increase R k −1 : R k = R k −1 + ∆R
4
5
Apply Newton’s method to DSCF(R k , N) starting from u k −1 . Obtain solution u k as an approximation to a solution of DSCF(R k , N). Iterate 3. and 4. until the desired Reynold’s number R # is reached.
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
22 / 26
Minimal kinetic energy solution for increasing R Observe: DSCF(R, N) is more difficult to solve for larger R. Question: How does u $ behave for increasing R? Compare the SDPR method results with
Naive homotopy-like continuation method 1
Choose R # , N and a step size ∆R.
2
Solve DSCF(0, N) and obtain its unique solution u 0 .
3
Increase R k −1 : R k = R k −1 + ∆R
4
5
Apply Newton’s method to DSCF(R k , N) starting from u k −1 . Obtain solution u k as an approximation to a solution of DSCF(R k , N). Iterate 3. and 4. until the desired Reynold’s number R # is reached.
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
22 / 26
Minimal kinetic energy solution for increasing R Observe: DSCF(R, N) is more difficult to solve for larger R. Question: How does u $ behave for increasing R? Compare the SDPR method results with
Naive homotopy-like continuation method 1
Choose R # , N and a step size ∆R.
2
Solve DSCF(0, N) and obtain its unique solution u 0 .
3
Increase R k −1 : R k = R k −1 + ∆R
4
5
Apply Newton’s method to DSCF(R k , N) starting from u k −1 . Obtain solution u k as an approximation to a solution of DSCF(R k , N). Iterate 3. and 4. until the desired Reynold’s number R # is reached.
Continuation method is not guaranteed to find the minimizer of F ! Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
22 / 26
CF(R,5) R 0 100 500 1000 2000 4000 6000 10000 20000 30000 100000
NC 1 37 37 37 37 37 36 35 35 35 34
Mevissen (Tokyo Tech)
NR 1 13 13 13 13 17 16 17 17 17 16
EC 0.0096 0.0030 6.2e-4 5.4e-4 6.2e-4 6.3e-4 5.7e-4 4.7e-4 4.5e-4 4.5e-4 4.5e-4
ESDPR(1) 0.0096 0.0030 6.2e-4 5e-4 6.2e-4 4.6e-4 4.5e-4 4.5e-4 4.5e-4 4.5e-4 4.5e-4
Solutions for Polynomial Systems
ESDPR(2) 0.0096 0.0030 6.2e-4 5e-4 6.2e-4 4.6e-4 4.5e-4 4.5e-4 3.3e-4 2.5e-4 8.8e-5
Emin 0.0096 0.0030 6.2e-4 5e-4 6.2e-4 4.6e-4 4.5e-4 4.5e-4 3.3e-4 2.5e-4 8.8e-5
ISSAC 2009
23 / 26
CF(R,5) R 0 100 500 1000 2000 4000 6000 10000 20000 30000 100000
NC 1 37 37 37 37 37 36 35 35 35 34
NR 1 13 13 13 13 17 16 17 17 17 16
EC 0.0096 0.0030 6.2e-4 5.4e-4 6.2e-4 6.3e-4 5.7e-4 4.7e-4 4.5e-4 4.5e-4 4.5e-4
ESDPR(1) 0.0096 0.0030 6.2e-4 5e-4 6.2e-4 4.6e-4 4.5e-4 4.5e-4 4.5e-4 4.5e-4 4.5e-4
ESDPR(2) 0.0096 0.0030 6.2e-4 5e-4 6.2e-4 4.6e-4 4.5e-4 4.5e-4 3.3e-4 2.5e-4 8.8e-5
Emin 0.0096 0.0030 6.2e-4 5e-4 6.2e-4 4.6e-4 4.5e-4 4.5e-4 3.3e-4 2.5e-4 8.8e-5
SDPR(1) yields the min energy solution for R ≤ 10000, SDPR(2) yields min energy solution for all R. Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
23 / 26
CF(R,5) 0.01
0.009
0.008
E (R) C
E
(R)
E
(R) / E
(R)
7
8
SDPR(1)
0.007
SDPR(2)
min
0.006
0.005
0.004
0.003
0.002
0.001
0
0
Mevissen (Tokyo Tech)
1
2
3
4
5 R
6
Solutions for Polynomial Systems
9
10 4
x 10
ISSAC 2009
24 / 26
CF(R,5) 0.01
0.009
0.008
E (R) C
E
(R)
E
(R) / E
(R)
7
8
SDPR(1)
0.007
SDPR(2)
min
0.006
0.005
0.004
0.003
0.002
0.001
0
0
1
2
3
4
5 R
6
9
10 4
x 10
uC (R) is not the minimum energy solution!
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
24 / 26
CF(R,5) 0.01
0.009
0.008
E (R) C
E
(R)
E
(R) / E
(R)
7
8
SDPR(1)
0.007
SDPR(2)
min
0.006
0.005
0.004
0.003
0.002
0.001
0
0
1
2
3
4
5 R
6
9
10 4
x 10
uC (R) is not the minimum energy solution! u $ (R) is obtained by SDPR(2) and Emin (R) tends to 0! Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
24 / 26
CF(R,5) 0.01
0.009
0.008
E (R) C
E
(R)
E
(R) / E
(R)
7
8
SDPR(1)
0.007
SDPR(2)
min
0.006
0.005
0.004
0.003
0.002
0.001
0
0
1
2
3
4
5 R
6
9
10 4
x 10
uC (R) is not the minimum energy solution! u $ (R) is obtained by SDPR(2) and Emin (R) tends to 0! Similar behavior can be observed for N ∈ {6, 7}. Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
24 / 26
Behavior for increasing R
Conjecture
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
25 / 26
Behavior for increasing R
Conjecture a) F (u0 (N)) = Emin (0, N) ≥ Emin (R, N) ≥ 0 ∀R ≥ 0.
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
25 / 26
Behavior for increasing R
Conjecture a) F (u0 (N)) = Emin (0, N) ≥ Emin (R, N) ≥ 0 ∀R ≥ 0. b) Emin (R, N) → 0 for R → ∞.
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
25 / 26
Behavior for increasing R
Conjecture a) F (u0 (N)) = Emin (0, N) ≥ Emin (R, N) ≥ 0 ∀R ≥ 0. b) Emin (R, N) → 0 for R → ∞. a) can be used as a certificate for non-optimality of a solution u # (R, N), if F (u # (R, N)) > Emin (0, N).
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
25 / 26
Behavior for increasing R
Conjecture a) F (u0 (N)) = Emin (0, N) ≥ Emin (R, N) ≥ 0 ∀R ≥ 0. b) Emin (R, N) → 0 for R → ∞. a) can be used as a certificate for non-optimality of a solution u # (R, N), if F (u # (R, N)) > Emin (0, N). ˜ (R, N), u # (R, N) is In the case u0 (N) can be continuated to u ˜ (R, N)). non-optimal if F (u # (R, N)) > F (u
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
25 / 26
Summary Min energy solution can be found for many R with SDPR(1) or SDPR(2).
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
26 / 26
Summary Min energy solution can be found for many R with SDPR(1) or SDPR(2). Algorithm for approximately enumerating all solutions of a polynomial system one by one.
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
26 / 26
Summary Min energy solution can be found for many R with SDPR(1) or SDPR(2). Algorithm for approximately enumerating all solutions of a polynomial system one by one. Convergence result.
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
26 / 26
Summary Min energy solution can be found for many R with SDPR(1) or SDPR(2). Algorithm for approximately enumerating all solutions of a polynomial system one by one. Convergence result. Discrete steady cavity flow problem for large R is a challenging test problem for new solvers in numerical algebra.
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
26 / 26
Summary Min energy solution can be found for many R with SDPR(1) or SDPR(2). Algorithm for approximately enumerating all solutions of a polynomial system one by one. Convergence result. Discrete steady cavity flow problem for large R is a challenging test problem for new solvers in numerical algebra. Future Research
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
26 / 26
Summary Min energy solution can be found for many R with SDPR(1) or SDPR(2). Algorithm for approximately enumerating all solutions of a polynomial system one by one. Convergence result. Discrete steady cavity flow problem for large R is a challenging test problem for new solvers in numerical algebra. Future Research Systematic procedure to determine $ and b.
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
26 / 26
Summary Min energy solution can be found for many R with SDPR(1) or SDPR(2). Algorithm for approximately enumerating all solutions of a polynomial system one by one. Convergence result. Discrete steady cavity flow problem for large R is a challenging test problem for new solvers in numerical algebra. Future Research Systematic procedure to determine $ and b. Prove Conjectures.
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
26 / 26
Summary Min energy solution can be found for many R with SDPR(1) or SDPR(2). Algorithm for approximately enumerating all solutions of a polynomial system one by one. Convergence result. Discrete steady cavity flow problem for large R is a challenging test problem for new solvers in numerical algebra. Future Research Systematic procedure to determine $ and b. Prove Conjectures. Apply alternative difference scheme to keep physical invariants under discretization.
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
26 / 26
Summary Min energy solution can be found for many R with SDPR(1) or SDPR(2). Algorithm for approximately enumerating all solutions of a polynomial system one by one. Convergence result. Discrete steady cavity flow problem for large R is a challenging test problem for new solvers in numerical algebra. Future Research Systematic procedure to determine $ and b. Prove Conjectures. Apply alternative difference scheme to keep physical invariants under discretization. Consider linear instead of quadratic constraints in enumeration algorithm to improve stability. Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
26 / 26
Kosuke Yokoyama2 1 Tokyo
Nobuki Takayama2
Institute of Technology
2 Kobe
University
ISSAC 2009, Seoul, Korea
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
1 / 26
The steady cavity flow problem 0 = 0 =
2 ∂2ψ + ∂∂yω2 + ω ∂x 2 ∂ψ ∂ω ∂ψ ∂ω 1 ∂x ∂y − ∂y ∂x + R
A
∀(x, y) ∈ [0, 1]2 , !
∂x 2
!
B
v1 =
∂2ω
∂ψ , ∂y
+
∂2ω ∂y 2
"
∀(x, y) ∈ [0, 1]2 .
D
C
v2 = −
∂ψ ∂x
Parameter R denotes the Reynolds number. Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
2 / 26
The steady cavity flow problem 0 = 0 =
2 ∂2ψ + ∂∂yω2 + ω ∂x 2 ∂ψ ∂ω ∂ψ ∂ω 1 ∂x ∂y − ∂y ∂x + R
A
∀(x, y) ∈ [0, 1]2 , !
∂x 2
!
B
v1 =
∂2ω
∂ψ , ∂y
+
∂2ω ∂y 2
"
∀(x, y) ∈ [0, 1]2 .
D
C
v2 = −
∂ψ ∂x
Parameter R denotes the Reynolds number. Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
2 / 26
The steady cavity flow problem 0 = 0 =
2 ∂2ψ + ∂∂yω2 + ω ∂x 2 ∂ψ ∂ω ∂ψ ∂ω 1 ∂x ∂y − ∂y ∂x + R
A
∀(x, y) ∈ [0, 1]2 , !
∂x 2
!
B
v1 =
∂2ω
∂ψ , ∂y
+
∂2ω ∂y 2
"
∀(x, y) ∈ [0, 1]2 .
D
C
v2 = −
∂ψ ∂x
Parameter R denotes the Reynolds number. Discretize the steady cavity flow problem via a finite difference scheme! Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
2 / 26
Discretization yields Discrete steady cavity flow problem DSCF(R,N) −4ψi,j + ψi+1,j + ψi−1,j + ψi,j+1 + ψi,j−1 +h2 ωi,j −4ωi,j + ωi+1,j + ωi−1,j + ωi,j+1 + ωi,j−1 + R4 (ψi+1,j − ψi−1,j )(ωi,j+1 − ωi,j−1) − R4 (ψi,j+1 − ψi,j−1 )(ωi+1,j − ωi−1,j ) ψ1,j = ψN,j = ψi,1 = ψi,N ψ ωN,j ω1,j = −2 h2,j2 , ψi,2 ωi,1 = −2 h2 , ωi,N
= 0 (i, j = 2, ..., N-1)
= 0 (i, j = 2, ..., N-1) = 0 (i, j = 1, ..., N) ψ = −2 N−1,j h2 ψ +h = −2 i,N−1 h2
Problem with two parameters R and N and dimension n = 2(N − 2)2 .
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
3 / 26
Discretization yields Discrete steady cavity flow problem DSCF(R,N) −4ψi,j + ψi+1,j + ψi−1,j + ψi,j+1 + ψi,j−1 +h2 ωi,j −4ωi,j + ωi+1,j + ωi−1,j + ωi,j+1 + ωi,j−1 + R4 (ψi+1,j − ψi−1,j )(ωi,j+1 − ωi,j−1) − R4 (ψi,j+1 − ψi,j−1 )(ωi+1,j − ωi−1,j ) ψ1,j = ψN,j = ψi,1 = ψi,N ψ ωN,j ω1,j = −2 h2,j2 , ψi,2 ωi,1 = −2 h2 , ωi,N
= 0 (i, j = 2, ..., N-1)
= 0 (i, j = 2, ..., N-1) = 0 (i, j = 1, ..., N) ψ = −2 N−1,j h2 ψ +h = −2 i,N−1 h2
Problem with two parameters R and N and dimension n = 2(N − 2)2 .
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
3 / 26
Discretization yields Discrete steady cavity flow problem DSCF(R,N) −4ψi,j + ψi+1,j + ψi−1,j + ψi,j+1 + ψi,j−1 +h2 ωi,j −4ωi,j + ωi+1,j + ωi−1,j + ωi,j+1 + ωi,j−1 + R4 (ψi+1,j − ψi−1,j )(ωi,j+1 − ωi,j−1) − R4 (ψi,j+1 − ψi,j−1 )(ωi+1,j − ωi−1,j ) ψ1,j = ψN,j = ψi,1 = ψi,N ψ ωN,j ω1,j = −2 h2,j2 , ψi,2 ωi,1 = −2 h2 , ωi,N
= 0 (i, j = 2, ..., N-1)
= 0 (i, j = 2, ..., N-1) = 0 (i, j = 1, ..., N) ψ = −2 N−1,j h2 ψ +h = −2 i,N−1 h2
Problem with two parameters R and N and dimension n = 2(N − 2)2 .
Conjecture DSCF(R, N) has finitely many complex solutions for any R and N.
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
3 / 26
Discretization yields Discrete steady cavity flow problem DSCF(R,N) −4ψi,j + ψi+1,j + ψi−1,j + ψi,j+1 + ψi,j−1 +h2 ωi,j −4ωi,j + ωi+1,j + ωi−1,j + ωi,j+1 + ωi,j−1 + R4 (ψi+1,j − ψi−1,j )(ωi,j+1 − ωi,j−1) − R4 (ψi,j+1 − ψi,j−1 )(ωi+1,j − ωi−1,j ) ψ1,j = ψN,j = ψi,1 = ψi,N ψ ωN,j ω1,j = −2 h2,j2 , ψi,2 ωi,1 = −2 h2 , ωi,N
= 0 (i, j = 2, ..., N-1)
= 0 (i, j = 2, ..., N-1) = 0 (i, j = 1, ..., N) ψ = −2 N−1,j h2 ψ +h = −2 i,N−1 h2
Problem with two parameters R and N and dimension n = 2(N − 2)2 .
Conjecture DSCF(R, N) has finitely many complex solutions for any R and N.
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
3 / 26
Discretization yields Discrete steady cavity flow problem DSCF(R,N) −4ψi,j + ψi+1,j + ψi−1,j + ψi,j+1 + ψi,j−1 +h2 ωi,j −4ωi,j + ωi+1,j + ωi−1,j + ωi,j+1 + ωi,j−1 + R4 (ψi+1,j − ψi−1,j )(ωi,j+1 − ωi,j−1) − R4 (ψi,j+1 − ψi,j−1 )(ωi+1,j − ωi−1,j ) ψ1,j = ψN,j = ψi,1 = ψi,N ψ ωN,j ω1,j = −2 h2,j2 , ψi,2 ωi,1 = −2 h2 , ωi,N
= 0 (i, j = 2, ..., N-1)
= 0 (i, j = 2, ..., N-1) = 0 (i, j = 1, ..., N) ψ = −2 N−1,j h2 ψ +h = −2 i,N−1 h2
Problem with two parameters R and N and dimension n = 2(N − 2)2 .
Conjecture DSCF(R, N) has finitely many complex solutions for any R and N. Aim: Enumerate all solutions of DSCF(R, N) w.r.t. kinetic energy! Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
3 / 26
Outline
1
The discrete steady cavity flow problem
2
Sparse SDP relaxation method Improving the accuracy of SDPR Gröbner basis method and SDPR Enumeration algorithm Numerical results
3
Relations of Reynolds number R and CF(R, N)
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
4 / 26
Sparse SDP relaxations for POP Idea of our approach: Apply the sparse semidefinite program relaxations (SDPR) to a polynomial optimization problem (POP) derived from DSCF(R, N).
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
5 / 26
Sparse SDP relaxations for POP Idea of our approach: Apply the sparse semidefinite program relaxations (SDPR) to a polynomial optimization problem (POP) derived from DSCF(R, N).
An SDP in standard form (P) min s.t.
$A0 , X % $Ai , X % = bi (i = 1, . . . , k) X !0 (X positive semidefinite) #k (D) max b y i=1 #i i s.t. A0 − ki=1 Ai yi ! 0 # A0 , . . . , Ak , X ∈ Sn , b, y ∈ Rk , $G, H% := i,j Gi,j Hi,j .
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
5 / 26
Sparse SDP relaxations for POP Idea of our approach: Apply the sparse semidefinite program relaxations (SDPR) to a polynomial optimization problem (POP) derived from DSCF(R, N).
An SDP in standard form (P) min s.t.
$A0 , X % $Ai , X % = bi (i = 1, . . . , k) X !0 (X positive semidefinite) #k (D) max b y i=1 #i i s.t. A0 − ki=1 Ai yi ! 0 # A0 , . . . , Ak , X ∈ Sn , b, y ∈ Rk , $G, H% := i,j Gi,j Hi,j .
¯ , y¯ ), min(P) = max(D) holds. If there exists interior feasible (X
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
5 / 26
Sparse SDP relaxations for POP Idea of our approach: Apply the sparse semidefinite program relaxations (SDPR) to a polynomial optimization problem (POP) derived from DSCF(R, N).
An SDP in standard form (P) min s.t.
$A0 , X % $Ai , X % = bi (i = 1, . . . , k) X !0 (X positive semidefinite) #k (D) max b y i=1 #i i s.t. A0 − ki=1 Ai yi ! 0 # A0 , . . . , Ak , X ∈ Sn , b, y ∈ Rk , $G, H% := i,j Gi,j Hi,j .
¯ , y¯ ), min(P) = max(D) holds. If there exists interior feasible (X (SDP) can be solved in poly time [Khachian 1979], efficiently by primal-dual interior point method [Nesterov, Nemirovski 1994].
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
5 / 26
Sparse SDP relaxations for POP POP min F (x) s.t. gj (x) ≥ 0 ∀ j ∈ {1, . . . , k} , hi (x) = 0 ∀ i ∈ {1, . . . , l} .
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
6 / 26
Sparse SDP relaxations for POP POP min F (x) s.t. gj (x) ≥ 0 ∀ j ∈ {1, . . . , k} , hi (x) = 0 ∀ i ∈ {1, . . . , l} . Sparse POP can be approximated by a hierarchy of sparse SDP relaxations SDPR(w) [Lasserre, Waki, Kim, Kojima]. min(SDPR(w )) → min(POP), for w → ∞, holds if compactness conditions for feas(POP) are satisfied.
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
6 / 26
Sparse SDP relaxations for POP POP min F (x) s.t. gj (x) ≥ 0 ∀ j ∈ {1, . . . , k} , hi (x) = 0 ∀ i ∈ {1, . . . , l} . Sparse POP can be approximated by a hierarchy of sparse SDP relaxations SDPR(w) [Lasserre, Waki, Kim, Kojima]. min(SDPR(w )) → min(POP), for w → ∞, holds if compactness conditions for feas(POP) are satisfied. POP derived from PDE satisfy structured sparsity patterns [Mevissen, Kojima, Nie, Takayama].
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
6 / 26
Sparse SDP relaxations for POP POP min F (x) s.t. gj (x) ≥ 0 ∀ j ∈ {1, . . . , k} , hi (x) = 0 ∀ i ∈ {1, . . . , l} . Sparse POP can be approximated by a hierarchy of sparse SDP relaxations SDPR(w) [Lasserre, Waki, Kim, Kojima]. min(SDPR(w )) → min(POP), for w → ∞, holds if compactness conditions for feas(POP) are satisfied. POP derived from PDE satisfy structured sparsity patterns [Mevissen, Kojima, Nie, Takayama]. Take DSCF(R,N) as constraints and choose an objective to obtain a POP! Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
6 / 26
An objective is needed to derive a POP: Kinetic energy of the flow:
$ $
[0,1]2
Discretize kinetic energy: F (ψ, ω) =
Mevissen (Tokyo Tech)
1 4
)
2≤i,j≤N−1
%&
∂ψ ∂y
'2
+
&
∂ψ ∂x
'2 (
dxdy
* +2 * +2 ψi+1,j − ψi−1,j + ψi,j+1 − ψi,j−1
Solutions for Polynomial Systems
ISSAC 2009
7 / 26
An objective is needed to derive a POP: Kinetic energy of the flow:
$ $
[0,1]2
Discretize kinetic energy: F (ψ, ω) =
1 4
)
2≤i,j≤N−1
%&
∂ψ ∂y
'2
+
&
∂ψ ∂x
'2 (
dxdy
* +2 * +2 ψi+1,j − ψi−1,j + ψi,j+1 − ψi,j−1
The cavity flow optimization problem CF(R, N) min F (ψ, ω) s.t. DSCF(R, N)
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
7 / 26
An objective is needed to derive a POP: Kinetic energy of the flow:
$ $
[0,1]2
Discretize kinetic energy: F (ψ, ω) =
1 4
)
2≤i,j≤N−1
%&
∂ψ ∂y
'2
+
&
∂ψ ∂x
'2 (
dxdy
* +2 * +2 ψi+1,j − ψi−1,j + ψi,j+1 − ψi,j−1
The cavity flow optimization problem CF(R, N) min F (ψ, ω) s.t. DSCF(R, N)
Proposition
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
7 / 26
An objective is needed to derive a POP: Kinetic energy of the flow:
$ $
[0,1]2
Discretize kinetic energy: F (ψ, ω) =
1 4
)
2≤i,j≤N−1
%&
∂ψ ∂y
'2
+
&
∂ψ ∂x
'2 (
dxdy
* +2 * +2 ψi+1,j − ψi−1,j + ψi,j+1 − ψi,j−1
The cavity flow optimization problem CF(R, N) min F (ψ, ω) s.t. DSCF(R, N)
Proposition a) CF(0, N) is a convex quadratic for any N.
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
7 / 26
An objective is needed to derive a POP: Kinetic energy of the flow:
$ $
[0,1]2
Discretize kinetic energy: F (ψ, ω) =
1 4
)
2≤i,j≤N−1
%&
∂ψ ∂y
'2
+
&
∂ψ ∂x
'2 (
dxdy
* +2 * +2 ψi+1,j − ψi−1,j + ψi,j+1 − ψi,j−1
The cavity flow optimization problem CF(R, N) min F (ψ, ω) s.t. DSCF(R, N)
Proposition a) CF(0, N) is a convex quadratic for any N. b) CF(R, N) is non-convex quadratic for any N, if R )= 0. Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
7 / 26
Sparsity pattern It can be shown that CF(R, N) satisfies a some structured sparsity pattern.
Figure: Chordal correlative sparsity pattern matrix for CF (R, 20)
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
8 / 26
Sparsity pattern It can be shown that CF(R, N) satisfies a some structured sparsity pattern.
Figure: Chordal correlative sparsity pattern matrix for CF (R, 20)
Therefore, the sparse SDP relaxations [Waki et al.] are efficient to solve CF(R, N)! Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
8 / 26
Sparsity pattern It can be shown that CF(R, N) satisfies a some structured sparsity pattern.
Figure: Chordal correlative sparsity pattern matrix for CF (R, 20)
Therefore, the sparse SDP relaxations [Waki et al.] are efficient to solve CF(R, N)! We will apply SDPR(w ) with order w ∈ {1, 2}. Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
8 / 26
Outline
1
The discrete steady cavity flow problem
2
Sparse SDP relaxation method Improving the accuracy of SDPR Gröbner basis method and SDPR Enumeration algorithm Numerical results
3
Relations of Reynolds number R and CF(R, N)
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
9 / 26
How to improve the accuracy of SDPR Problem SDPR(1) and SDPR(2) may not yield accurate approximation to the global optimizers of CF(R, N).
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
10 / 26
How to improve the accuracy of SDPR Problem SDPR(1) and SDPR(2) may not yield accurate approximation to the global optimizers of CF(R, N).
Technique 1 Impose tight lower and upper bounds for all ψi and ωi , ψ ω ω lbdψ i ≤ ψi ≤ ubdi and lbdi ≤ ωi ≤ ubdi
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
∀ 1 ≤ i ≤ N 2.
ISSAC 2009
10 / 26
How to improve the accuracy of SDPR Problem SDPR(1) and SDPR(2) may not yield accurate approximation to the global optimizers of CF(R, N).
Technique 1 Impose tight lower and upper bounds for all ψi and ωi , ψ ω ω lbdψ i ≤ ψi ≤ ubdi and lbdi ≤ ωi ≤ ubdi
∀ 1 ≤ i ≤ N 2.
Technique 2 Apply locally convergent optimization techniques like Newton’s method for nonlinear systems or sequential quadratic programming (SQP) starting from the SDPR(w ) solution.
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
10 / 26
Combining these techniques yields
The SDPR method
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
11 / 26
Combining these techniques yields
The SDPR method 1
Choose the two parameters R and N.
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
11 / 26
Combining these techniques yields
The SDPR method 1
Choose the two parameters R and N.
2
˜ ω ˜ := (ψ, Apply SDPR(w ) to CF(R, N) and obtain solution u ˜ ).
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
11 / 26
Combining these techniques yields
The SDPR method 1
Choose the two parameters R and N.
2
˜ ω ˜ := (ψ, Apply SDPR(w ) to CF(R, N) and obtain solution u ˜ ).
3
Apply sequential quadratic programming (SQP) to CF(R, N) or ˜, Newton’s method to DSCF(R, N), each of them starting from u and obtain u := (ψ, ω).
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
11 / 26
Outline
1
The discrete steady cavity flow problem
2
Sparse SDP relaxation method Improving the accuracy of SDPR Gröbner basis method and SDPR Enumeration algorithm Numerical results
3
Relations of Reynolds number R and CF(R, N)
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
12 / 26
Gröbner basis method Use Gröbner basis method to tune SDPR method and validate its numerical results!
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
13 / 26
Gröbner basis method Use Gröbner basis method to tune SDPR method and validate its numerical results!
Apply rational univariate representation (RUR) to DSCF(R, 5): Find all complex solutions of a system with 18 variables, 9 appear as linear and 9 appear as quadratic variables. Then, all real solutions can be enumerated w.r.t. their discretized kinetic energy.
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
13 / 26
Gröbner basis method Use Gröbner basis method to tune SDPR method and validate its numerical results!
Apply rational univariate representation (RUR) to DSCF(R, 5): Find all complex solutions of a system with 18 variables, 9 appear as linear and 9 appear as quadratic variables. Then, all real solutions can be enumerated w.r.t. their discretized kinetic energy.
Claim: Results obtained by RUR for N = 5 can be used for tuning SDPR method for N > 5, too!
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
13 / 26
Outline
1
The discrete steady cavity flow problem
2
Sparse SDP relaxation method Improving the accuracy of SDPR Gröbner basis method and SDPR Enumeration algorithm Numerical results
3
Relations of Reynolds number R and CF(R, N)
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
14 / 26
Algorithm to enumerate all solutions of DSCF(R, N) Iterate the 5 steps to approximate the k smallest energy solutions:
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
15 / 26
Algorithm to enumerate all solutions of DSCF(R, N) Iterate the 5 steps to approximate the k smallest energy solutions: 1
Given u (k −1) , the approximation to the (k − 1)th smallest energy solution obtained by solving SDPRk −1 (w ).
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
15 / 26
Algorithm to enumerate all solutions of DSCF(R, N) Iterate the 5 steps to approximate the k smallest energy solutions: 1
2
Given u (k −1) , the approximation to the (k − 1)th smallest energy solution obtained by solving SDPRk −1 (w ). , Choose $k1 , $k2 > 0 and integers b1k , b2k ∈ 1, . . . , (N − 2)2 .
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
15 / 26
Algorithm to enumerate all solutions of DSCF(R, N) Iterate the 5 steps to approximate the k smallest energy solutions: 1
2 3
Given u (k −1) , the approximation to the (k − 1)th smallest energy solution obtained by solving SDPRk −1 (w ). , Choose $k1 , $k2 > 0 and integers b1k , b2k ∈ 1, . . . , (N − 2)2 . Add the following quadratic constraints to SDPRk −1 (w ) and denote the resulting (tighter) SDP relaxation as SDPRk (w ). (k −1) 2 )
(uj − uj
(uj+(N−2)2 −
Mevissen (Tokyo Tech)
≥ $k1 (k −1) uj+(N−2)2 )2
∀1 ≤ j ≤ b1k , ≥ $k2 ∀1 ≤ j ≤ b2k .
Solutions for Polynomial Systems
ISSAC 2009
(1)
15 / 26
Algorithm to enumerate all solutions of DSCF(R, N) Iterate the 5 steps to approximate the k smallest energy solutions: 1
2 3
Given u (k −1) , the approximation to the (k − 1)th smallest energy solution obtained by solving SDPRk −1 (w ). , Choose $k1 , $k2 > 0 and integers b1k , b2k ∈ 1, . . . , (N − 2)2 . Add the following quadratic constraints to SDPRk −1 (w ) and denote the resulting (tighter) SDP relaxation as SDPRk (w ). (k −1) 2 )
(uj − uj
(uj+(N−2)2 − 4
≥ $k1 (k −1) uj+(N−2)2 )2
∀1 ≤ j ≤ b1k , ≥ $k2 ∀1 ≤ j ≤ b2k .
(1)
˜ (k ) . Solve SDPRk (w ) and obtain a first approximation u
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
15 / 26
Algorithm to enumerate all solutions of DSCF(R, N) Iterate the 5 steps to approximate the k smallest energy solutions: 1
2 3
Given u (k −1) , the approximation to the (k − 1)th smallest energy solution obtained by solving SDPRk −1 (w ). , Choose $k1 , $k2 > 0 and integers b1k , b2k ∈ 1, . . . , (N − 2)2 . Add the following quadratic constraints to SDPRk −1 (w ) and denote the resulting (tighter) SDP relaxation as SDPRk (w ). (k −1) 2 )
(uj − uj
(uj+(N−2)2 − 4 5
≥ $k1 (k −1) uj+(N−2)2 )2
∀1 ≤ j ≤ b1k , ≥ $k2 ∀1 ≤ j ≤ b2k .
(1)
˜ (k ) . Solve SDPRk (w ) and obtain a first approximation u ˜ (k ) as starting point. Obtain Apply Newton’s method or SQP with u u (k ) as an approximation to u (k )$ .
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
15 / 26
Algorithm to enumerate all solutions of DSCF(R, N) Iterate the 5 steps to approximate the k smallest energy solutions: 1
2 3
Given u (k −1) , the approximation to the (k − 1)th smallest energy solution obtained by solving SDPRk −1 (w ). , Choose $k1 , $k2 > 0 and integers b1k , b2k ∈ 1, . . . , (N − 2)2 . Add the following quadratic constraints to SDPRk −1 (w ) and denote the resulting (tighter) SDP relaxation as SDPRk (w ). (k −1) 2 )
(uj − uj
(uj+(N−2)2 − 4 5
≥ $k1 (k −1) uj+(N−2)2 )2
∀1 ≤ j ≤ b1k , ≥ $k2 ∀1 ≤ j ≤ b2k .
(1)
˜ (k ) . Solve SDPRk (w ) and obtain a first approximation u ˜ (k ) as starting point. Obtain Apply Newton’s method or SQP with u u (k ) as an approximation to u (k )$ .
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
15 / 26
Algorithm to enumerate all solutions of DSCF(R, N) Iterate the 5 steps to approximate the k smallest energy solutions: 1
2 3
Given u (k −1) , the approximation to the (k − 1)th smallest energy solution obtained by solving SDPRk −1 (w ). , Choose $k1 , $k2 > 0 and integers b1k , b2k ∈ 1, . . . , (N − 2)2 . Add the following quadratic constraints to SDPRk −1 (w ) and denote the resulting (tighter) SDP relaxation as SDPRk (w ). (k −1) 2 )
(uj − uj
(uj+(N−2)2 − 4 5
≥ $k1 (k −1) uj+(N−2)2 )2
∀1 ≤ j ≤ b1k , ≥ $k2 ∀1 ≤ j ≤ b2k .
(1)
˜ (k ) . Solve SDPRk (w ) and obtain a first approximation u ˜ (k ) as starting point. Obtain Apply Newton’s method or SQP with u u (k ) as an approximation to u (k )$ .
Constraints (1) have shown to be superior to alternatives. Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
15 / 26
Convergence result for the enumeration algorithm Proposition 2 Let R and N be fixed, (u (1) , . . . , u (k −1) ) be the output of the first (k − 1) iterations. If this output is a sufficiently close approximation of (u (1)$ , . . . , u (k −1)$ ), and if DSCF(R, N) is finite and distinct in terms of F , i.e. F (u (1)$ ) < F (u (2)$ ) < . . ., then there exist b ∈ {1, . . . , n} and $ ∈ Rb such that u (k ) from the kth iteration satisfies u (k ) (w ) → u (k )$
Mevissen (Tokyo Tech)
when w → ∞.
Solutions for Polynomial Systems
ISSAC 2009
16 / 26
Convergence result for the enumeration algorithm Proposition 2 Let R and N be fixed, (u (1) , . . . , u (k −1) ) be the output of the first (k − 1) iterations. If this output is a sufficiently close approximation of (u (1)$ , . . . , u (k −1)$ ), and if DSCF(R, N) is finite and distinct in terms of F , i.e. F (u (1)$ ) < F (u (2)$ ) < . . ., then there exist b ∈ {1, . . . , n} and $ ∈ Rb such that u (k ) from the kth iteration satisfies u (k ) (w ) → u (k )$
when w → ∞.
Proof: Apply of Lasserre’s theorem to sequence of POPs generated by the algorithm.
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
16 / 26
Convergence result for the enumeration algorithm Proposition 2 Let R and N be fixed, (u (1) , . . . , u (k −1) ) be the output of the first (k − 1) iterations. If this output is a sufficiently close approximation of (u (1)$ , . . . , u (k −1)$ ), and if DSCF(R, N) is finite and distinct in terms of F , i.e. F (u (1)$ ) < F (u (2)$ ) < . . ., then there exist b ∈ {1, . . . , n} and $ ∈ Rb such that u (k ) from the kth iteration satisfies u (k ) (w ) → u (k )$
when w → ∞.
Proof: Apply of Lasserre’s theorem to sequence of POPs generated by the algorithm.
But: In numerical experiments we are restricted to w ∈ {1, 2}.
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
16 / 26
Convergence result for the enumeration algorithm Proposition 2 Let R and N be fixed, (u (1) , . . . , u (k −1) ) be the output of the first (k − 1) iterations. If this output is a sufficiently close approximation of (u (1)$ , . . . , u (k −1)$ ), and if DSCF(R, N) is finite and distinct in terms of F , i.e. F (u (1)$ ) < F (u (2)$ ) < . . ., then there exist b ∈ {1, . . . , n} and $ ∈ Rb such that u (k ) from the kth iteration satisfies u (k ) (w ) → u (k )$
when w → ∞.
Proof: Apply of Lasserre’s theorem to sequence of POPs generated by the algorithm.
But: In numerical experiments we are restricted to w ∈ {1, 2}.
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
16 / 26
Convergence result for the enumeration algorithm Proposition 2 Let R and N be fixed, (u (1) , . . . , u (k −1) ) be the output of the first (k − 1) iterations. If this output is a sufficiently close approximation of (u (1)$ , . . . , u (k −1)$ ), and if DSCF(R, N) is finite and distinct in terms of F , i.e. F (u (1)$ ) < F (u (2)$ ) < . . ., then there exist b ∈ {1, . . . , n} and $ ∈ Rb such that u (k ) from the kth iteration satisfies u (k ) (w ) → u (k )$
when w → ∞.
Proof: Apply of Lasserre’s theorem to sequence of POPs generated by the algorithm.
But: In numerical experiments we are restricted to w ∈ {1, 2}. Use results from RUR for N = 5 to tune $ and b for N ≥ 5. Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
16 / 26
Outline
1
The discrete steady cavity flow problem
2
Sparse SDP relaxation method Improving the accuracy of SDPR Gröbner basis method and SDPR Enumeration algorithm Numerical results
3
Relations of Reynolds number R and CF(R, N)
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
17 / 26
CF(4000,5) k 0 1 2
$k1 1e-3 1e-3
w 1 1 1
b1k 3 3
b2k 0 0
tC 2 5 8
F (u (k ) ) 4.6e-4 6.3e-4 1.0e-3
$sc 2e-10 5e-4 5e-4
1
1
1
0.9
0.9
0.9
0.8
0.8
0.8
0.7
0.7
0.7
0.6
0.6
0.6
0.5
0.5
0.5
0.4
0.4
0.4
0.3
0.3
0.3
0.2
0.2
0.2
0.1
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
solution u (0) u (1) u (2)
0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Can verify by Gröbner basis method [RUR]: u (0) , u (1) and u (2) are indeed the 3 smallest energy solutions! Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
18 / 26
CF(20000,7) k 0 1 1 1 2
$k1 1e-3 1e-6 1e-5 1e-5
w 1 1 1 1 1
b1k 1 5 5 5
b2k 0 0 0 0
tC 2 5 5 9 14
F (u (k ) ) 3.4e-4 3.4e-4 3.4e-4 5.9e-4 5.2e-3
$sc 3e-7 5e-4 6e-6 5e-6 5e-6
1
1
1
0.9
0.9
0.9
0.8
0.8
0.8
0.7
0.7
0.7
0.6
0.6
0.6
0.5
0.5
0.5
0.4
0.4
0.4
0.3
0.3
0.3
0.2
0.2
0.2
0.1
0.1
0
0
0
0.2
0.4
0.6
0.8
u (0) Mevissen (Tokyo Tech)
1
solution u (0) u (0) u (0) u (1) u (2)
0.1
0
0.2
0.4
0.6
0.8
1
u (1) Solutions for Polynomial Systems
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
u (2) ISSAC 2009
19 / 26
CF(20000,7) k 0 1 1 1 2
$k1 1e-3 1e-6 1e-5 1e-5
w 1 1 1 1 1
b1k 1 5 5 5
b2k 0 0 0 0
tC 2 5 5 9 14
F (u (k ) ) 3.4e-4 3.4e-4 3.4e-4 5.9e-4 5.2e-3
$sc 3e-7 5e-4 6e-6 5e-6 5e-6
1
1
1
0.9
0.9
0.9
0.8
0.8
0.8
0.7
0.7
0.7
0.6
0.6
0.6
0.5
0.5
0.5
0.4
0.4
0.4
0.3
0.3
0.3
0.2
0.2
0.2
0.1
0.1
0
0
0
0.2
0.4
0.6
0.8
1
solution u (0) u (0) u (0) u (1) u (2)
0.1
0
0.2
u (0)
0.4
0.6
0.8
1
u (1)
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
u (2)
Right choice for $ and b crucial! Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
19 / 26
CF(40000,7) k 0 1 2 3 0
$k1 5e-6 5e-6 8e-6 -
w 1 1 1 1 2
b1k 5 5 5 -
b2k 0 0 0 -
tC 3 7 11 16 5872
$sc 2e-7 6e-9 3e-6 5e-6 8e-10
F (u (k ) ) 3.4e-4 7.3e-4 5.9e-4 2.3e-4 2.6e-4
1
1
1
0.9
0.9
0.9
0.8
0.8
0.8
0.7
0.7
0.7
0.6
0.6
0.6
0.5
0.5
0.5
0.4
0.4
0.4
0.3
0.3
0.3
0.2
0.2
0.2
0.1
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
u (0) (1)
1
0
solution u (0) (1) u (1) (1) u (2) (1) u (3) (1) u (0) (2)
0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
u (0) (2)
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
u (3) (1)
Enumeration algorithm more efficient in approximating global minimizer than classical SDP relaxation! Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
20 / 26
Outline
1
The discrete steady cavity flow problem
2
Sparse SDP relaxation method Improving the accuracy of SDPR Gröbner basis method and SDPR Enumeration algorithm Numerical results
3
Relations of Reynolds number R and CF(R, N)
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
21 / 26
Minimal kinetic energy solution for increasing R Observe: DSCF(R, N) is more difficult to solve for larger R.
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
22 / 26
Minimal kinetic energy solution for increasing R Observe: DSCF(R, N) is more difficult to solve for larger R. Question: How does u $ behave for increasing R?
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
22 / 26
Minimal kinetic energy solution for increasing R Observe: DSCF(R, N) is more difficult to solve for larger R. Question: How does u $ behave for increasing R?
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
22 / 26
Minimal kinetic energy solution for increasing R Observe: DSCF(R, N) is more difficult to solve for larger R. Question: How does u $ behave for increasing R? Compare the SDPR method results with
Naive homotopy-like continuation method
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
22 / 26
Minimal kinetic energy solution for increasing R Observe: DSCF(R, N) is more difficult to solve for larger R. Question: How does u $ behave for increasing R? Compare the SDPR method results with
Naive homotopy-like continuation method 1
Choose R # , N and a step size ∆R.
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
22 / 26
Minimal kinetic energy solution for increasing R Observe: DSCF(R, N) is more difficult to solve for larger R. Question: How does u $ behave for increasing R? Compare the SDPR method results with
Naive homotopy-like continuation method 1
Choose R # , N and a step size ∆R.
2
Solve DSCF(0, N) and obtain its unique solution u 0 .
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
22 / 26
Minimal kinetic energy solution for increasing R Observe: DSCF(R, N) is more difficult to solve for larger R. Question: How does u $ behave for increasing R? Compare the SDPR method results with
Naive homotopy-like continuation method 1
Choose R # , N and a step size ∆R.
2
Solve DSCF(0, N) and obtain its unique solution u 0 .
3
Increase R k −1 : R k = R k −1 + ∆R
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
22 / 26
Minimal kinetic energy solution for increasing R Observe: DSCF(R, N) is more difficult to solve for larger R. Question: How does u $ behave for increasing R? Compare the SDPR method results with
Naive homotopy-like continuation method 1
Choose R # , N and a step size ∆R.
2
Solve DSCF(0, N) and obtain its unique solution u 0 .
3
Increase R k −1 : R k = R k −1 + ∆R
4
Apply Newton’s method to DSCF(R k , N) starting from u k −1 . Obtain solution u k as an approximation to a solution of DSCF(R k , N).
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
22 / 26
Minimal kinetic energy solution for increasing R Observe: DSCF(R, N) is more difficult to solve for larger R. Question: How does u $ behave for increasing R? Compare the SDPR method results with
Naive homotopy-like continuation method 1
Choose R # , N and a step size ∆R.
2
Solve DSCF(0, N) and obtain its unique solution u 0 .
3
Increase R k −1 : R k = R k −1 + ∆R
4
5
Apply Newton’s method to DSCF(R k , N) starting from u k −1 . Obtain solution u k as an approximation to a solution of DSCF(R k , N). Iterate 3. and 4. until the desired Reynold’s number R # is reached.
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
22 / 26
Minimal kinetic energy solution for increasing R Observe: DSCF(R, N) is more difficult to solve for larger R. Question: How does u $ behave for increasing R? Compare the SDPR method results with
Naive homotopy-like continuation method 1
Choose R # , N and a step size ∆R.
2
Solve DSCF(0, N) and obtain its unique solution u 0 .
3
Increase R k −1 : R k = R k −1 + ∆R
4
5
Apply Newton’s method to DSCF(R k , N) starting from u k −1 . Obtain solution u k as an approximation to a solution of DSCF(R k , N). Iterate 3. and 4. until the desired Reynold’s number R # is reached.
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
22 / 26
Minimal kinetic energy solution for increasing R Observe: DSCF(R, N) is more difficult to solve for larger R. Question: How does u $ behave for increasing R? Compare the SDPR method results with
Naive homotopy-like continuation method 1
Choose R # , N and a step size ∆R.
2
Solve DSCF(0, N) and obtain its unique solution u 0 .
3
Increase R k −1 : R k = R k −1 + ∆R
4
5
Apply Newton’s method to DSCF(R k , N) starting from u k −1 . Obtain solution u k as an approximation to a solution of DSCF(R k , N). Iterate 3. and 4. until the desired Reynold’s number R # is reached.
Continuation method is not guaranteed to find the minimizer of F ! Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
22 / 26
CF(R,5) R 0 100 500 1000 2000 4000 6000 10000 20000 30000 100000
NC 1 37 37 37 37 37 36 35 35 35 34
Mevissen (Tokyo Tech)
NR 1 13 13 13 13 17 16 17 17 17 16
EC 0.0096 0.0030 6.2e-4 5.4e-4 6.2e-4 6.3e-4 5.7e-4 4.7e-4 4.5e-4 4.5e-4 4.5e-4
ESDPR(1) 0.0096 0.0030 6.2e-4 5e-4 6.2e-4 4.6e-4 4.5e-4 4.5e-4 4.5e-4 4.5e-4 4.5e-4
Solutions for Polynomial Systems
ESDPR(2) 0.0096 0.0030 6.2e-4 5e-4 6.2e-4 4.6e-4 4.5e-4 4.5e-4 3.3e-4 2.5e-4 8.8e-5
Emin 0.0096 0.0030 6.2e-4 5e-4 6.2e-4 4.6e-4 4.5e-4 4.5e-4 3.3e-4 2.5e-4 8.8e-5
ISSAC 2009
23 / 26
CF(R,5) R 0 100 500 1000 2000 4000 6000 10000 20000 30000 100000
NC 1 37 37 37 37 37 36 35 35 35 34
NR 1 13 13 13 13 17 16 17 17 17 16
EC 0.0096 0.0030 6.2e-4 5.4e-4 6.2e-4 6.3e-4 5.7e-4 4.7e-4 4.5e-4 4.5e-4 4.5e-4
ESDPR(1) 0.0096 0.0030 6.2e-4 5e-4 6.2e-4 4.6e-4 4.5e-4 4.5e-4 4.5e-4 4.5e-4 4.5e-4
ESDPR(2) 0.0096 0.0030 6.2e-4 5e-4 6.2e-4 4.6e-4 4.5e-4 4.5e-4 3.3e-4 2.5e-4 8.8e-5
Emin 0.0096 0.0030 6.2e-4 5e-4 6.2e-4 4.6e-4 4.5e-4 4.5e-4 3.3e-4 2.5e-4 8.8e-5
SDPR(1) yields the min energy solution for R ≤ 10000, SDPR(2) yields min energy solution for all R. Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
23 / 26
CF(R,5) 0.01
0.009
0.008
E (R) C
E
(R)
E
(R) / E
(R)
7
8
SDPR(1)
0.007
SDPR(2)
min
0.006
0.005
0.004
0.003
0.002
0.001
0
0
Mevissen (Tokyo Tech)
1
2
3
4
5 R
6
Solutions for Polynomial Systems
9
10 4
x 10
ISSAC 2009
24 / 26
CF(R,5) 0.01
0.009
0.008
E (R) C
E
(R)
E
(R) / E
(R)
7
8
SDPR(1)
0.007
SDPR(2)
min
0.006
0.005
0.004
0.003
0.002
0.001
0
0
1
2
3
4
5 R
6
9
10 4
x 10
uC (R) is not the minimum energy solution!
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
24 / 26
CF(R,5) 0.01
0.009
0.008
E (R) C
E
(R)
E
(R) / E
(R)
7
8
SDPR(1)
0.007
SDPR(2)
min
0.006
0.005
0.004
0.003
0.002
0.001
0
0
1
2
3
4
5 R
6
9
10 4
x 10
uC (R) is not the minimum energy solution! u $ (R) is obtained by SDPR(2) and Emin (R) tends to 0! Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
24 / 26
CF(R,5) 0.01
0.009
0.008
E (R) C
E
(R)
E
(R) / E
(R)
7
8
SDPR(1)
0.007
SDPR(2)
min
0.006
0.005
0.004
0.003
0.002
0.001
0
0
1
2
3
4
5 R
6
9
10 4
x 10
uC (R) is not the minimum energy solution! u $ (R) is obtained by SDPR(2) and Emin (R) tends to 0! Similar behavior can be observed for N ∈ {6, 7}. Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
24 / 26
Behavior for increasing R
Conjecture
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
25 / 26
Behavior for increasing R
Conjecture a) F (u0 (N)) = Emin (0, N) ≥ Emin (R, N) ≥ 0 ∀R ≥ 0.
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
25 / 26
Behavior for increasing R
Conjecture a) F (u0 (N)) = Emin (0, N) ≥ Emin (R, N) ≥ 0 ∀R ≥ 0. b) Emin (R, N) → 0 for R → ∞.
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
25 / 26
Behavior for increasing R
Conjecture a) F (u0 (N)) = Emin (0, N) ≥ Emin (R, N) ≥ 0 ∀R ≥ 0. b) Emin (R, N) → 0 for R → ∞. a) can be used as a certificate for non-optimality of a solution u # (R, N), if F (u # (R, N)) > Emin (0, N).
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
25 / 26
Behavior for increasing R
Conjecture a) F (u0 (N)) = Emin (0, N) ≥ Emin (R, N) ≥ 0 ∀R ≥ 0. b) Emin (R, N) → 0 for R → ∞. a) can be used as a certificate for non-optimality of a solution u # (R, N), if F (u # (R, N)) > Emin (0, N). ˜ (R, N), u # (R, N) is In the case u0 (N) can be continuated to u ˜ (R, N)). non-optimal if F (u # (R, N)) > F (u
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
25 / 26
Summary Min energy solution can be found for many R with SDPR(1) or SDPR(2).
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
26 / 26
Summary Min energy solution can be found for many R with SDPR(1) or SDPR(2). Algorithm for approximately enumerating all solutions of a polynomial system one by one.
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
26 / 26
Summary Min energy solution can be found for many R with SDPR(1) or SDPR(2). Algorithm for approximately enumerating all solutions of a polynomial system one by one. Convergence result.
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
26 / 26
Summary Min energy solution can be found for many R with SDPR(1) or SDPR(2). Algorithm for approximately enumerating all solutions of a polynomial system one by one. Convergence result. Discrete steady cavity flow problem for large R is a challenging test problem for new solvers in numerical algebra.
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
26 / 26
Summary Min energy solution can be found for many R with SDPR(1) or SDPR(2). Algorithm for approximately enumerating all solutions of a polynomial system one by one. Convergence result. Discrete steady cavity flow problem for large R is a challenging test problem for new solvers in numerical algebra. Future Research
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
26 / 26
Summary Min energy solution can be found for many R with SDPR(1) or SDPR(2). Algorithm for approximately enumerating all solutions of a polynomial system one by one. Convergence result. Discrete steady cavity flow problem for large R is a challenging test problem for new solvers in numerical algebra. Future Research Systematic procedure to determine $ and b.
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
26 / 26
Summary Min energy solution can be found for many R with SDPR(1) or SDPR(2). Algorithm for approximately enumerating all solutions of a polynomial system one by one. Convergence result. Discrete steady cavity flow problem for large R is a challenging test problem for new solvers in numerical algebra. Future Research Systematic procedure to determine $ and b. Prove Conjectures.
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
26 / 26
Summary Min energy solution can be found for many R with SDPR(1) or SDPR(2). Algorithm for approximately enumerating all solutions of a polynomial system one by one. Convergence result. Discrete steady cavity flow problem for large R is a challenging test problem for new solvers in numerical algebra. Future Research Systematic procedure to determine $ and b. Prove Conjectures. Apply alternative difference scheme to keep physical invariants under discretization.
Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
26 / 26
Summary Min energy solution can be found for many R with SDPR(1) or SDPR(2). Algorithm for approximately enumerating all solutions of a polynomial system one by one. Convergence result. Discrete steady cavity flow problem for large R is a challenging test problem for new solvers in numerical algebra. Future Research Systematic procedure to determine $ and b. Prove Conjectures. Apply alternative difference scheme to keep physical invariants under discretization. Consider linear instead of quadratic constraints in enumeration algorithm to improve stability. Mevissen (Tokyo Tech)
Solutions for Polynomial Systems
ISSAC 2009
26 / 26
