10.34, Numerical Methods Applied to Chemical Engineering
Professor William H. Green
Lecture #18: Optimization. Sensitivity Analysis. Introduction: Boundary Value
Problems (BVPs).
Summary: Optimization with Constraints
minx f(x) such that
cm(x) - sm = 0
sm ≥ 0 m = 1 … Ninequalities
2
minx f(x) + ξ(c-s)
sm = 0 m > Ninequalities
penalty method, second term ξ(c-s)2 is optional
KKT conditions: at constrained (local) minimum:
Augmented
Vxf – Σm(λm Vxcm) = 0 Æ
Lagrangian
cm – sm = 0
(LA)
λmcm = 0
{see book}
sm ≥ 0
m = 1 … Ninequalities
sm = 0
equalities
⎛ ∇f − ∑ λ m ∇ c m ⎞
⎜
⎟
cm − sm
⎜
⎟
F ( x) = 0 = ⎜
⎟
λm cm
⎜
⎟
⎜
⎟
s
m
⎝
⎠
⎛c⎞
⎜ ⎟
x = ⎜λ ⎟
⎜s⎟
⎝ ⎠
Newton Æ SQP
If everything is linear: Æ SIMPLEX (i.e. many business problems)
g(x) = 0 Æ xN = G(x1, …, xN-1)
Unconstrained Æ trust region Newton-type BFGS
gigantic Æ conjugate gradient
In Chemical Engineering, the problems often involve models with differential equations:
cost
f(x) =
∑ w (Y (t
i
i
o
return
; x) − Yi (t f ; x) )
i
knobs
what we need
what we produce
(can adjust)
feed composition
Need Jacobian of G with respect to Y; need in stiff solver to solve.
dY
= G (Y ; x)
dt
Y (t 0 ) = Y 0 ( x)
Cite as: William Green, Jr., course materials for 10.34 Numerical Methods Applied to Chemical
Engineering, Fall 2006. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of
Technology. Downloaded on [DD Month YYYY].
Need gradient and f.
To use all of our methods, we need to be able to compute:
⎛ ∂Yσ ,i ∂Yi (t f ) ⎞
∂f
⎟
= ∑ wi ⎜
−
⎜ ∂x
⎟
∂x j
∂
x
i
j
j
⎝
⎠
how do you compute this?
∂ ⎛ ∂Yi ⎞ ⎛⎜
∂G ∂Yn ⎞⎟ ∂Gi
+
⎜
⎟ = ⎜∑
∂x j ⎝ ∂t ⎠ ⎝ n ∂Yn ∂x j ⎟⎠ ∂x j
{“sensitivity of Yi(tf) to xj”}
chain rule
∂ ⎛⎜ ∂Yi
∂t ⎜⎝ ∂x j
⎞
⎟
⎟
⎠
sij
⎛ ∂G
⎞ ∂G
d
sij = ⎜⎜ ∑ i s nj ⎟⎟ + i
dt
⎝ n ∂Yn
⎠ ∂x j
Î
Vf
(for every x we get an Vf that can be used for optimization)
solve this with initial conditions
{Jacobian of G}
J
Have n2 differential equations; stiff; linear in s.
Sensitivity Analysis
Programs to do this: DASPK
SOLVE for s and f simultaneously
DAEPACK
DSL485
DASAC
⎛ ∂G
⎞ ∂G
d
sij = ⎜⎜ ∑ i s nj ⎟⎟ + i
dt
⎝ n ∂Yn
⎠ ∂x j
Initial Conditions
What is sji(t0)?
sji(t0) = 0
{most knobs}
sji(t0) = 1
{for adjustment of Y0}
Professor Barton teaches an advanced course in optimization.
10.34, Numerical Methods Applied to Chemical Engineering
Prof. William Green
Lecture 18
Page 2 of 4
Cite as: William Green, Jr., course materials for 10.34 Numerical Methods Applied to Chemical
Engineering, Fall 2006. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of
Technology. Downloaded on [DD Month YYYY].
Boundary Value Problems (BVPs)
Conservation Laws:
∂φ
/∂t = -V·(φv) - V·JD + S(φ)
convection diffusion reaction
JD = Γ+Vφ
isotropic: JD = -cVφ
for steady-state, isotropic: 0 = -V·(φv) - cV2φ + S(φ)
∀x
Laplacian
Boundary conditions:
Dirichlet
φ(boundary) = number
von Neumann
Vφ(boundary) = number or 0
Symmetry
∂φ
=0
∂x j
φ(x) infinite
{rare to find exact}
adjust: large finite number (104)
φapprox(x) = f(x; c)
N basis
Basis function expansions
φapprox =
∑ c Ψ ( x)
n =1
n
n
(
∫
m =1, N basis
)
Ψm ( x) − ∇ ⋅ (φ approx v) + c∇ 2φ approx + s (φ approx ) = 0
ℜ ( x ,φapprox ,c )
B .C .
( x)
= f ( x;{φi })
φ approx
φ approx ( xi )
called “Residual”
{xi } = mesh grid
some interpolation
Finite difference
approximation
to differential equation
Dirichlet
φ(boundary)
∂φ
∂x
=
x1
φ2 − φ0
x2 − x0
∂φ
∂x
≈
xi
φi +1 − φi −1
xi +1 − xi −1
State how you did
approximation because there
are many ways to do it
{φ} i = 1,N
φ B .C .
von Neumann
10.34, Numerical Methods Applied to Chemical Engineering
Prof. William Green
Lecture 18
Page 3 of 4
Cite as: William Green, Jr., course materials for 10.34 Numerical Methods Applied to Chemical
Engineering, Fall 2006. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of
Technology. Downloaded on [DD Month YYYY].
∂φ
∂x
Usual Æ 2nd order polynomials
given φ0?
x0
∂φ
φ ( x) = φ ( x 0 ) +
∂x
unknown
∂φ
∂x
φ 0 = f (φ1 , φ 2 )
φ ( x1 ) = φ 0 +
φ ( x2 ) = φ0 +
∂φ
∂x
∂φ
∂x
known
=
x0
unknown
φ 2 − f (φ1 , φ 2 )
x 2 − x0
x1
( x1 − x 0 ) +
x0
( x 2 − x0 ) +
x0
φ0 =
( x − x0 ) 2
9 9
( x 2 − x0 ) 2
− φ2
φ1
( x1 − x0 ) 2
φ0 =
( x2 − x0 ) 2
−1
( x1 − x0 ) 2
If Δx uniform,
1 ∂ 2φ
( x − x0 ) +
2 ∂x 2
x0
1 ∂ 2φ
2 ∂x 2
1 ∂ 2φ
2 ∂x 2
for
( x1 − x0 ) 2 ...
x0
( x 2 − x 0 ) 2 ...
x0
∂φ
∂x
=0
x0
4φ1 − φ 2
3
This is how you find out B.C. with second order polynomial schemes and a finite difference
approximation.
10.34, Numerical Methods Applied to Chemical Engineering
Prof. William Green
Lecture 18
Page 4 of 4
Cite as: William Green, Jr., course materials for 10.34 Numerical Methods Applied to Chemical
Engineering, Fall 2006. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of
Technology. Downloaded on [DD Month YYYY].