Optimization and Computational Fluid Dynamics - Department of ...

3 Mathematical Aspects of CFD-based Optimization 71
Consider the following (locally) convex optimization problem
min gk(pk):=fk(yk(pk),pk) (3.35)
pk
where k =1, 2,...,L is the resolution or discretization parameter, where
L denotes the finest resolution, and pk is the (unconstrained) optimization
variable in the space Vk. For the resolution k, one chooses an appropriate
discretized state variable yk and an accordingly resolved objective evaluation
fk. For variables defined on Vk, we introduce the inner product (·, ·)k with
associated norm �y�k =(y,y) 1/2
k . Between spaces Vk, restriction operators
I k−1
k
: Vk → Vk−1 and prolongation operators I k k−1 : Vk−1 → Vk are defined.
We require that (I k−1
k y,v)k−1 =(y,I k k−1 v)k for all y ∈ Vk and v ∈ Vk−1.
On each space, denote with Sk an optimization algorithm, e.g., a gradient
of the solution to (3.35),
based technique. Given an initial approximation p0 k
the application of Sk results in gk(Sk(p0 k )) 1 :
1. Pre-optimization. Define p1 k = Sk(p0 k ).
2. Set up and solve a coarse-grid minimization problem. Define p1 k−1 =
I k−1
k p1 k and σk−1 = ∇gk−1(y1 k−1 ) − Ik−1
k ∇gk(y1 k ). The coarse-grid minimization
problem is given by
�
min gk−1(pk−1) − σ
pk−1
T �
k−1 pk−1 . (3.36)
Apply one cycle of MG/OPT(k-1) to Eq. (3.36) to obtain p 2 k−1 .
3. Line-search and coarse-grid correction. Perform a line search in the I k k−1 (p2 k−1 −
p 1 k−1 ) direction to obtain τk. The coarse-grid correction is given by
p 2 k = p 1 k + τk I k k−1(p 2 k−1 − I k−1
k p 1 k) .
4. Post-optimization. Define p 3 k = Sk(p 2 k ).
Roughly speaking, the essential guideline for constructing gk on coarse
levels is that it must sufficiently well approximate the convexity properties
of the functional f(y(·), ·) at finest resolution. In addition, we have that the
gradient of the coarse-grid functional at p1 k−1 = Ik−1
k p1 k equals the restriction
of the gradient of the fine-grid functional at p1 k . In fact, by adding the term
−σT k−1 pk−1 in Step 2, we have that
∇ � gk−1(pk−1) − σ T k−1 pk−1
�
|p1 = I
k−1 k−1
k ∇gk(p 1 k ) .