NUMERICAL OPTIMISATION ASSIGNMENT 6
MARTA BETCKE KIKO RUL·LAN
Minimal Surface Cost Function
[Adapted from Exercise 7.7 from Nocedal-Wright]
The minimum surface problem is a classical application of the calculus of variations and can be found in many textbooks. We wish to find the surface of minimum area, defined on the unit square, that inter- polates a prescribed continuous function on the boundary of the square. In the standard discretization of this problem, the unknowns are the values of the sought-after function z(x, y) on a q × q rectangular mesh of points over the unit square.
More specifically, we divide each edge of the square into q intervals of equal length, yielding (q + 1)2 grid points. We label the grid points as:
x(i−1)(q+1)+1, . . . , xi(q+1) for i = 1, 2, …, q + 1,
so that each value of i generates a line. With each point we associate a variable zi that represents the height of the surface at this point. The values of the function are fixed by the boundary conditions at the 4q grid points on the boundary. The optimization problem is to determine the other (q + 1)2 − 4q variables zi so that the total surface area is minimized.
A typical subsquare in this partition looks as follows:
We denote this square by Aj and note that its area is 1/q2. The desired function is z(x,y), and we wish to compute its surface over Aj. Calculus books show that the area of the surface is given by:
fj(x)=
Approximating the derivatives by finite differences, fj has the form
∂z2 ∂z2
1+ ∂x + ∂y dxdy
(x,y)∈Aj
1 1 q2 2
fj (x) = 2 1 + (z(xj ) − z(xj+q+2))2 + (z(xj+1) − z(xj+q+1))2 q2
. (1)
The function F in surfaceFunc vector.m implements the sum of all fj over the unit square. The gradient and the Hessian of the cost function F are provided in surfaceGrad.m and surfaceHess.m, respectively.
EXERCISE 1
Implement the Line Search Newton – Conjugate Gradient (LS-Newton-CG) algorithm (Algorithm 7.1 from Nocedal-Wright). More help is provided in Cody Coursework.
Submit your implementation via Cody Coursework. [30pt]
EXERCISE 2
You are given an adaptation of the trust region SR-1 function in trustRegionLS.m and the 2D sub- space in solverCM2dSubspaceExtLS.m. Point out and explain the relevant modifications in the solver solverCM2dSubspaceExtLS.m.
Submit your solution via TurnitIn.
EXERCISE 3
Compute and plot
⋆ ⋆ ⋆ ⋆
a minimal area surface function with given the boundary conditions:
Bottom edge: Left edge: Top edge: Right edge:
BB (t) = sin(πt), t ∈ [0, 1]. BL(t) = sin(πt + π), t ∈ [0, 1]. BT (t) = sin(3πt), t ∈ [0, 1]. BR(t) = sin(3πt + π), t ∈ [0, 1].
(a) using the LS-Newton-CG algorithm, (b) using the BFGS algorithm (provided),
(c) using the Trust Region-SR1 algorithm (provided in Ex2).
Discretization with q = 14 yields a problem of reasonable size. Provide any parameters and explanations that you consider relevant in the minimisation.
Submit your solution via TurnitIn. [50pt]
Remark. The submission to TurnitIn should not exceed 6 pages. Avoid submitting code unless explicitly asked for and focus on explaining your results.
[20pt]