The energy stored in a stretched elastic sheet near its rest position is modelled by

PHYSICS · Multi-part · Fermi University Physics · Computational Physics · Scientific programming for physicists · Scientific programming for physicists

Scientific Computing for Physicists, Ch Minimisation with guarantees, with constraints and with randomness

Problem

The energy stored in a stretched elastic sheet near its rest position is modelled by f(x,y)=5x2+45y2f(x,y) = 5x^2 + 45y^2, in units of ϵ\epsilon with x,yx,y in units of σ\sigma. A steepest-descent search sits at (x0,y0)=(2,1)(x_0,y_0) = (2, 1). ((a)) Compute the Hessian condition number κ\kappa at the minimum, the rest position (0,0)(0,0). ((b)) Compute the step α∗\alpha^\ast that an exact line search would take along d=−∇f(x0,y0)\mathbf d = -\nabla f(x_0,y_0), using α∗=gTg/gTHg\alpha^\ast = \mathbf g^{\mathsf T}\mathbf g / \mathbf g^{\mathsf T}H\mathbf g with g=∇f(x0,y0)\mathbf g = \nabla f(x_0,y_0). ((c)) Using backtracking from α=1\alpha=1 (halving, c1=10−4c_1 = 10^{-4}), find the number of halvings needed before the Armijo condition is satisfied, and the accepted α\alpha.

Browse the problem library