A calibration stage reports its position every seconds

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Numerical and computational mathematics · Numerical and computational mathematics

Error, Chance and the Machine, Ch Approximating a function: interpolation and differentiation

Problem

A calibration stage reports its position every τ\tau seconds, and its velocity is wanted. Positions are the running sum of velocities, xi=τ∑k≤ivkx_i=\tau\sum_{k\le i}v_k, so recovering the velocities means solving Av=x\mathsf A\mathbf v=\mathbf x, where A\mathsf A is the N×NN\times N lower-triangular matrix with every entry on and below the diagonal equal to τ\tau. For N=50N=50 its largest and smallest singular values are σmax⁡=32.2 τ\sigma_{\max}=32.2\,\tau and σmin⁡=0.500 τ\sigma_{\min}=0.500\,\tau. Tikhonov regularisation minimises ∥Av−x∥2+λ2∥v∥2\|\mathsf A\mathbf v-\mathbf x\|^2+\lambda^2\|\mathbf v\|^2, which multiplies the contribution of each singular component to the solution by σi2/(σi2+λ2)\sigma_i^2/(\sigma_i^2+\lambda^2). ((a)) Find the condition number of A\mathsf A in the 2-norm. ((b)) For λ=5τ\lambda=5\tau, evaluate the damping factor for the largest and for the smallest singular value. ((c)) The singular vector that belongs to σmin⁡\sigma_{\min} changes sign at every step. Explain why damping it acts like smoothing the record before differencing, and how the width of a smoothing window corresponds to λ\lambda.

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