Heat Source Identification Based on L 1 Optimization

Transcription

1 Heat Source Identification Based on L 1 Optimization Yingying Li, Stanley Osher and Richard Tsai August 27, 2009 Yingying Li, Stanley Osher and Richard Tsai () L 1 Heat Source Identification Aug 27, / 19

2 Outline Heat source identification Solution by L 1 minimization and Bregman iteration Strategies to reduce computational cost Adaptive solution with successive samples Yingying Li, Stanley Osher and Richard Tsai () L 1 Heat Source Identification Aug 27, / 19

3 Heat Source Identification Consider the heat equation with sparse initial condition, { u t = ( a(x)u x )x t > 0, u 0 = k c kδ(x x k ) t = 0, c k > 0. Suppose at time T we measure samples f ij = u(x i, y j ). Can we recover u 0 knowing that it is a sum of delta functions? We can define a linear operator A T such that A T [u 0 ] = u t=t, so essentially the problem is inverting A T (but A T is ill-conditioned). Yingying Li, Stanley Osher and Richard Tsai () L 1 Heat Source Identification Aug 27, / 19

4 Applications The initial condition of a diffusive process can have real-life importance. Consider finding the location of buried pollutants. The buried pollutants are source conditions that diffuse over time. With concentration measurements at enough points, we can deduce where the pollution is hidden. Yingying Li, Stanley Osher and Richard Tsai () L 1 Heat Source Identification Aug 27, / 19

5 L 0 L 1 The goal is to use as few delta functions as possible to match up with the measurements, which is an L 0 minimization problem. As in compressed sensing, we approach the inverse by solving an L 1 minimization problem min u u 1 subject to (Au) i = f i. We shall use Bregman iteration for the minimization. Yingying Li, Stanley Osher and Richard Tsai () L 1 Heat Source Identification Aug 27, / 19

6 Bregman iterative method The constrained problem: The unconstrained problem: min u u R n 1 subject to Au = f. min E(u) = u u R n l 1 + λ Au f 2 l, 2 Algorithm: 1: Initialize: k = 0, u 0 = 0, f 0 = f. 2: while Au f 2 f 2 > tolerance do 3: Solve u k+1 arg min u u 1 + λ Au f k 2 2 by coordinate descent 4: f k+1 f k + (f Au k+1 ) 5: k k + 1 6: end while Yingying Li, Stanley Osher and Richard Tsai () L 1 Heat Source Identification Aug 27, / 19

7 Support Restriction Define D k = supp(u k ), S k = D k {neighboring points of D k }. u k+1 = arg min{ u l 1 + λ Au f 2 2 : supp(u) S k }. Yingying Li, Stanley Osher and Richard Tsai () L 1 Heat Source Identification Aug 27, / 19

8 Exclusion Region Suppose the spike amplitudes are bounded from below by α min > 0, then k s.t. A[α min δ y ](x k ) > f (x k ) = u 0 (y) = 0. periodic boundry (a = 1) zero boundry Yingying Li, Stanley Osher and Richard Tsai () L 1 Heat Source Identification Aug 27, / 19

9 Numerical Results Exact u 0 f Recovery The orange shows the distribution of a(x). In the left figure, the blue dots indicate the heat sources and the red stars are samples. Here a(x) is a nonnegative smooth function. In the middle figure, the blue shows the distribution of u at T = The last figure shows the recovery result. Yingying Li, Stanley Osher and Richard Tsai () L 1 Heat Source Identification Aug 27, / 19

10 Numerical Results Exact u 0 f Recovery The orange circle indicates the distribution of a(x), and a(x) = 0.2 inside of the circle and a(x) = 1 the rest of the area. Yingying Li, Stanley Osher and Richard Tsai () L 1 Heat Source Identification Aug 27, / 19

11 Successive Greedy Solution 1 Solve the heat source identification problem with k samples; 2 Using the solution u k to choose the k + 1 sample; 3 Iterate. Why solve in a successive greedy manner? 1 We want to recover all heat sources by using as few measurements as possible under unknown the total number of heat sources. 2 Increase the stability of the L 1 minimization problem. The drawback of L 1 minimum: produce ghost solution under few measurements. With the undetermined constaint, the L 1 solution tends to be closer to the measurements than it should. Yingying Li, Stanley Osher and Richard Tsai () L 1 Heat Source Identification Aug 27, / 19

12 Covering Region Suppose the spike amplitudes are bounded from below by α min > 0, then k s.t. A[α min δ y ](x k ) threshold = y {Covering region}. We define a way to measure how much a point is covered by samples, V (x) = A( j δ xj ). The bigger V (x) is, the more information available at x. Yingying Li, Stanley Osher and Richard Tsai () L 1 Heat Source Identification Aug 27, / 19

13 Refine locally or Explore further Refine locally: If u k varies significantly from u k 1, then choose the next sampling location x k+1 by x k+1 = arg max G σ u k G σ u k 1. x:x B r (x j ) Explore further: Otherwise, x k+1 = arg min x:x B r (x j ) V (x). Yingying Li, Stanley Osher and Richard Tsai () L 1 Heat Source Identification Aug 27, / 19

14 Exact u 0 f Yingying Li, Stanley Osher and Richard Tsai () L 1 Heat Source Identification Aug 27, / 19

15 Recovery ingying Li, Stanley Osher and Richard Tsai () L 1 Heat Source Identification Aug 27, / 19

16 Recovery ingying Li, Stanley Osher and Richard Tsai () L 1 Heat Source Identification Aug 27, / 19

17 Recovery ingying Li, Stanley Osher and Richard Tsai () L 1 Heat Source Identification Aug 27, / 19

18 Recovery ingying Li, Stanley Osher and Richard Tsai () L 1 Heat Source Identification Aug 27, / 19

19 Recovery ingying Li, Stanley Osher and Richard Tsai () L 1 Heat Source Identification Aug 27, / 19

20 Recovery ingying Li, Stanley Osher and Richard Tsai () L 1 Heat Source Identification Aug 27, / 19

21 Recovery ingying Li, Stanley Osher and Richard Tsai () L 1 Heat Source Identification Aug 27, / 19

22 Recovery ingying Li, Stanley Osher and Richard Tsai () L 1 Heat Source Identification Aug 27, / 19

23 Recovery ingying Li, Stanley Osher and Richard Tsai () L 1 Heat Source Identification Aug 27, / 19

24 Recovery ingying Li, Stanley Osher and Richard Tsai () L 1 Heat Source Identification Aug 27, / 19

25 Recovery ingying Li, Stanley Osher and Richard Tsai () L 1 Heat Source Identification Aug 27, / 19

26 Recovery ingying Li, Stanley Osher and Richard Tsai () L 1 Heat Source Identification Aug 27, / 19

27 Recovery ingying Li, Stanley Osher and Richard Tsai () L 1 Heat Source Identification Aug 27, / 19

28 Recovery ingying Li, Stanley Osher and Richard Tsai () L 1 Heat Source Identification Aug 27, / 19

29 Recovery ingying Li, Stanley Osher and Richard Tsai () L 1 Heat Source Identification Aug 27, / 19

30 Recovery ingying Li, Stanley Osher and Richard Tsai () L 1 Heat Source Identification Aug 27, / 19

31 Recovery ingying Li, Stanley Osher and Richard Tsai () L 1 Heat Source Identification Aug 27, / 19

32 Recovery Done! ingying Li, Stanley Osher and Richard Tsai () L 1 Heat Source Identification Aug 27, / 19

33 Comparison With the same number of random samples, the solutions of least square and L 1 minimization are not as accurate as the successive greedy solution. Exact u 0 Least square L 1 Yingying Li, Stanley Osher and Richard Tsai () L 1 Heat Source Identification Aug 27, / 19

34 Future work In the successive greedy method, we add the total length of the path into consideration. So we also N min x k+1 x k under the minimum N. k=1 In practice, it is like there is one moving sensor that tries to find out all the heat sources with the minimum measurements and minimum moving distance. Unknown geometrical environment with obstacles, our sense can discover where its view is not blocked, the goal is to figure out the locations of obstacles. Yingying Li, Stanley Osher and Richard Tsai () L 1 Heat Source Identification Aug 27, / 19

35 Questions? Yingying Li, Stanley Osher and Richard Tsai () L 1 Heat Source Identification Aug 27, / 19

36 Thanks for your attention! Yingying Li, Stanley Osher and Richard Tsai () L 1 Heat Source Identification Aug 27, / 19