IMSP_MUMPS_Solver

MUMPS Solver for the following 2D Inverse medium scattering problem(IMSP)

• $\phi_0$ : total field
• $\phi^i$ : inident field
• $\psi$ : scattered field
• $q$ : scatterer, with compact support in $\mathbb{R}^2$

Boundary Condition

Summerfield Radiation Condition

Reduce the problem by an artificial surface: First-order Absorbing Boundary Condition

Forward problem

For fixed $\phi^i,q,k$

Induce

Inverse problem

Determine the scatter $q(x)$ from the measurements of $\psi|_{\partial \Omega}$

Incident Wave (plane wave)

Uniformly, Define $\phi^i$ from $m$ different angles:

Optimization Model

• $M$ : the matrix to generate $\psi|_{\partial \Omega}$ from $\psi$
• $q_t$ : the ground truth of the scatterer $q$
• $\operatorname{Data}(q_t)$ : $\psi|_{\partial \Omega}$ aroused by $q_t$

Method

• Solve the Forward problem(PDE) by FDM to generate the equation and MUMPS to solve it
• Derive $\frac{\partial \mathscr{F}_k}{\partial q}$ and $\frac{\partial J}{\partial q}$ through functional analysis
• Use L-BFGS to solve the total optimization problem

Scatterer

Parameters

• k : the frequency of the incident wave
• m : the number of incident angles
• maxq : the strength of the scatterer
• nosielevel : the nosie level of the collected boundary data
  We use a grid of 64 on $[0,1]^2$ to discrete the problem. The Initial gauess of $q$ is zero.
  As for L-BFGS, we set gtol = 1e-10 and maxiter=50.

Usage

• MUMPS Install
• bash scripts/xxx.sh

Results

• iter : iter after iteration termination
• $J_0$ : initial value of $J$
• rel-J : $J_{res}$ / $J_0$
• rel-err : relative error between $q_{res}$ and $q_t$ after iteration termination
• time : time per iter after iteration termination

T shape

k	m	maxq	noise	iter	$J_0$	rel-J	rel-err	time
80	64	0.1	0.0	50	1.71e2	5.28e-7	8.61%	4.77
10,20,40,60,80					3.13e2	6.16e-7	8.63%	19.27
40,60,80					2.99e2	6.07e-7	8.62%	12.59
60,80,100					6.01e2	5.28e-8	1.05%	12.61
	16				1.72e2	1.93e-5	13.70%	1.18
	32				1.71e2	1.00e-6	8.65%	2.14
	128				1.71e2	4.86e-7	8.61%	8.54
		0.01			2.15e1	4.39e-7	9.48%	5.71
		0.3			1.74e2	5.72e-2	154.17%	5.44
		0.5			5.62e2	1.98e-1	138.90%	5.38
		0.7			4.84e1	1.35e-1	116.30%	5.62
		1.0			3.47e1	1.39e-1	113.09%	5.52
			0.1		1.75e2	2.57e-2	25.75%	5.75
			0.3		2.14e2	1.87e-1	71.51%	5.44
			0.5		2.91e2	3.81e-1	134.54%	5.43

Gauss shape

k	m	maxq	noise	iter	$J_0$	rel-J	rel-err	time
20	64	0.1	0.0	50	1.26e0	7.86e-8	1.50%	3.99
4					3.98e-2	4.18e-5	60.75%	3.92
10					3.53e-1	5.62e-6	21.36%	4.16
15					7.30e-1	6.24e-7	6.59%	3.91
40					4.98e0	1.59e-9	0.10%	3.92
10,15					1.08e0	5.80e-7	6.66%	7.43
10,15,20					2.34e0	4.58e-8	1.46%	12.05
10,20,40					6.59e0	3.44e-9	0.10%	11.89
	16				1.26e0	7.36e-8	1.52%	1.13
	32				1.26e0	7.86e-8	1.50%	2.08
	128				1.26e0	7.86e-8	1.50%	7.75
		0.01			1.27e-1	1.67e-7	1.83%	6.55
		0.2			2.47e0	5.41e-8	1.20%	5.89
		0.3			3.63e0	3.64e-8	0.93%	5.64
		0.4			4.73e0	2.24e-8	0.68%	5.63
		0.5			5.76e0	1.72e-8	0.54%	5.48
			0.1		1.27e0	1.18e-2	22.12%	6.07
			0.3		1.38e0	9.55e-2	71.12%	5.82
			0.5		1.63e0	2.29e-1	117.78%	5.65

Multi-Gauss shape

k	m	maxq	noise	iter	$J_0$	rel-J	rel-err	time
20	64	0.1	0.0	50	1.84e0	7.88e-8	1.39%	3.97
4					4.50e-2	7.97e-5	56.69%	4.21
10					5.19e-1	6.35e-6	21.04%	4.08
15					1.09e0	7.28e-7	5.85%	4.03
40					7.24e0	1.14e-8	0.42%	4.01
10,15					1.60e0	7.20e-7	5.96%	7.85
10,15,20					3.45e0	7.50e-8	1.48%	12.12
10,20,40					9.61e0	9.42e-9	0.42%	12.55
	16				1.84e0	8.88e-8	1.48%	1.13
	32				1.84e0	7.83e-8	1.38%	2.01
	128				1.84e0	7.88e-8	1.39%	9.49
		0.01			1.86e-1	2.49e-7	1.77%	4.13
		0.2			3.66e0	6.36e-8	1.24%	3.89
		0.3			5.42e0	6.42e-8	1.19%	4.66
		0.4			7.13e0	6.59e-8	1.12%	4.01
		0.5			8.76e0	6.45e-8	1.19%	4.48
			0.1		1.86e0	1.25e-2	20.98%	5.59
			0.3		2.05e0	9.95e-2	69.01%	5.94
			0.5		2.43e0	2.36e-1	101.88%	5.86