A Particle Method for Solving Fredholm Equations of the First Kind

Figures & data

Fig. 1 Average $\text{ISE}(f_{n+1}^N)$ and runtime for 1,000 repetitions of discretized EMS, SMC and DKDE. The number of bins and of particles/samples N ranges between 10² and 10⁴.

Table 1 Estimates of mean, variance, ISE, 95th-percentile of MSE, KL-divergence and runtime for 1,000 repetitions of EM, EMS, IB, SMC and DKDE for the Gaussian mixture example. The mean of f is 0.43333, the variance is 0.010196. Bold indicates best values.

	Mean	Variance	$\text{ISE}(f_{n+1}^N)$	$\text{MSE}(x_c)$	KL	${\hspace{0.17em}\text{log}\hspace{0.17em}}_{10}$
				(95th)		Runtime / s
EM	0.36667	0.010	3.26	16.32	2299	-6.01
EMS (K)	0.36646	0.012	2.41	8.20	2355	−5.90
EMS (3-point)	0.3668	0.011	1.58	13.04	2303	−5.88
IB	0.43304	0.011	1.71	10.17	2489	−5.29
SMC (500)	0.43303	0.011	0.90	3.42	2484	0.63
SMC (1000)	0.43302	0.011	0.78	3.33	2483	1.87
SMC (5000)	0.43302	0.011	0.55	2.17	2485	3.47
DKDE-pi	0.43288	0.012	0.96	3.38	2483	0.81
DKDE-cv	0.43287	0.014	1.5	4.76	2503	4.15

Fig. 2 Reconstruction of the Shepp–Logan phantom with N = 20, 000 particles and $\epsilon =0.001$. The stopping criterion (18) is satisfied at iteration 15.