A Two-Stage Method for Piecewise-Constant Solution for Fredholm Integral Equations of the First Kind
Abstract
:1. Introduction
2. The Objective Functionals
2.1. Discretization
- (1)
- The discretization of is given by:
- (2)
- Let be the i-th column of the identity matrix, and let:
- (3)
- As for , we simply approximate it by:Here, we omit the factor for the sake of simplicity.
3. Numerical Implementation
Algorithm 1: Damped Newton (DN) method for the minimization of . |
Input an initial guess ; ; Begin iterations Compute and ; Solve to obtain ; Obtain the minimum point of the one-dimensional nonnegative function by using line search to get ; Update the approximate solution: ; Check the termination condition: If , break; ; End iterations Output . |
Algorithm 2: Weighted total variation Modica-Mortola (WTVMM) method for estimating piecewise-constant solution of Equation (2). |
Obtain the minimizer of by using the MGN method or the DN method; Obtain a minimizer of by using the DN method with as the initial guess; Output . |
4. Numerical Examples
- (1)
- Since we use as an approximation of , we set β to a very small positive number. We note that if β is significantly larger than , then:
- (2)
- If θ is much larger than , then the weighting function is close to one, i.e.,
- (3)
- We choose the value for α by the help of the discrepancy principle and observe if the numerical solution is near piecewise-constant. After choosing a value for α, we consider adjustment of β and θ: if the numerical solution is over-smooth, we choose a smaller value for β or θ; on the other hand, if the numerical solution is oscillating, we choose a larger value for β or θ.
- (1)
- For the WTVMM method, the main cost lies in obtaining the numerical solution , i.e., minimization of defined by (8).
- (2)
- The numerical solutions obtained by using the WTV method are better than those obtained by the TV method. The numerical solutions obtained by using the WTVMM method can be quite accurate, which can be clearly seen from the relative errors shown in Table 1 and Table 2, the numerical solutions shown in Figure 2c and the point-wise errors shown in Figure 2d.
- (3)
- If the position of the break points in the solution can be identified by the previous step, the approximate solution obtained by the WTVMM method can be very accurate. In the other hand, since the local minimizer of obtained by the DN method is sensitive to the initial guess, we may not be able to obtain a good numerical solution if the approximate solution obtained in the previous step cannot identify the break points well.
5. Concluding Remarks
Acknowledgments
Author Contributions
Conflicts of Interest
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Error | |||
---|---|---|---|
TV | (, , -, -) | 24 | |
WTV | (, , , -) | 11 | |
WTVMM | (, , , 1) | 11 + 4 |
Error | |||
---|---|---|---|
TV | (, , -, -) | 10 | |
WTV | (, , , -) | 13 | |
WTVMM | (, , , 1) | 10+11 |
5.0 | (, , ) | (, , ) | (, , ) | |
(, , ) | (, , ) | (, , ) | ||
(, , ) | (, , ) | (, , ) |
(, , ) | (, , ) | (, , ) | ||
(, , ) | (, , ) | (, , ) | ||
(, , ) | (, , ) | (, , ) |
Error | |||
---|---|---|---|
TV | (, , -, -) | 14 | |
WTV | (, , , -) | 8 | |
WTVMM | (, , , 1) | 8+3 |
Error | |||
---|---|---|---|
TV | (, , -, -) | 16 | |
WTV | (, , , -) | 11 | |
WTVMM | (, , , 1) | 11+4 |
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Lin, F.-R.; Yang, S.-W. A Two-Stage Method for Piecewise-Constant Solution for Fredholm Integral Equations of the First Kind. Mathematics 2017, 5, 28. https://doi.org/10.3390/math5020028
Lin F-R, Yang S-W. A Two-Stage Method for Piecewise-Constant Solution for Fredholm Integral Equations of the First Kind. Mathematics. 2017; 5(2):28. https://doi.org/10.3390/math5020028
Chicago/Turabian StyleLin, Fu-Rong, and Shi-Wei Yang. 2017. "A Two-Stage Method for Piecewise-Constant Solution for Fredholm Integral Equations of the First Kind" Mathematics 5, no. 2: 28. https://doi.org/10.3390/math5020028
APA StyleLin, F.-R., & Yang, S.-W. (2017). A Two-Stage Method for Piecewise-Constant Solution for Fredholm Integral Equations of the First Kind. Mathematics, 5(2), 28. https://doi.org/10.3390/math5020028