Optimal error bound and a generalized Tikhonov regularization method for identifying an unknown source in the Poisson equation. (English) Zbl 1280.65048
Summary: We prove a stability estimate for an inverse heat source problem. Based on the obtained stability estimate, we present a generalized Tikhonov regularization method and obtain the error estimate. Numerical experiment shows that the generalized Tikhonov regularization works well.
MSC:
| 65J20 | Numerical solutions of ill-posed problems in abstract spaces; regularization |
| 35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |
References:
| [1] | DOI: 10.1090/surv/034 · doi:10.1090/surv/034 |
| [2] | Isakov V., Appl. Math. Sci. 127, in: Inverse Problems in Partial Differential Equations (2005) |
| [3] | DOI: 10.1103/PhysRevD.8.1044 · doi:10.1103/PhysRevD.8.1044 |
| [4] | DOI: 10.1080/1068276042000219912 · doi:10.1080/1068276042000219912 |
| [5] | DOI: 10.1088/0266-5611/12/5/002 · Zbl 0858.35131 · doi:10.1088/0266-5611/12/5/002 |
| [6] | DOI: 10.1016/0898-1221(96)00130-7 · Zbl 0858.65098 · doi:10.1016/0898-1221(96)00130-7 |
| [7] | DOI: 10.1016/0893-9659(88)90084-5 · Zbl 0696.35187 · doi:10.1016/0893-9659(88)90084-5 |
| [8] | DOI: 10.1063/1.531978 · Zbl 0876.35130 · doi:10.1063/1.531978 |
| [9] | DOI: 10.1006/enfo.2001.0055 · doi:10.1006/enfo.2001.0055 |
| [10] | DOI: 10.1029/2001WR000223 · doi:10.1029/2001WR000223 |
| [11] | Atmadja J., Water Resources Research 39 pp 1038– |
| [12] | DOI: 10.1088/0266-5611/14/3/010 · Zbl 0917.35156 · doi:10.1088/0266-5611/14/3/010 |
| [13] | DOI: 10.1088/0266-5611/2/4/007 · Zbl 0624.35078 · doi:10.1088/0266-5611/2/4/007 |
| [14] | DOI: 10.1007/BF02404048 · Zbl 0901.65083 · doi:10.1007/BF02404048 |
| [15] | DOI: 10.1088/0266-5611/22/4/011 · Zbl 1112.35147 · doi:10.1088/0266-5611/22/4/011 |
| [16] | DOI: 10.1016/j.jmaa.2005.03.037 · Zbl 1087.35095 · doi:10.1016/j.jmaa.2005.03.037 |
| [17] | DOI: 10.1016/0895-7177(93)90081-9 · Zbl 0799.35228 · doi:10.1016/0895-7177(93)90081-9 |
| [18] | DOI: 10.1007/s10665-005-9023-0 · Zbl 1146.80007 · doi:10.1007/s10665-005-9023-0 |
| [19] | Hào D. N., Numerische Mathematik 68 pp 469– |
| [20] | DOI: 10.1093/imamat/hxm024 · Zbl 1135.65034 · doi:10.1093/imamat/hxm024 |
| [21] | DOI: 10.1016/j.cam.2006.10.026 · Zbl 1135.35097 · doi:10.1016/j.cam.2006.10.026 |
| [22] | DOI: 10.1088/0266-5611/22/4/011 · Zbl 1112.35147 · doi:10.1088/0266-5611/22/4/011 |
| [23] | DOI: 10.1016/j.jmaa.2005.03.037 · Zbl 1087.35095 · doi:10.1016/j.jmaa.2005.03.037 |
| [24] | DOI: 10.1016/j.camwa.2002.11.025 · Zbl 1063.65102 · doi:10.1016/j.camwa.2002.11.025 |
| [25] | DOI: 10.1016/j.cam.2009.01.008 · Zbl 1219.65100 · doi:10.1016/j.cam.2009.01.008 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
