Table 1.
Overview of the regularization tuning methods considered. We have indicated with a ✓ sign for each method, (i) which data/information is required (residual vector, estimated kinetic model parameters or the Jacobian of the residual vector), and (ii) whether the regularization method utilizes further tuning parameters, an estimate of the measurement noise level or a limit for the maximal/minimal regularization parameter. Finally, the last three columns indicate if a computationally expensive procedure is involved, which can be an issue for large scale problems. SVD denotes singular value decomposition
| Regularization method | Computation involves | Further required inputs | Involved computation | ||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Method | Short ID | Refs | Residuals | Estimated | Jacobian | Tuning | Meas. error | α max/α min | Matrix | SVD | Trace |
| parameters | parameter | estimate | inverse | ||||||||
| Discrepancy principle | DP | [113] | ✓ | - | - | ✓ | ✓ | - | - | - | - |
| Modified DP | MDP | [114] | ✓ | - | ✓ | ✓ | ✓ | - | ✓ | - | - |
| Transformed DP | TDP | [115] | ✓ | - | ✓ | ✓ | ✓ | - | ✓ | - | - |
| Monotone Error Rule | MER | [116] | ✓ | ✓ | ✓ | ✓ | ✓ | - | ✓ | - | - |
| Balancing Principle | BP | [117] | - | ✓ | ✓ | ✓ | ✓ | - | - | ✓ | - |
| Hardened Balancing | HBP | [118] | - | ✓ | ✓ | - | - | - | - | ✓ | - |
| Quasi optimality | QO | [80] | - | ✓ | - | - | - | ✓ | - | - | - |
| L–curve method (curvature) | LCC | [105] | ✓ | ✓ | - | - | - | ✓ | - | - | - |
| L–curve method (Reginska) | LCR | [119] | ✓ | ✓ | - | - | - | ✓ | - | - | - |
| Extrapolated Error Rule | EER | [120] | ✓ | - | ✓ | - | - | - | - | - | - |
| Residual Method | RM | [121] | ✓ | - | ✓ | - | - | ✓ | ✓ | - | ✓ |
| Generalized Cross-validation | GCV | [122] | ✓ | - | ✓ | - | - | - | ✓ | - | ✓ |
| GCV (Golub) | GCVG | [107] | ✓ | - | ✓ | - | - | - | ✓ | - | ✓ |
| Robust GCV | RGCV | [108] | ✓ | - | ✓ | ✓ | - | - | ✓ | - | ✓ |
| Strong RGCV | SRGCV | [109] | ✓ | - | ✓ | ✓ | - | - | ✓ | - | ✓ |