An improved numerical method for balanced truncation for symmetric second-order systems

Figures & data

Table 1. Left and right transformation matrices of balanced truncation for second-order systems.

Type	Right transformation	Left transformation
PP
PV
VP
VV

Table

Algorithm 1 Generalized Low-rank ADI iteration (G-LR-ADI)
Input: and G as in (14) and shift parameters . Output: , such that
1: for do
2:  if then
3:   Solve for .
4:   .
5:  else
6:   Solve for .
7:   .
8:   Update low-rank factor .
9:  end if
10: end for

Table

Algorithm 2 Second-Order Low-rank ADI iteration (SO-LR-ADI)
Input: M, D, K, B defining (1) and shift parameters .
Output: Real low-rank solution factor , such that in (14).
1: for do
2:  if then
3:   Solve for .
4:   Set .
5:  else
6:   Solve for .
7:   Set .
8:   Construct iterate
9:  end if
10:  if then
11:   .
12:  else
13:   .
14:   .
15:   Update low-rank solution factor
16:   Set .
17:  end if
18: end for

Table 2. Specifications of the used examples as well as parameters and results of the SO-LR-ADI runs. All timings are given in seconds.

Example	5.1	5.2	5.3	5.4
	3000, 1	3000, 1	17,361, 12	150,001, 5
, (Hz)	1, 10^4	10^−1, 10^4	1, 10^5	10^−3, 10
	30, 45, 45	60, 50, 80	40, 60, 50	60, 100, 80
	10^−8	10^−8	10^−6	10^−6
	57	353	100	369
	0.46	3.25	157.69	1441.54
(avg.)	0.049 (0.001)	0.12 (0.0006)	2.17 (0.041)	15.66 (0.043)
(avg.)	0.43 (0.01)	2.56 (0.013)	92.70 (1.78)	7199.12 (19.88)
	10^−8	10^−8	10^−8	10^−8
	20	35	50	400
	PP	VV	PP	VV
	22	65	34	82
	0.18	0.42	51.92	181.55
(avg.)	0.0073 (0.0024)	0.03 (0.0016)	3.25 (0.2032)	11.69 (3.89)

Figure 1. Reduction results for the Bernoulli beam model resulting from the residual based stopping criterion. (a) Bode plot. (b) Relative error.

Table 3. Largest relative errors of the reduced order models resulting from residual and singular value-based termination of SO-LR-ADI.

	Example
	5.1	5.2	5.3	5.4
Residual-based	1.213 × 10^−10	1.696 × 10^−6	1.025 × 10^−5	5.422 × 10^−4
-based	1.213 × 10^−10	2.679 × 10^−6	3.519 × 10^−5	4.554 × 10^−2

Figure 2. (a) History of normalized residual norm and relative change (26) of singular values for Example 5.2. The black dashed line refers to the tolerance of . (b) Relative errors obtained from both approaches using VV second-order balanced truncation.

Figure 3. History of relative change (26) of different singular values for Example 5.3 (a) and 5.4 (b). The black dashed line refers to the tolerance of .