# Solving¶

1. How do I evaluate the Jacobian for a solved problem?

2. How do I choose the right linear solver?

When using the TRUST_REGION minimizer, the choice of linear solver is an important decision. It affects solution quality and runtime. Here is a simple way to reason about it.

1. For small (a few hundred parameters) or dense problems use DENSE_QR.

2. For general sparse problems (i.e., the Jacobian matrix has a substantial number of zeros) use SPARSE_NORMAL_CHOLESKY. This requires that you have SuiteSparse or CXSparse installed.

3. For bundle adjustment problems with up to a hundred or so cameras, use DENSE_SCHUR.

4. For larger bundle adjustment problems with sparse Schur Complement/Reduced camera matrices use SPARSE_SCHUR. This requires that you build Ceres with support for SuiteSparse, CXSparse or Eigen’s sparse linear algebra libraries.

If you do not have access to these libraries for whatever reason, ITERATIVE_SCHUR with SCHUR_JACOBI is an excellent alternative.

5. For large bundle adjustment problems (a few thousand cameras or more) use the ITERATIVE_SCHUR solver. There are a number of preconditioner choices here. SCHUR_JACOBI offers an excellent balance of speed and accuracy. This is also the recommended option if you are solving medium sized problems for which DENSE_SCHUR is too slow but SuiteSparse is not available.

Note

If you are solving small to medium sized problems, consider setting Solver::Options::use_explicit_schur_complement to true, it can result in a substantial performance boost.

If you are not satisfied with SCHUR_JACOBI‘s performance try CLUSTER_JACOBI and CLUSTER_TRIDIAGONAL in that order. They require that you have SuiteSparse installed. Both of these preconditioners use a clustering algorithm. Use SINGLE_LINKAGE before CANONICAL_VIEWS.

3. Use Solver::Summary::FullReport() to diagnose performance problems.

When diagnosing Ceres performance issues - runtime and convergence, the first place to start is by looking at the output of Solver::Summary::FullReport. Here is an example

./bin/bundle_adjuster --input ../data/problem-16-22106-pre.txt

iter      cost      cost_change  |gradient|   |step|    tr_ratio  tr_radius  ls_iter  iter_time  total_time
0  4.185660e+06    0.00e+00    2.16e+07   0.00e+00   0.00e+00  1.00e+04       0    7.50e-02    3.58e-01
1  1.980525e+05    3.99e+06    5.34e+06   2.40e+03   9.60e-01  3.00e+04       1    1.84e-01    5.42e-01
2  5.086543e+04    1.47e+05    2.11e+06   1.01e+03   8.22e-01  4.09e+04       1    1.53e-01    6.95e-01
3  1.859667e+04    3.23e+04    2.87e+05   2.64e+02   9.85e-01  1.23e+05       1    1.71e-01    8.66e-01
4  1.803857e+04    5.58e+02    2.69e+04   8.66e+01   9.93e-01  3.69e+05       1    1.61e-01    1.03e+00
5  1.803391e+04    4.66e+00    3.11e+02   1.02e+01   1.00e+00  1.11e+06       1    1.49e-01    1.18e+00

Ceres Solver v1.12.0 Solve Report
----------------------------------
Original                  Reduced
Parameter blocks                        22122                    22122
Parameters                              66462                    66462
Residual blocks                         83718                    83718
Residual                               167436                   167436

Minimizer                        TRUST_REGION

Sparse linear algebra library    SUITE_SPARSE
Trust region strategy     LEVENBERG_MARQUARDT

Given                     Used
Linear solver                    SPARSE_SCHUR             SPARSE_SCHUR
Threads                                     1                        1
Linear solver threads                       1                        1
Linear solver ordering              AUTOMATIC                22106, 16

Cost:
Initial                          4.185660e+06
Final                            1.803391e+04
Change                           4.167626e+06

Minimizer iterations                        5
Successful steps                            5
Unsuccessful steps                          0

Time (in seconds):
Preprocessor                            0.283

Residual evaluation                   0.061
Jacobian evaluation                   0.361
Linear solver                         0.382
Minimizer                               0.895

Postprocessor                           0.002
Total                                   1.220

Termination:                   NO_CONVERGENCE (Maximum number of iterations reached.)


Let us focus on run-time performance. The relevant lines to look at are

Time (in seconds):
Preprocessor                            0.283

Residual evaluation                   0.061
Jacobian evaluation                   0.361
Linear solver                         0.382
Minimizer                               0.895

Postprocessor                           0.002
Total                                   1.220


Which tell us that of the total 1.2 seconds, about .3 seconds was spent in the linear solver and the rest was mostly spent in preprocessing and jacobian evaluation.

The preprocessing seems particularly expensive. Looking back at the report, we observe

Linear solver ordering              AUTOMATIC                22106, 16


Which indicates that we are using automatic ordering for the SPARSE_SCHUR solver. This can be expensive at times. A straight forward way to deal with this is to give the ordering manually. For bundle_adjuster this can be done by passing the flag -ordering=user. Doing so and looking at the timing block of the full report gives us

Time (in seconds):
Preprocessor                            0.051

Residual evaluation                   0.053
Jacobian evaluation                   0.344
Linear solver                         0.372
Minimizer                               0.854

Postprocessor                           0.002
Total                                   0.935


The preprocessor time has gone down by more than 5.5x!.