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BaSpaCho

BaSpaCho (Batched Sparse Cholesky) is a state of the art direct solver for symmetric positive-definite sparse matrices. It uses supernodal Cholesky decomposition to achieve state of the art performance, turning portions of the sparse matrix into dense blocks and invoking high-performance BLAS/lapack libraries. It is designed with optimization libraries for Levenberg-Marquardt in mind, and aims at reducing part of the complexity offering the best tool for the job. Compared to the library currently considered state of the art (CHOLMOD from SuiteSparse) it supports:

• pure-CUDA mode with support for batching, ie. solving a batch of matrices with identical structure. This is to support differentiable optimization in Theseus library.
• parallel elimination of independent sparse small elimination nodes, essentially the operation done via "Schur-elimination trick" in mathematical optimization libraries such as Ceres. This is a workaround to the supernodal algorithm being a bad fit for the problem structure, so the solver is fed a partially computed decomposition. The Schur-elimination trick and Cholesky decomposition are essentially the same thing, and it makes sense to have a solver library detect (or being hinted) the cases where the supernodal algorithm would not perform well, so that a different strategy can be used.
• Block-structured matrix data types, and facilities for working with them, such as matrix-vector multiplication for the benefit of iterative methods such as preconditioned conjugate gradient.
• Partial factor/solve, allowing to easily compute marginal probability distributions, and allowing mixed solvers to use an iterative method on the result of a partial Cholesky factorization.

Requirements:

• BLAS (ATLAS/OpenBLAS/MKL, see below)

Libraries fetched automatical by build:

• gtest
• Eigen
• dispenso, for multithreading
• Sophus (only used in BA demo)

Optional libraries:

• CUDA toolkit (tested with CUDA 10.2/11.7), if not available must explicitly disable GPU support, see below.
• AMD, from SuiteSparse, can be used instead of Eigen for block reordering algorithm.
• CHOLMOD, from SuiteSparse, used in benchmark as a reference for performance of sparse solvers.

Configure and Install

Configuring with system blas (eg OpenBLAS):

cmake -S . -B build -DCMAKE_BUILD_TYPE=Release -DCMAKE_CUDA_COMPILER=/usr/local/cuda/bin/nvcc

Configuring with MKL:

. /opt/intel/oneapi/setvars.sh
cmake -S . -B build -DCMAKE_BUILD_TYPE=Release -DBLA_VENDOR=Intel10_64lp

Compiling and testing:

cmake --build build -- -j16 && ctest --test-dir build

Benchmarking all configurations (using CHOLMOD as baseline):

build/baspacho/benchmarking/bench -B 1_CHOLMOD

Benchmarking on a problem from Bundle Adjustment in the Large

build/baspacho/benchmarking/BAL_bench -i ~/BAL/problem-871-527480-pre.txt

Show tests:

ctest --test-dir build --show-only

Cuda

Cuda is enabled by default with BASPACHO_USE_CUBLAS option (on by default), add -DBASPACHO_USE_CUBLAS=0 to disable in build. May have to add -DCMAKE_CUDA_COMPILER=/usr/local/cuda/bin/nvcc to allow build to find the cuda compiler. The Cuda architectures can be specified with e.g. -DBASPACHO_CUDA_ARCHS="60;70;75", which also supports the options 'detect' (default) which detects the installed GPU arch, and 'torch' which fills in the architectures supported by PyTorch and >=60 (see below).

Blas

The library used is specified in the CMake variable BLA_VENDOR, a few possibilities are:

• ATLAS
• OpenBLAS
• Intel10_64{i}lp{_seq} (if 'i' indices are 64bits, if '_seq' sequential ie single thread) Blas is enabled by default with BASPACHO_USE_BLAS option (on by default), add -DBASPACHO_USE_BLAS=0 to disable in build. Use BLA_STATIC to link statically to BLAS.

For the full list check CMake docs at: https://cmake.org/cmake/help/latest/module/FindBLAS.html#blas-lapack-vendors

Reordering algorithm Approximate Minimum Degree (AMD)

BaSpaCho can use either the implementation in Eigen of AMD (by default), or the version in library AMD as part of SuiteSparse. Add -DBASPACHO_USE_SUITESPARSE_AMD=1 to the build step to use the implementation in SuiteSparse instead of Eigen.

Benchmarking and fine-tuning

BaSpaCho uses models of the timings of BLAS operations in order to decide on the best node merges selected by the supernodal algorithm. The trade-offs might change dramatically depending on the architecture, the number of cores, the BLAS implementation used and CPU vs GPU. Timings required to fit a model can be collected with the bench tool when provided the -Z command line argument. Here is a description of the benchmarking tools provided.

bench

The tool build/baspacho/benchmarking/bench can be used to compare the speed of different configurations on a varied set of synthetic problems. For instance if CHOLMOD is installed you can use it a reference sparse solver to compare to, and the following compares the most time-consuming operation factor using CHOLMOD as baseline:

build/baspacho/benchmarking/bench -B 1_CHOLMOD

(other operations are analysis and solve-X where X can be any number of right-hand sides). Running with -h gives a view of all operations, solvers and problems available (and arguments accepting regular expressions in order to test a subselection of solvers/problems). When running default factor operation with -Z statistics are saved to files of the form <config>_<op>.csv where the <config> looks like stats_cpu_f8 and discriminates the configuration, while <op> is one of the underlying blas-like operations used in factorization (ie one of potrf, trsm, syge, asmbl, where syge stands for syrk/gemm for people familiar with BLAS). The operations generated with a specific configuration (eg cpu_f64) can be processed as

build/baspacho/examples/opt_comp_model -p stats_cpu_f64_potrf.csv -a stats_cpu_f64_asmbl.csv 
                                       -t stats_cpu_f64_trsm.csv -g stats_cpu_f64_syge.csv

on order to fit a computation model that can be provided as a setting to createSolver in order to fine-tune for your architecture.

BAL_bench

It's possible to benchmark on a bundle-adjustment problem from Bundle Adjustment in the Large as

build/baspacho/benchmarking/BAL_bench -i ~/BAL/problem-871-527480-pre.txt

ie. the solver is tested on a linear problem which is identical to the Again, if installed, CHOLMOD is tested as a baseline, but only on the 'reduced' camera-camera problem. BaSpaCho is tested on both the point elimination and the reduced problem.

Examples

A few examples are provided to illustrate what the library is capable of.

• Optimizer.h provides a fully featured Levenberg-Marquardt optimizer, with support for direct solver and flavors of mixed direct/iterative solvers. The class is used in:
  • OptimizeSimple.cpp, solving a simple problem of points connected with springs
  • OptimizeBaAtLarge.cpp, solving a bundle adjustment problem from Bundle Adjustment in the Large
  • OptimizeCompModel.cpp, where a computation model params are fit to timings of BLAS operations.
• PCG_Sample.cpp contains a demonstration of how it's possible to perform a partial elimination and complete the solution of a linear system iteratively using PCG.

Possible future improvements:

• handle singularity reporting size of non-SPD minor, possibly discard non-definite sparse-elim diagonal blocks as "bad" blocks (eg unconstrained points in BA)
• optimize analyse and solve steps, and mat-x-vec multiplication
• investigate possible support of update/downdate ops, supernodal structure make it hard when the sparse structure changes but it should be possible in case of no sparse structure change.

Caveats

• Only supernodal Cholesky is implemented. We focused on problems having at least a certain degree of interconnectedness, which will benefit of Blas libraries and parallelism. If working with a problem where simplicial would have better performance (eg banded matrix) just use Cholmod or Eigen. Also, there is no support for update/downdate operations of factors. Notice that given the strongly sequential nature of simplicial decomposition the GPU will not be a good fit either.
• Only reordering method currently supported is Approximate Minimum Degree (AMD), this will probably be expanded and made more customizable in the future but this is the current status of things.
• The block-structured type of matrices is builtin in the library, and while this presents advantages in common circumstances the library may prove slighly slower if the matrix lacks completely a block structure, as the entries will be represented as 1x1 blocks. This is hardly a problem in most practical problems because matrices naturally have a block-structure depending on parameters of dimensions >1. Also, thread and cuda-kernel operations are designed around blocks so it's not ideal if you have some huge parameter blocks, the library will work best when the parameter blocks have sizes 1 to 12 (in a factor graph, generally you have many parameter blocks of the same type).
• About determinism: assuming BLAS is deterministic, BaSpaCho will be 100% deterministic on the CPU, but not on CUDA if there is any "sparse elimination" set of parameters, because both factor and solve operations use atomic addition for parallelism on the GPU. Also, a Cuda architecture >=6 is needed for atomicAdd on double numbers (this is the compute hardware architecture and not the version of Cuda, arch >=6 means you need Tesla P100 or GTX 1080-family, or newer. See Cuda Architectures). Otherwise you will have to add define CUDA_DOUBLE_ATOMIC_ADD_WORKAROUND in order to enable the workaround in CudaAtomic.cuh.