# Russell Sparse - Solvers for large sparse linear systems (wraps MUMPS and UMFPACK) <!-- omit from toc -->
[](https://docs.rs/russell_sparse/)
_This crate is part of [Russell - Rust Scientific Library](https://github.com/cpmech/russell)_
## Contents <!-- omit from toc -->
- [Introduction](#introduction)
- [Documentation](#documentation)
- [Installation](#installation)
- [Setting Cargo.toml](#setting-cargotoml)
- [Optional features](#optional-features)
- [🌟 Examples](#-examples)
- [Solve a tiny sparse linear system using UMFPACK](#solve-a-tiny-sparse-linear-system-using-umfpack)
- [Tools](#tools)
- [MUMPS + OpenBLAS issue](#mumps--openblas-issue)
- [References](#references)
- [For developers](#for-developers)
## Introduction
This library implements tools for handling sparse matrices and functions to solve large sparse systems using the best libraries out there, such as [UMFPACK (recommended)](https://github.com/DrTimothyAldenDavis/SuiteSparse) and [MUMPS (for very large systems)](https://mumps-solver.org).
This library implements three storage formats for sparse matrices:
* COO: COOrdinates matrix, also known as a sparse triplet.
* CSC: Compressed Sparse Column matrix
* CSR: Compressed Sparse Row matrix
Additionally, to unify the handling of the above sparse matrix data structures, this library implements:
* SparseMatrix: Either a COO, CSC, or CSR matrix
The COO matrix is the best when we need to update the values of the matrix because it has easy access to the triples (i, j, aij). For instance, the repetitive access is the primary use case for codes based on the finite element method (FEM) for approximating partial differential equations. Moreover, the COO matrix allows storing duplicate entries; for example, the triple `(0, 0, 123.0)` can be stored as two triples `(0, 0, 100.0)` and `(0, 0, 23.0)`. Again, this is the primary need for FEM codes because of the so-called assembly process where elements add to the same positions in the "global stiffness" matrix. Nonetheless, the duplicate entries must be summed up at some stage for the linear solver (e.g., MUMPS, UMFPACK). These linear solvers also use the more memory-efficient storage formats CSC and CSR. See the [russell_sparse documentation](https://docs.rs/russell_sparse) for further information.
This library also provides functions to read and write Matrix Market files containing (huge) sparse matrices that can be used in performance benchmarking or other studies. The [read_matrix_market()] function reads a Matrix Market file and returns a [CooMatrix]. To write a Matrix Market file, we can use the function [write_matrix_market()], which takes a [SparseMatrix] and, thus, automatically convert COO to CSC or COO to CSR, also performing the sum of duplicates. The `write_matrix_market` also writes an SMAT file (almost like the Matrix Market format) without the header and with zero-based indices. The SMAT file can be given to the fantastic [Vismatrix](https://github.com/cpmech/vismatrix) tool to visualize the sparse matrix structure and values interactively; see the example below.
### Documentation
* [](https://docs.rs/russell_sparse/) — [russell_sparse documentation](https://docs.rs/russell_sparse/)
## Installation
This crate depends on some non-rust high-performance libraries. [See the main README file for the steps to install these dependencies.](https://github.com/cpmech/russell)
### Setting Cargo.toml
[](https://crates.io/crates/russell_sparse)
👆 Check the crate version and update your Cargo.toml accordingly:
```toml
[dependencies]
russell_sparse = "*"
```
### Optional features
The following (Rust) features are available:
* `intel_mkl`: Use Intel MKL instead of OpenBLAS
* `local_suitesparse`: Use a locally compiled version of SuiteSparse
* `with_mumps`: Enable the MUMPS solver (locally compiled)
Note that the [main README file](https://github.com/cpmech/russell) presents the steps to compile the required libraries according to each feature.
## 🌟 Examples
This section illustrates how to use `russell_sparse`. See also:
* [More examples on the documentation](https://docs.rs/russell_sparse/)
* [Examples directory](https://github.com/cpmech/russell/tree/main/russell_sparse/examples)
### Solve a tiny sparse linear system using UMFPACK
```rust
use russell_lab::{vec_approx_eq, Vector};
use russell_sparse::prelude::*;
use russell_sparse::StrError;
fn main() -> Result<(), StrError> {
// constants
let ndim = 3; // number of rows = number of columns
let nnz = 5; // number of non-zero values
// allocate solver
let mut umfpack = SolverUMFPACK::new()?;
// allocate the coefficient matrix
let mut coo = SparseMatrix::new_coo(ndim, ndim, nnz, Sym::No)?;
coo.put(0, 0, 0.2)?;
coo.put(0, 1, 0.2)?;
coo.put(1, 0, 0.5)?;
coo.put(1, 1, -0.25)?;
coo.put(2, 2, 0.25)?;
// print matrix
let a = coo.as_dense();
let correct = "┌ ┐\n\
│ 0.2 0.2 0 │\n\
│ 0.5 -0.25 0 │\n\
│ 0 0 0.25 │\n\
└ ┘";
assert_eq!(format!("{}", a), correct);
// call factorize
umfpack.factorize(&mut coo, None)?;
// allocate two right-hand side vectors
let b = Vector::from(&[1.0, 1.0, 1.0]);
// calculate the solution
let mut x = Vector::new(ndim);
umfpack.solve(&mut x, &coo, &b, false)?;
let correct = vec![3.0, 2.0, 4.0];
vec_approx_eq(&x, &correct, 1e-14);
Ok(())
}
```
See [russell_sparse documentation](https://docs.rs/russell_sparse) for more examples.
See also the folder `examples`.
## Tools
This crate includes a tool named `solve_matrix_market` to study the performance of the available sparse solvers (currently MUMPS and UMFPACK).
`solve_matrix_market` reads a [Matrix Market file](https://math.nist.gov/MatrixMarket/formats.html) and solves the linear system:
```text
A ⋅ x = b
```
where the right-hand side (b) is a vector containing only ones.
The data directory contains an example of a Matrix Market file named `bfwb62.mtx`, and you may download more matrices from https://sparse.tamu.edu/
For example, run the command:
```bash
cargo run --release --bin solve_matrix_market -- ~/Downloads/matrix-market/bfwb62.mtx
```
Or
```bash
cargo run --release --bin solve_matrix_market -- --help
```
to see the options.
The default solver of `solve_matrix_market` is UMFPACK. To run with MUMPS, use the `--genie` (-g) flag:
```bash
cargo run --release --bin solve_matrix_market -- -g mumps ~/Downloads/matrix-market/bfwb62.mtx
```
The output looks like this:
```json
{
"main": {
"platform": "Russell",
"blas_lib": "OpenBLAS",
"solver": "MUMPS-local"
},
"matrix": {
"name": "bfwb62",
"nrow": 62,
"ncol": 62,
"nnz": 202,
"complex": false,
"symmetric": "YesLower"
},
"requests": {
"ordering": "Auto",
"scaling": "Auto",
"mumps_num_threads": 0
},
"output": {
"effective_ordering": "Amf",
"effective_scaling": "RowColIter",
"effective_mumps_num_threads": 1,
"openmp_num_threads": 24,
"umfpack_strategy": "Unknown",
"umfpack_rcond_estimate": 0.0
},
"determinant": {
"mantissa_real": 0.0,
"mantissa_imag": 0.0,
"base": 2.0,
"exponent": 0.0
},
"verify": {
"max_abs_a": 0.0001,
"max_abs_ax": 1.0000000000000004,
"max_abs_diff": 5.551115123125783e-16,
"relative_error": 5.550560067119071e-16
},
"time_human": {
"read_matrix": "43.107µs",
"initialize": "266.59µs",
"factorize": "196.81µs",
"solve": "166.87µs",
"total_ifs": "630.27µs",
"verify": "2.234µs"
},
"time_nanoseconds": {
"read_matrix": 43107,
"initialize": 266590,
"factorize": 196810,
"solve": 166870,
"total_ifs": 630270,
"verify": 2234
},
"mumps_stats": {
"inf_norm_a": 0.0,
"inf_norm_x": 0.0,
"scaled_residual": 0.0,
"backward_error_omega1": 0.0,
"backward_error_omega2": 0.0,
"normalized_delta_x": 0.0,
"condition_number1": 0.0,
"condition_number2": 0.0
}
}
```
## MUMPS + OpenBLAS issue
We found that MUMPS + OpenBLAS becomes very, very slow when the number of OpenMP threads is left automatic, i.e., using the available number of threads. Thus, with OpenBLAS, it is recommended to set LinSolParams.mumps_num_threads = 1 (this is automatically set when using OpenBLAS).
This issue has also been discovered by Reference #1, who states (page 72) _"We have observed that multi-threading of OpenBLAS library in MUMPS leads to multiple thread conflicts which sometimes result in significant slow-down of the solver."_
Therefore, we have to take one of the two approaches:
* If fixing the number of OpenMP threads for MUMPS, set the number of OpenMP threads for OpenBLAS to 1
* If fixing the number of OpenMP threads for OpenBLAS, set the number of OpenMP threads for MUMPS to 1
This issue has **not** been noticed with MUMPS + Intel MKL.
Command to reproduce the issue:
```bash
OMP_NUM_THREADS=20 ~/rust_modules/release/solve_matrix_market -g mumps ~/Downloads/matrix-market/inline_1.mtx -m 0 -v --override-prevent-issue
```
Also, to reproduce the issue, we need:
* Git hash = e020d9c8486502bd898d93a1998a0cf23c4d5057
* Remove Debian OpenBLAS, MUMPS, and etc.
* Install the compiled MUMPS solver with `02-ubuntu-openblas-compile.bash`
### References
1. Dorozhinskii R (2019) [Configuration of a linear solver for linearly implicit time integration and efficient data transfer in parallel thermo-hydraulic computations](https://mediatum.ub.tum.de/doc/1486743/1486743.pdf). _Master's Thesis in Computational Science and Engineering._ Department of Informatics Technical University of Munich.
## For developers
* The `c_code` directory contains a thin wrapper to the sparse solvers (MUMPS, UMFPACK)
* The `build.rs` file uses the crate `cc` to build the C-wrappers
* The `zscripts` directory also contains following:
* `memcheck.bash`: Checks for memory leaks on the C-code using Valgrind
* `run-examples`: Runs all examples in the `examples` directory
* `run-solve-matrix-market.bash`: Runs the solve-matrix-market tool from the `bin` directory