LinearSolve.jl is the higher level interface for solving linear systems `Ax=b`

which allows algorithms like ODE solvers and nonlinear solvers to easily switch between using sparse direct methods, Krylov methods, and more. However, automating the solution of linear systems when `A`

is a distributed matrix across an HPC cluster would allow many large applications to become easy. The global of this project would be to integrate with libraries such as Elemental.jl, PartitionedArrays.jl, and PETSc.jl to make this easy.

**Recommended Skills**: Background knowledge in numerical linear algebra and parallel computing.

**Expected Results**: New parallel algorithms wrapped into LinearSolve.jl

**Mentors**: Chris Rackauckas

**Expected Project Size**: 175 hour or 350 hour depending on the chosen subtasks.

**Difficulty**: Easy to Medium depending on the chosen subtasks.

There are many ways to solve `f(x)=0`

, and there are many applications, from partial differential equation solvers to optimizers, which use nonlinear solvers in the inner loop. NonlinearSolve.jl is the core nonlinear solver library of the Julia programming language and it currently exists mostly as a wrapper to older codes (some written in Fortran), but this has many downsides. Those wrapped libraries do not support GPU-acceleration, they do not have fast paths with static arrays for small equations, they do not support higher precision arithmetic, etc. The goal of this project would be to extend the native Julia nonlinear solvers beyond Newton-Raphson and demonstrate the capabilities with this extended feature set. Features can include globalizing methods, Quasi-Newton (Broyden etc.), nonlinear preconditioning, line searches, and more.

**Recommended Skills**: Background knowledge in numerical analysis.

**Expected Results**: New tutorials in SciMLTutorials and benchmarks in SciMLBenchmarks.

**Mentors**: Chris Rackauckas and Utkarsh

**Expected Project Size**: 175 hour or 350 hour depending on the chosen subtasks.

**Difficulty**: Easy to Medium depending on the chosen subtasks.

Many university classes use the SciML ecosystem for its teaching, and thus classrooms all over the world will be improved. Tutorials that capture more domains will allow professors teaching biological modeling courses to not have to manually rewrite physics-based tutorials to match their curriculum, and conversion of READMEs to documentation will help such professors link to reference portions for these tools in their lecture notes.

Additionally, these benchmarks are a widely referenced cross-language benchmark of differential equations, which gives a standard between Python, R, Julia, MATLAB, and many C++ and Fortran packages. Improving the technical writing around the benchmarks can make this set of documents more widely accessible, and enlarging the scope of topics will help individuals of all programming languages better assess the methods they should be choosing for their problems.

Note that this will include authorship for SciML publications which use the benchmarks.

**Recommended Skills**: Background knowledge in numerical analysis and modeling.

**Expected Results**: New tutorials in SciMLTutorials and benchmarks in SciMLBenchmarks.

**Mentors**: Chris Rackauckas

**Expected Project Size**: 175 hour or 350 hour depending on the chosen subtasks.

**Difficulty**: Easy.