## Batched Dense Tensor Contractions

The numerical kernels of efficient high-order operator evaluation reduce to many small dense tensor contractions, one for each element of the computational mesh. These contractions can be performed in parallel over the elements and can be implemented as a sequence of small matrix-matrix multiplications (dgemms).

CEED is developing efficient algorithms for these numerical kernels on a variety of advanced hardware, both by directly working on the specifying tensor contraction found in high-order finite element kernels, as well as by recasting to batched interface of LAPACK-like linear algebra libraries providing efficient dgemms for matrices of small-to-medium size.

CEED scientists are also involved in the standardization efforts for batched versions of Basic Linear Algebra Software Library (BLAS) routines, which are the foundations for LAPACK-type libraries.

*This is an active area of research for our team and we are interested in
collaboration.*

## Scalable Matrix-Free Solvers

CEED is developing matrix-free linear solvers and preconditioners for partially
assembled high-order operators utilizing expertise from solver
libraries like *hypre* and
PETSc.

*This is an active area of research for our team and we are interested in
collaboration.*

*Stay tuned for more details...*