Avenues for Future Research

Avenues for Ongoing/Future Research

• Continuing efforts on linear solvers:

– Broaden algorithmic and architectural scope of performance studies.

– Autotuning and composition of linear solvers.

– Application studies (structures, fluids, nanoscale modeling)

• Linear and non-linear Eigenvalue Problems

• Examine role of emerging language concepts (atomics, transactions) in linear algebra kernels.

Eigenvalue Problems

• Along with linear system solvers, eigenvalue problems are among the most frequently used HPC kernels.

• They are essential to diverse applications in structures, fluids, electromagnetics, nanoelectronics, etc.

Eigenvalue Problems: Example

• Electronic structure calculation requires minimization of the Kohn-Sham (KS) total energy functional associated with an atomistic system with respect to its electron orbitals.

• The total energy can be expressed as:

E_{tot}(X) = E_{kin}(X) + E_{ion}(X) + E_{H}(X) + E_{XC}(X)  (1)

Here X = (x_{1}, …, x_{k}) represents the electronic wavefunctions corresponding to k occupied states, and E corresponds to various energy terms.

• The total energy must be minimized under the orthonormality constraint X^{T} X = I.

• The Karush-Kuhn-Tucker (KKT) condition yields the following nonlinear eigenvalue problem:

H(X)X = XΛ_{k}  (2)

• H(X) is the Kohn-Sham Hamiltonian that depends on X.

• The most commonly used algorithm is the SCF (self consistent field) iteration (fixed point of the KKT condition).

• While the parallelization characteristics of SCF are identical to the linear eigenvalue problem, its convergence is often suspect.

• Other techniques such as direct inversion of iterative subspaces (DIIS) have been proposed to improve robustness.

• While there is some work on parallelization of these schemes, their scaling properties are ill-understood.

• Scaling solutions to this eigenvalue problem to petascale platforms would represent the most significant development in materials modeling.