Fast and stable algorithms for large-scale computation

Scientific computing and data analytics have become the third and
fourth pillars of scientific discovery. Their success is tightly linked to
a rapid increase in the size and complexity of problems and datasets
of interest. In this talk, I will discuss our recent efforts in the development
of novel numerical algorithms for tackling these challenges.
In the fi rst part, I will present a stochastic Lanczos algorithm for estimating
the spectrum without computing any eigenvalues. This is
one of the key ingredients in the new breed of “spectrum slicing”-type
eigensolvers for electronic structure calculations. In the second part, I
will present our newly developed fast structured direct solvers for kernel
systems and its applications in accelerating the learning process.
By exploiting inherited low-rank property, these structured solvers
provide a new framework for performing matrix operations in linear
complexity and are highly suitable for large-scale computation. At the
end of this talk, I will demonstrate their performance by solving an 8
million by 8 million fully dense matrix problem in only a few minutes
on a desktop.