PD DrSci. Boris Khoromskij (PhD),
FIELD OF RESEARCH
TensorStructured Numerical Methods in HighDimensional Scientific Computing

Development of efficient multilinear algebra in
Quantics Tensor Train/Chain and
Tensor Networks formats.

Tensorproduct approximation of
multidimensional nonlocal (integral)
operators , e^{tA}, A^{1}, ,
the convolution
product and Green's functions in R^{d} , d > 3.
 Numerical tensorstructured methods for
the HartreeFock and post HartreeFock equations, molecular dynamics, DMRG optimization,
multidimensional FEM/BEM and stochastic PDEs.

Fast and robust numerical multilinear algebra
based on the lowrank Tucker, canonical and mixed Tuckercanonical tensor formats.
 Tensor iterative methods and lowrank spectrally close preconditioners for multidimensional elliptic problems with application to PDEconstrained optimal control problems and for the PoissonBoltzmann equation in protein modeling.

Machine learning for tensor approximation in many dimensions.
Former research topics
 Datasparse approximation to the integral and pseudodifferential operators
using the H matrix and related techniques.
Boundary/edge concentrated FEM/BEM.

Efficient
O(N log^{q} N)complexity representation of dense
N x N matrices approximating boundary/volume potentials (BEM).
O(N log^{q} N) computation and storage of the FEM elliptic
inverse and Schurcomplement matrices (datasparse direct methods).
 Robust domain decomposition methods
for elliptic/parabolic
problems, multilevel interface preconditioners. Coupling of FEM and BEM. Application
to the skin modeling problem (medicine).

Functional analysis and datasparse O(N log^{q} N) approximation
to the elliptic PoincareSteklov
operators.
Low order h and
hp FEM methods on the boundary concentrated meshes.
Application to the Laplace, biharmonic, Lame and Stokes equations
as well as to elliptic equations with piecewise analytic coefficients.

High order finite difference and finite element schemes by the
Richardson extrapolation
techniques on a sequence of grids.

Localglobal convergence of Newtontype methods
for solving nonlinear operator equations. Applications to
the quasilinear elliptic boundary
value problems, inverse scattering problems and to
eigenvalue problems for
integraldifferential operators arising in magnetostatics and theoretical
physics.