PD DrSci. Boris Khoromskij (PhD),

FIELD OF RESEARCH

Tensor-Structured Numerical Methods in High-Dimensional Scientific Computing

  • Development of efficient multilinear algebra in Quantics Tensor Train/Chain and Tensor Networks formats.
  • Tensor-product approximation of multi-dimensional nonlocal (integral) operators , e-tA, A-1, , the convolution product and Green's functions in Rd , d > 3.
  • Numerical tensor-structured methods for the Hartree-Fock and post Hartree-Fock equations, molecular dynamics, DMRG optimization, multi-dimensional FEM/BEM and stochastic PDEs.
  • Fast and robust numerical multi-linear algebra based on the low-rank Tucker, canonical and mixed Tucker-canonical tensor formats.
  • Tensor iterative methods and low-rank spectrally close preconditioners for multidimensional elliptic problems with application to PDE-constrained optimal control problems and for the Poisson-Boltzmann equation in protein modeling.
  • Machine learning for tensor approximation in many dimensions.

Former research topics

  • Data-sparse approximation to the integral and pseudodifferential operators using the H -matrix and related techniques. Boundary/edge concentrated FEM/BEM.
  • Efficient O(N logq N)-complexity representation of dense N x N matrices approximating boundary/volume potentials (BEM). O(N logq N) -computation and storage of the FEM elliptic inverse and Schur-complement matrices (data-sparse 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 data-sparse O(N logq N) -approximation to the elliptic Poincare-Steklov 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.
  • Local-global convergence of Newton-type methods for solving nonlinear operator equations. Applications to the quasi-linear elliptic boundary value problems, inverse scattering problems and to eigenvalue problems for integral-differential operators arising in magnetostatics and theoretical physics.