Conceitos Básicos
The author presents a novel method for solving nonlinear tensor differential equations efficiently using low-rank approximation on manifolds.
Resumo
A new method is introduced to solve nonlinear tensor differential equations by projecting them onto low-rank tensor manifolds. Traditional orthogonal projections are replaced with interpolatory projections, making computations easier for nonlinear equations. The proposed algorithm selects indices based on the discrete empirical interpolation method (DEIM) to parameterize tensors and their tangent spaces. By integrating the solution on the manifold, the approach minimizes residuals efficiently. The paper provides detailed insights into time integration schemes, rank adaptation, and numerical examples like the Allen-Cahn equation.
Estatísticas
At any time t, the solution tensor u(t) has O(nd) degrees of freedom.
The proposed TT-cross-DEIM algorithm requires O(dnr3) FLOPS.
The condition number of matrices Uk in rank increase criterion is small.