High order approximations of the operator Lyapunov equation have low rank
نویسندگان
چکیده
Abstract We present a low-rank greedily adapted hp -finite element algorithm for computing an approximation to the solution of Lyapunov operator equation. show that there is hidden regularity in eigenfunctions equation which can be utilized justify use high order finite spaces. Our numerical experiments indicate we achieve eight figures accuracy trace posed dumbbell-domain using space dimension only $$10^4$$ 10 4 degrees freedom. Even more surprising observation -refinement has effect reducing rank solution.
منابع مشابه
Low rank approximations of infinite-dimensional Lyapunov equations and applications
Lyapunov equation. We analyze the approximation properties of solutions of abstract Lyapunov equations in the setting of a scale of Hilbert spaces associated to an unbounded diagonalizable operator which satisfies the Kato’s square root theorem. We call an (unbounded) operator A diagonalizable if there exists a bounded operator Q, with a bounded inverse, such that the (unbounded) operator Q−1AQ...
متن کاملComplex Tensors Almost Always Have Best Low-rank Approximations
Low-rank tensor approximations are plagued by a well-known problem — a tensor may fail to have a best rank-r approximation. Over R, it is known that such failures can occur with positive probability, sometimes with certainty: in R2×2×2, every tensor of rank 3 fails to have a best rank-2 approximation. We will show that while such failures still occur over C, they happen with zero probability. I...
متن کاملGeneralized Low-Rank Approximations
We study the frequent problem of approximating a target matrix with a matrix of lower rank. We provide a simple and efficient (EM) algorithm for solving weighted low rank approximation problems, which, unlike simple matrix factorization problems, do not admit a closed form solution in general. We analyze, in addition, the nature of locally optimal solutions that arise in this context, demonstra...
متن کاملWeighted Low-Rank Approximations
We study the common problem of approximating a target matrix with a matrix of lower rank. We provide a simple and efficient (EM) algorithm for solving weighted low-rank approximation problems, which, unlike their unweighted version, do not admit a closedform solution in general. We analyze, in addition, the nature of locally optimal solutions that arise in this context, demonstrate the utility ...
متن کاملThe geometry of weighted low-rank approximations
The low-rank approximation problem is to approximate optimally, with respect to some norm, a matrix by one of the same dimension but smaller rank. It is known that under the Frobenius norm, the best low-rank approximation can be found by using the singular value decomposition (SVD). Although this is no longer true under weighted norms in general, it is demonstrated here that the weighted low-ra...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Bit Numerical Mathematics
سال: 2022
ISSN: ['0006-3835', '1572-9125']
DOI: https://doi.org/10.1007/s10543-022-00917-z