Multiscale interpolative constructions#
The MPS representation of polynomial interpolants can be efficiently constructed using multiscale interpolative constructions, following Lindsey’s idea (see Ref. https://arxiv.org/pdf/2311.12554). On the current formulation, these methods are based on the Lagrange interpolation framework, interpolating the function on Chebyshev-Lobatto nodes.
The SeeMPS library implements these interpolative constructions for multivariate functions. Multiscale resolution features have yet not been implemented. The method requires sampling a tensor of coefficients from the input function, which becomes computationally intractable for high-dimensional problems. Hence, current implementations provide an efficient tool for encoding smooth, low-dimensional functions in MPS form with high efficiency and accuracy.
Several algorithmic variants follow. Basic construct using mps_lagrange_chebyshev_basic() assemble three distinct tensor cores, two of which are placed at the edges of the MPS construction and one is copied and placed in the bulk. Only the left-most core is function-dependent, while all the rest only depend on the interpolation order and can therefore be precomputed and reused. Their combination forms a Lagrange-Chebyshev interpolant across two Chebyshev-Lobatto grids, which presents spectral approximation convergence rates for sufficiently smooth functions. However, this approach tends to overestimate the required bond dimension of the interpolant, requiring a large-scale final truncation.
The method’s performance can be enhanced through rank-revealing optimizations using the SVD decomposition, avoiding the need for a large-scale final simplification. This is implemented in the mps_lagrange_chebyshev_rr() routine. Moreover, local interpolation can be performed, resulting in highly sparse MPS cores and enhancing performance. This is implemented in mps_lagrange_chebyshev_lrr().
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Performs a "basic" Lagrange MPS Chebyshev interpolation of a function. |
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Performs a Lagrange rank-revealing MPS Chebyshev interpolation of a function. |
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Performs a local rank-revealing Lagrange MPS Chebyshev interpolation of a function. |