Parallel cross interpolation for high-precision calculation of high-dimensional integrals

Abstract

We propose a parallel version of the cross interpolation algorithm and apply it to calculate high-dimensional integrals motivated by Ising model in quantum physics. In contrast to mainstream approaches, such as Monte Carlo and quasi Monte Carlo, the samples calculated by our algorithm are neither random nor form a regular lattice. Instead we calculate the given function along individual dimensions (modes) and use this data to reconstruct its behaviour in the whole domain. The positions of the calculated univariate fibers are chosen adaptively for the given function. The required evaluations can be executed in parallel both along each mode (variable) and over all modes.To demonstrate the efficiency of the proposed method, we apply it to compute highdimensional Ising susceptibility integrals, arising from asymptotic expansions for the spontaneous magnetisation in two-dimensional Ising model of ferromagnetism. We observe strong superlinear convergence of the proposed method, while the MC and qMC algorithms converge sublinearly. Using multiple precision arithmetic, we also observed exponential convergence of the proposed algorithm. Combining high-order convergence, almost perfect scalability up to hundreds of processes, and the same flexibility as MC and qMC, the proposed algorithm can be a new method of choice for problems involving high-dimensional integration, e.g. in statistics, probability, and quantum physics. ≈ −1 I I J J I I I J J J Figure 1: Cross interpolation for matrices. The full matrix A is approximated by a low-rank decompositionà based on a small number of columns and rows computed in A. Note that the approximation (1) is exact in the positions of computed rows and columns. One of the first examples of the latter was the parallel density matrix renormalization group (DMRG) algorithm [81] for ground state computations in quantum physics. In mathematical community this research direction started with dimension-parallel linear solver [30] and cross algorithms in HT format [44]. The main difficulty of parallelisation over dimension is the need of algorithmic modifications, since state of the arts tensor algorithms were designed in intrinsically sequential way. Ideally, such modifications should not compromise numerical stability or convergence for the sake of parallel efficiency. In this paper we develop a parallel version of the TT cross interpolation algorithm [74]. The parallel algorithm is adaptive and converges with the same rate as the sequential version, but involves only local communications with constant loading of processes, and demonstrates almost perfect scaling up to the ultimate partitioning where each process is responsible for a single direction (mode, variable). Moreover, further speedup can be achieved using OpenMP parallelisation of tensor algebra in each process.The rest of the paper is organised as follows. In Sec. 2 we recall the cross interpolation method for matrices and provide necessary definitions. In Sec. 3 we discuss how the matrix interpolation can be applied for high-dimensi...

Solution of the Fokker–Planck Equation by Cross Approximation Method in the Tensor Train Format

“…Then this function is passed as an argument to the standard rank adaptive method from the ttpy package. The CAM is described in more detail in the original papers ( Oseledets and Tyrtyshnikov, 2010 ; Savostyanov and Oseledets, 2011 ), as well as in a recent work ( Dolgov and Savostyanov, 2020 ), which formulates a computationally efficient parallel implementation of the algorithm.…”

“…1 backwards in time, and then to find the PDF value, we integrate a system of eqs 1 , 3 . Since we can evaluate the value of at any , we can use the cross approximation method (CAM) ( Oseledets and Tyrtyshnikov, 2010 ; Savostyanov and Oseledets, 2011 ; Dolgov and Savostyanov, 2020 ) in the TT-format to recover a supposedly low-rank tensor from its samples. In this way we do not need to have any compact representation of f , but only numerically solve the corresponding ODE.…”

TTRISK: Tensor Train Decomposition Algorithm for Risk Averse Optimization

“…Applying this coordinate descent idea to the problem of interpolating a given function with a TT decomposition yields a family of TT-Cross algorithms [28,34,5]. Suppose we are given a procedure to evaluate a continuous function f (ξ) at any given ξ.…”

“…A particularly simple instance of such is the Tensor Train (TT) decomposition [27] that admits efficient numerical computations. One of the most powerful algorithms of this kind is the cross approximation, as well as its variants [28,34,5]. Those allow one to compute a TT approximation to potentially any function, using a number of samples from the sought function that is a small multiple of the number of degrees of freedom in the tensor decomposition.…”

Solution of the Fokker-Planck equation by cross approximation method in the tensor train format

“…To find the preimage, we need to integrate the equation (1) backwards in time, and then to find the PDF value, we integrate a system of equations ( 1) and (3). Since we can evaluate the value of ρ(x, t) at any x, we can use the cross approximation method (CAM) [7,8,9] in the TTformat to recover a supposedly low-rank tensor from its samples. In this way we do not need to have any compact representation of f , but only numerically solve the corresponding ODE.…”