ttcross

Cross interpolation of high-dimensional arrays in tensor train format. This code implements the parallel version of the tensor cross interpolation algorithm, see

• Dmitry Savostyanov, Quasioptimality of maximum-volume cross interpolation of tensors, Linear Algebra and its Applications, 2014.
• Sergey Dolgov and Dmitry Savostyanov, Parallel cross interpolation for high-precision calculation of high-dimensional integrals, arXiv: 1903.11554, 2019.

If you benefit from this code, please consider citing our papers in your work.

Basic usage

0. Read the articles.
1. Download the code.
2. Edit the Makefile to match your system. Note: this software makes use of BLAS and LAPACK libraries. Please set the BLASLIB variable in the Makefile accordingly.
3. Compile the code by make command.
4. Run appropriate ./test_* command to execute an experiment. Use mpirun -np NN ./test_* to run the experiment in parallel on NN processors. Use OMP system variables (e.g. OMP_NUM_THREADS) to control the number of OMP threads.

Basic tests

• test_mc_ising.exe KIND INDEX Q REP: calculate Ising susceptibility integrals using Monte Carlo algorithm
• test_qmc_ising.exe KIND INDEX Q REP: calculate Ising susceptibility integrals using quasi Monte Carlo algorithm
• test_crs_ising.exe KIND INDEX N RANK PIV : calculate Ising susceptibility integrals using tensor cross interpolation algorithm

Parameters

• KIND: which integral to calculate, 'C', 'D' or 'E'
• INDEX: the sub-index of the integral, e.g. 6 to calculate C_6, etc.
• Q: the size of the MC sampling or QMC lattice, the number of evaluations will be 2^Q
• REP: the number of repeats for MC or QMC to estimate standard deviation (1<=REP)
• N: the one-dimensional quadrature size for the integration
• RANK: the maximum TT-rank in the cross interpolation method (1<=RANK)
• PIV: the number of rook pivoting searches in the cross interpolation method (0<=PIV)

Quadruple precision

Double precision calculations are accurate to approximately 15 decimal digits. To reach higher precision, you can compile the code enabling -fdefault-real-8 or -fdefault-real-16 in the compiler. Note that you also will need to obtain and link quadruple-precision BLAS/LAPACK libraries. Use test_crs_ising.exe with the same parameters, adjusting the quadrature size N to the desired accuracy.

Multiple precision

For those who desires even higher precision, we provide a multiple-precision version of the code. It relies on the MPFUN2015 library written by David H. Bailey. The source code of MPFUN2015 is included in the mpfun-mpfr-v08 directory. The code uses GNU MPFR Library https://www.mpfr.org/. Please install it on your system and set the MPFLIB variable in the Makefile accordingly. Compile the multi-precision version with make test_mpf_ising.exe, noting the following:

• No special BLAS/LAPACK libraries are needed for this experiment --- all multiple-precision BLAS/LAPACK subroutines are provided in the mpblas.f90 file. You will still need a standard BLAS/LAPACK library to link and run the code.
• The -fdefault-real-XX flag has to be disabled.
• The precision is adjusted via the parameter mpipl in ./mpfun-mpfr-v08/mpfunf.f90. After compiling, run the test_mpf_ising.exe to calculate Ising susceptibility integrals using tensor cross interpolation algorithm in multiple precision. Use the same parameters as in test_crs_ising.exe, adjusting the quadrature size N to reach the desired accuracy.