Updated on 26/10/2023.

This repository hosts the code for the deep inverse Rosenblatt transport methods.

Load both src and external folders to use the code. The MultiIndices class in the external folder is used by SparseFun for sparse polynomial approximations.

See the repository deep-tensor-examples for more examples and TT-IRT for a different implementation.

Important change of interfaces:

• All polynomial classes have name started with Capital letters.
• All diagonal map classes also have name started with Capital letters. This affects EmpiricalMap, UniformMap, and the abstract DiagonalMap.
• The DIRT (including TTDIRT and SparseDIRT) class now has a simplified interface.
  • In the simplest case, one only needs to passin the density function, the parameter dimension and the approximation domain. This automatically override the default FTT option (in DIRT class), use a default 2nd order Lagrange polynimal basis, and use the Gaussian reference. See example_dirt_default.m.
  • Alternatively, one can specify the density function, the parameter dimension and the approximation polynomial basis (with a given domain). This automatically override the default FTT option (in DIRT class) and use the Gaussian reference. See example_conditional_dirt.m.
  • One can also have the full specification, with polynomial basis for each layer, reference measure, FTT options, etc. See example_dirt.m.
• All the help files are due to updates.

References:

• For SIRT, DIRT, and IRT:
  • Cui, Dolgov and Zahm (2023). Self-reinforced polynomial approximation methods for concentrated probability densities. arXiv preprint: 2303.02554
  • Cui, Dolgov and Scheichl (2022). Deep importance sampling using tensor-trains with application to a priori and a posteriori rare event estimation. arXiv preprint: 2209.01941
  • Cui, Dolgov and Zahm (2023). Scalable conditional deep inverse Rosenblatt transports using tensor-trains and gradient-based dimension reduction. Journal of Computational Physics, 112103
  • Cui and Dolgov (2022). Deep composition of tensor trains using squared inverse Rosenblatt transports. Foundation of Computational Mathematics, Foundations of Computational Mathematics 22 (6), 1863-1922
  • Dolgov, Anaya-Izquierdo, Fox and Scheichl (2020). Approximation and sampling of multivariate probability distributions in the tensor train decomposition. Statistics and Computing 30(3), 603-625.
• For building the tensor train using AMEN:
  • Dolgov, Savostyanov (2014). Alternating minimal energy methods for linear systems in higher dimensions. SIAM Journal on Scientific Computing 36(5), A2248-A2271.
• For functional tensor train:
  • Gorodetsky, Karaman and Marzouk (2018). A continuous analogue of the tensor-train decomposition. Computer Methods in Applied Mechanics and Engineering 347, 59-84.
  • Bigoni, Engsig-Karup, Marzouk (2016). Spectral tensor-train decomposition. SIAM Journal on Scientific Computing 38(4), A2405-A2439.

The MultiIndices class is from the ApproximationToolbox package of Nouy.

For TTIRT:

• CIRT and Jacobian need to be implemented
• Examples 3 and 4 of TT_IRT_OU need to be need to be tested
• TT IRT with domain transformation need to tested