Fast low-rank approximations of multidimensional integrals in ion-atomic collisions modelling. (English) Zbl 1399.65101
Summary: An efficient technique based on low-rank separated approximations is proposed for the computation of three-dimensional integrals arising in the energy deposition model that describes ion-atomic collisions. Direct tensor-product quadrature requires grids of size \(4000^3\) that is unacceptable. Moreover, several of such integrals have to be computed simultaneously for different values of the parameters. To reduce the complexity, we use the structure of the integrand and apply numerical linear algebra techniques for the construction of low-rank approximation. The resulting algorithm is \(10^3\) faster than spectral quadratures in spherical coordinates used in the original DEPOSIT code. The approach can be generalized to other multidimensional problems in physics.
MSC:
| 65D30 | Numerical integration |
| 65F15 | Numerical computation of eigenvalues and eigenvectors of matrices |
| 82C22 | Interacting particle systems in time-dependent statistical mechanics |
| 65D32 | Numerical quadrature and cubature formulas |
Keywords:
low-rank approximation; 2D cross; separated representation; exponential sums; 3D integration; Slater wave functionSoftware:
DEPOSITReferences:
| [1] | Beylkin G, Mohlenkamp MJ. Numerical operator calculus in higher dimensions. Proceedings of the National Academy of Sciences of the United States of America. 2002; 99(16): 10 246-10 251. · Zbl 1008.65026 |
| [2] | Khoromskaia V, Khoromskij BN, Schneider R. Tensor‐structured factorized calculation of two‐electron integrals in a general basis. SIAM Journal on Scientific Computing. 2013; 35(2): A987-A1010. · Zbl 1266.65069 |
| [3] | Kolda TG, Bader BW. Tensor decompositions and applications. SIAM Review. 2009; 51(3): 455-500. · Zbl 1173.65029 |
| [4] | Tyrtyshnikov EE. Incomplete cross approximation in the mosaic-skeleton method. Computing. 2000; 64(4): 367-380. · Zbl 0964.65048 |
| [5] | Bebendorf M. Approximation of boundary element matrices. Numerische Mathematik. 2000; 86(4): 565-589. · Zbl 0966.65094 |
| [6] | Oseledets IV, Savostianov DV, Tyrtyshnikov EE. Tucker dimensionality reduction of three‐dimensional arrays in linear time. SIAM Journal on Matrix Analysis and Applications. 2008; 30(3): 939-956. · Zbl 1180.15025 |
| [7] | Rakhuba MV, Oseledets IV. Fast multidimensional convolution in low‐rank formats via cross approximation. SIAM Journal on Scientific Computing. 2015; 37(2): A565-A582. · Zbl 1320.65197 |
| [8] | Oseledets IV, Tyrtyshnikov EE. Breaking the curse of dimensionality, or how to use SVD in many dimensions. SIAM Journal on Scientific Computing. 2009; 31(5): 3744-3759. · Zbl 1200.65028 |
| [9] | Oseledets IV. Tensor‐train decomposition. SIAM Journal on Scientific Computing. 2011; 33(5): 2295-2317. · Zbl 1232.15018 |
| [10] | Harshman RA. Foundation of the PARAFAC procedure: model and conditions for an explanatory multimode factor analysis. UCLA Working papers in phonetics. 1970; 16: 1-84. |
| [11] | Oseledets IV. Constructive representation of functions in low‐rank tensor formats. Constructive Approximation. 2013; 37(1): 1-18. · Zbl 1282.15021 |
| [12] | Beylkin G, Mohlenkamp MJ. Algorithms for numerical analysis in high dimensions. SIAM Journal on Scientific Computing. 2005; 26(6): 2133-2159. · Zbl 1085.65045 |
| [13] | Hackbusch W, Khoromskij BN. Low‐rank Kronecker‐product approximation to multi‐dimensional nonlocal operators. I. Separable approximation of multi‐variate functions. Computing. 2006; 76(3‐4): 177-202. · Zbl 1087.65049 |
| [14] | Hackbusch W, Khoromskij BN. Low‐rank Kronecker‐product approximation to multi‐dimensional nonlocal operators. II. HKT representation of certain operators. Computing. 2006; 76(3‐4): 203-225. · Zbl 1087.65050 |
| [15] | Khoromskij BN. Fast and accurate tensor approximation of multivariate convolution with linear scaling in dimension. Journal of Computational and Applied Mathematics. 2010; 234(11): 3122-3139. · Zbl 1197.65216 |
| [16] | Khoromskij BN. \( \mathcal{O}(d \log n)\)-quantics approximation of N-d tensors in high‐dimensional numerical modeling. Constructive Approximation. 2011; 34(2): 257-280. · Zbl 1228.65069 |
| [17] | Khoromskij B. Tensor‐structured preconditioners and approximate inverse of elliptic operators in \(\mathbb{R}^d\). Constructive Approximation. 2009; 30: 599-620. · Zbl 1185.65051 |
| [18] | Khoromskij BN, Khoromskaia V. Multigrid accelerated tensor approximation of function related multidimensional arrays. SIAM Journal on Scientific Computing. 2009; 31(4): 3002-3026. · Zbl 1197.65215 |
| [19] | Khoromskij BN, Khoromskaia V, Chinnamsetty SR, Flad HJ. Tensor decomposition in electronic structure calculations on 3D Cartesian grids. Journal of Computational Physics. 2009; 228(16): 5749-5762. · Zbl 1171.82017 |
| [20] | Bohr N. On the decrease of velocity of swiftly moving electrified particles in passing through matter. Philosophical Magazine Series 6 581; 30(178). |
| [21] | Russek A, Meli J. Ionization phenomena in high‐energy atomic collisions. Physica. 1970; 46(2): 222-243. |
| [22] | Cocke CL. Production of highly charged low‐velocity recoil ions by heavy‐ion bombardment of rare‐gas targets. Physical Review A. 1979; 20: 749-758. |
| [23] | Shevelko VP, Litsarev MS, Tawara H. Multiple ionization of fast heavy ions by neutral atoms in the energy deposition model. Journal of Physics B: Atomic, Molecular and Optical Physics. 2008; 41(11): 115-204. |
| [24] | Shevelko VP, Kato D, Litsarev MS, Tawara H. The energy‐deposition model: electron loss of heavy ions in collisions with neutral atoms at low and intermediate energies. Journal of Physics B: Atomic, Molecular and Optical Physics. 2010; 43(21): 215-202. |
| [25] | Song MY, Litsarev MS, Shevelko VP, Tawara H, Yoon JS. Single‐ and multiple‐electron loss cross‐sections for fast heavy ions colliding with neutrals: semi‐classical calculations. Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms. 2009; 267(14): 2369-2375. |
| [26] | Shevelko VP, Litsarev MS, Song MY, Tawara H, Yoon JS. Electron loss of fast many‐electron ions colliding with neutral atoms: possible scaling rules for the total cross sections. Journal of Physics B: Atomic, Molecular and Optical Physics. 2009; 42(6): 065-202. |
| [27] | Shevelko V, Litsarev M, Stohlker T, Tawara H, Tolstikhina I, Weber G. Electron loss and capture processes in collisions of heavy many‐electron ions with neutral atoms. In Atomic Processes in Basic and Applied Physics Springer Series on Atomic, Optical, and Plasma Physics, Shevelko V (ed.), Tawara H (ed.) (eds)., vol. 68, 2012. |
| [28] | Litsarev MS, Shevelko VP. Multiple‐electron losses of highly charged ions colliding with neutral atoms. Physica Scripta. 2013; 2013(T156): 014-037. |
| [29] | Tolstikhina IY, Shevelko VP. Collision processes involving heavy many‐electron ions interacting with neutral atoms. Physics‐Uspekhi. 2013; 56(3): 213-242. |
| [30] | Tolstikhina IY, Litsarev MS, Kato D, Song MY, Yoon JS, Shevelko V. Collisions of Be, Fe, Mo and W atoms and ions with hydrogen isotopes: electron capture and electron loss cross sections. Journal of Physics B: Atomic, Molecular and Optical Physics. 2014; 47(3): 035-206. |
| [31] | Litsarev MS. The deposit computer code: calculations of electron‐loss cross‐sections for complex ions colliding with neutral atoms. Computer Physics Communications. 2013; 184(2): 432-439. |
| [32] | Litsarev MS, Oseledets IV. The deposit computer code based on the low‐rank approximations. Computer Physics Communications. 2014; 185(10): 2801-2802. · Zbl 1360.65011 |
| [33] | Beylkin G, Monzón L. On approximation of functions by exponential sums. Applied and Computational Harmonic Analysis. 2005; 19(1): 17-48. · Zbl 1075.65022 |
| [34] | Beylkin G, Monzón L. Approximation by exponential sums revisited. Applied and Computational Harmonic Analysis. 2010; 28(2): 131-149. · Zbl 1189.65035 |
| [35] | Tyrtyshnikov EE. Mosaic‐skeleton approximations. Calcolo. 1996; 33(1): 47-57. · Zbl 0906.65048 |
| [36] | Goreinov SA, Tyrtyshnikov EE, Zamarashkin NL. A theory of pseudo-skeleton approximations. Numerical Linear Algebra with Applications. 1997; 261: 1-21. · Zbl 0877.65021 |
| [37] | Goreinov SA, Zamarashkin NL, Tyrtyshnikov EE. Pseudo-skeleton approximations by matrices of maximum volume. Mathematical Notes. 1997; 62(4): 515-519. · Zbl 0916.65040 |
| [38] | Hackbusch W, Braess D. Approximation of \(\frac{ 1}{ x}\) by exponential sums in [1,∞]. IMA Journal of Numerical Analysis; 25(4): 685-697. · Zbl 1082.65025 |
| [39] | Gavrilyuk IP, Hackbusch W, Khoromskij BN. Tensor‐product approximation to the inverse and related operators in high‐dimensional elliptic problems. Computing; 74: 131-157. · Zbl 1071.65032 |
| [40] | C++ code: Deposit 2014. (Available from: http://bitbucket.org/appl/_m729/code‐deposit) [Accessed on 2 July 2014]. |
| [41] | Stoer J, Bulirsch R. Introduction to Numerical Analysis, 3rd edn. Springer, 2002. · Zbl 1004.65001 |
| [42] | Abramowitz M, Stegun IA. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. tenth Dover printing, tenth GPO printing edn., Dover: New York, 1972. · Zbl 0543.33001 |
| [43] | Salvat F, Martnez JD, Mayol R, Parellada J. Analytical Dirac-Hartree-Fock-Slater screening function for atoms (z=1-92). Physical Review A. 1987; 36: 467-474. |
| [44] | Slater J. Quantum Theory of Atomic Structure. International Series in Pure and Applied Physics. McGraw‐Hill: New York, 1960. · Zbl 0094.44702 |
| [45] | Shevelko VP, Vainshtein LA. Atomic Physics for Hot Plasmas. Institute of Physics Pub.: Bristol and Philadelphia, 1993. |
| [46] | Khoromskij B. Structured rank‐(r1,...,rd) decomposition of function‐related tensors in Rd. Computational Methods in Applied Mathematics. 2006; 6: 194-220. · Zbl 1120.65052 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
