7.2 Six-Dimensional Problems
7.2.3 Helium
We finally consider the two-electron atomic system of helium as in (2.14), HHeu :=−1
2∆u− 2
1
|x1|+ 1
|x2|
u + 1
|x1− x2|u = λu . (He) As discussed in Section 2.3, there is no known closed-form solution for the ground state eigenpair (λ0, u0), but highly accurate approximations to λ0 ≈ −2.903724, as given in (2.15), are known.
The ingredients for approximating the operators arising in this problem have already appeared in the previous examples: The interaction potential is treated as in the case of hookium, and the nuclear potentials are, up to the tensorization with three-dimensional identity operators, handled in the same way as in the example of hydrogen.
7.2 Six-Dimensional Problems
10−2 10−1 100
101 102 103 104 105
Figure 7.14. Problem (Hk): total number of nonzero coefficients in mode frames, in dependence on residual estimate. The line has slope−23.
0 20 40 60 80 100 120
Figure 7.15. Problem (Hk): maximum ranks on subdivision elements (rows) in dependence on the iteration number (columns). Black corresponds to the maximum value 6.
7 Numerical Realization and Experiments
Table7.7.Problem(Hk):multilinearranksofiteratesonthefirst20appearingparts¯Λ6,nofthesubdivision.Iterationswithcoarseningareinitalics.
totaliterationnumber107108109110111112113114115116117118119outeriterationnumber22232323232324242424242525[0,0]×[0,0]×[0,0](3,3,3)(4,4,4)(5,5,5)(5,5,5)(5,5,5)(3,3,3)(5,4,4)(5,5,5)(5,5,5)(5,5,5)(4,4,3)(5,5,5)(5,5,5)[1,1]×[0,0]×[0,0](1,1,1)(2,2,2)(2,2,2)(2,2,2)(2,2,2)(1,1,1)(4,4,4)(3,3,3)(3,3,3)(3,3,3)(1,1,1)(4,3,4)(4,3,4)[0,0]×[1,1]×[0,0](1,1,1)(2,2,2)(2,2,2)(2,2,2)(2,2,2)(1,1,1)(2,3,2)(2,2,2)(2,2,2)(2,3,2)(1,1,1)(3,4,3)(3,3,3)[0,0]×[0,0]×[1,1](2,2,2)(2,2,2)(2,2,2)(2,2,2)(1,1,1)(4,4,4)(3,3,3)(3,3,3)(3,3,3)(1,1,1)(3,4,4)(3,4,4)[0,0]×[1,1]×[1,1](1,1,1)(1,2,2)(1,2,2)(1,2,2)(1,1,1)(1,2,2)(1,2,2)(1,2,2)(1,1,1)(1,2,2)[1,1]×[1,1]×[0,0](1,1,1)(1,1,1)(2,2,1)(2,2,1)(1,1,1)(2,2,1)(2,2,1)(2,2,1)(1,1,1)(2,2,1)[1,1]×[0,0]×[1,1](1,1,1)(2,1,2)(2,1,2)(2,1,2)(1,1,1)(2,1,2)(2,1,2)(2,1,2)(1,1,1)(2,1,2)[1,1]×[1,1]×[1,1](1,1,1)(1,1,1)(1,1,1)(1,1,1)(1,1,1)(1,1,1)(1,1,1)(1,1,1)(1,1,1)(1,1,1)[0,1]×[2,3]×[0,1](1,1,1)(1,1,1)(1,1,1)(1,1,1)(1,1,1)(1,1,1)(2,2,2)(2,2,2)(1,1,1)(1,1,1)[2,3]×[0,1]×[0,1](1,1,1)(1,1,1)(1,1,1)(1,1,1)(1,1,1)(1,1,1)(2,1,2)(2,2,2)(1,1,1)(1,1,1)[0,1]×[0,1]×[2,3](1,1,1)(1,1,1)(1,1,1)(1,1,1)(1,1,1)(1,1,1)(2,2,2)(2,2,2)(1,1,1)(1,1,1)[0,1]×[2,2]×[2,2](1,1,1)(1,1,1)(1,1,1)(1,1,1)(1,1,1)(1,1,1)(1,1,1)(1,1,1)(1,1,1)(1,1,1)[2,2]×[0,1]×[2,2](1,1,1)(1,1,1)(1,1,1)(1,1,1)(1,1,1)(1,1,1)(1,1,1)(1,1,1)(1,1,1)(1,1,1)[2,2]×[2,2]×[0,1](1,1,1)(1,1,1)(1,1,1)(1,1,1)(1,1,1)(1,1,1)(1,1,1)(1,1,1)(1,1,1)(1,1,1)[2,2]×[2,2]×[2,2](1,1,1)(1,1,1)(1,1,1)(1,1,1)(1,1,1)(1,1,1)(1,1,1)(1,1,1)(1,1,1)[0,1]×[2,2]×[3,3](1,1,1)(1,1,1)(1,1,1)(1,1,1)(1,1,1)(1,1,1)(1,1,1)(1,1,1)[0,1]×[3,3]×[2,2](1,1,1)(1,1,1)(1,1,1)(1,1,1)(1,1,1)(1,1,1)(1,1,1)(1,1,1)[2,2]×[0,1]×[3,3](1,1,1)(1,1,1)(1,1,1)(1,1,1)(1,1,1)(1,1,1)(1,1,1)(1,1,1)[3,3]×[0,1]×[2,2](1,1,1)(1,1,1)(1,1,1)(1,1,1)(1,1,1)(1,1,1)(1,1,1)(1,1,1)[2,2]×[3,3]×[0,1](1,1,1)(1,1,1)(1,1,1)(1,1,1)(1,1,1)(1,1,1)(1,1,1)(1,1,1)
Table7.8.Problem(Hk):modesizesofiteratescorrespondingtoTable7.7.
totaliterationnumber107108109110111112113114115116117118119outeriterationnumber22232323232324242424242525[0,0]×[0,0]×[0,0](246,247,246)(742,754,754)(924,923,925)(1013,987,988)(1048,1026,1028)(277,277,277)(814,814,814)(978,970,974)(1048,1077,1052)(1103,1114,1104)(307,307,307)(847,847,847)(991,994,993)[1,1]×[0,0]×[0,0](37,14,14)(598,251,251)(844,325,328)(1137,332,334)(1327,334,340)(54,28,28)(802,295,295)(985,352,353)(1359,391,389)(1494,391,389)(78,35,35)(920,351,351)(1213,383,382)[0,0]×[1,1]×[0,0](14,37,14)(255,679,251)(332,902,329)(332,1077,330)(332,1278,330)(28,54,28)(289,664,285)(345,905,343)(366,1369,365)(370,1476,367)(35,79,35)(351,918,351)(390,1237,388)[0,0]×[0,0]×[1,1](256,256,737)(332,332,916)(346,347,1106)(349,349,1298)(28,28,56)(297,295,802)(353,353,999)(370,369,1355)(370,369,1407)(35,35,77)(351,351,918)(383,382,1161)[0,0]×[1,1]×[1,1](199,229,229)(227,336,336)(227,336,336)(271,346,348)(230,255,257)(232,273,269)(274,463,465)(274,472,465)(264,323,323)(266,433,418)[1,1]×[1,1]×[0,0](230,229,198)(327,325,230)(339,337,230)(343,337,230)(257,255,230)(373,371,232)(377,375,243)(438,510,269)(326,323,266)(416,435,266)[1,1]×[0,0]×[1,1](230,198,229)(343,230,342)(349,230,346)(349,231,346)(257,230,257)(371,232,381)(371,244,383)(382,275,402)(325,264,323)(422,266,425)[1,1]×[1,1]×[1,1](175,178,178)(210,208,212)(210,208,212)(210,208,212)(184,184,184)(184,184,184)(249,311,311)(259,323,318)(243,243,243)(243,243,243)[0,1]×[2,3]×[0,1](255,325,255)(323,720,323)(372,1380,370)(374,1951,374)(309,440,308)(358,971,362)(453,1852,458)(457,2034,458)(385,709,385)(457,1320,457)[2,3]×[0,1]×[0,1](325,255,255)(719,324,324)(1160,366,366)(1912,370,372)(440,308,308)(899,382,384)(1897,460,460)(2004,464,460)(709,383,384)(1323,457,457)[0,1]×[0,1]×[2,3](255,255,325)(323,323,719)(354,354,1302)(365,363,1901)(309,308,440)(358,362,943)(454,458,1850)(456,458,2001)(385,385,709)(457,457,1371)[0,1]×[2,2]×[2,2](205,122,122)(251,235,234)(251,235,234)(251,235,234)(240,175,175)(255,282,278)(255,403,409)(305,409,416)(297,261,259)(299,345,343)[2,2]×[0,1]×[2,2](124,206,122)(235,253,234)(235,253,234)(235,253,234)(175,239,175)(276,255,278)(276,255,278)(407,255,396)(259,297,259)(341,297,339)[2,2]×[2,2]×[0,1](124,122,206)(235,235,253)(235,235,253)(430,428,253)(175,175,239)(276,278,255)(276,278,255)(276,278,255)(259,261,297)(345,347,301)[2,2]×[2,2]×[2,2](169,169,169)(169,169,169)(169,169,169)(113,113,113)(206,208,208)(206,208,208)(206,208,208)(191,191,191)(243,243,243)[0,1]×[2,2]×[3,3](148,80,137)(148,80,137)(249,184,453)(188,117,214)(188,117,214)(270,227,544)(146,80,129)(204,144,331)[0,1]×[3,3]×[2,2](148,136,78)(148,136,78)(249,453,182)(188,214,117)(251,493,199)(251,493,199)(146,129,79)(204,331,144)[2,2]×[0,1]×[3,3](78,148,137)(78,148,137)(184,249,454)(119,192,214)(119,192,214)(220,264,545)(79,146,129)(142,204,334)[3,3]×[0,1]×[2,2](134,148,78)(324,218,146)(324,218,146)(214,192,117)(214,192,117)(546,266,220)(129,146,79)(334,204,144)[2,2]×[3,3]×[0,1](78,136,148)(78,136,148)(182,454,249)(119,214,192)(119,214,192)(222,550,266)(79,129,146)(142,334,204)
7.2 Six-Dimensional Problems
−3 −2 −1 0 1 2 3
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045
−3 −2 −1 0 1 2 3
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045
Figure 7.16. Problem (Hk): convergence of iterates at the end of outer iterations 15, 20, and 25 (total iteration numbers 71, 96, and 122) to the exact solution on the section (x1, 0, 0, 0, 0, 0). Left:
before coarsening step, right: after coarsening step.
Note that in contrast to the previous example of hookium, HHe: H1(R6)→ H−1(R6) satisfies the assumptions of our convergence analysis. In particular, the diagonal rescaling provides similarly effective preconditioning as in the case of hydrogen, and the iteration step size can be chosen substantially larger than for hookium.
The numerical treatment of (He), in particular the approximate application of the Hamiltonian, is more expensive than in the other examples due to a combination of several difficulties: the operator ranks required for a given target tolerance are more than three times as large as for hydrogen; the approximation of the nuclear cusps, which are more pronounced than in the case of hydrogen, already requires a substantial number of wavelet coefficients; and the approximate application of the interaction potential to the iterates accordingly becomes more expensive than in the case of hookium.
Our results therefore do not cover a range of error tolerances as low as in the simpler test cases. The obtained residual approximations and eigenvalue errors are shown in Figure 7.17.
The convergence pattern shows some variations, in particular after steps with a switch to more accurate potential approximations. In the later iterations, however, the eigenvalue error remains below 4× 10−3.
For the helium ground state we do not have a tensor approximability result as the one for hookium in Theorem 4.34. The observed growth of the total number of entries in the mode frames as shown in Figure 7.18, however, is consistent with the same asymptotic rate that we have obtained for hookium. The evolution of multilinear ranks and mode frame sizes over the course of the iteration is shown in Figure 7.19 and Tables 7.9 and 7.10; again we observe a gradual increase in the ranks of iterates, where the maximum arising multilinear rank on a subdivision element is (9, 9, 9).
7 Numerical Realization and Experiments
5 10 15 20 25 30
10−3 10−2 10−1 100
Figure 7.17. Problem (He): ◦ residual, × eigenvalue approximation for each iteration step; the line gives the current error tolerance η.
10−1 100
103 104 105
Figure 7.18. Problem (He): total number of nonzero coefficients in mode frames, in dependence on residual estimate. The line has slope−23.
7.2 Six-Dimensional Problems
Table7.9.Problem(He):multilinearranksofiteratesonthefirst20appearingparts
¯ Λ6
,nofthesubdivision.Iterationswithcoarseningareinitalics. totaliterationnumber202122232425262728293031 outeriterationnumber455566667777 [−2,−2]×[−2,−2]×[−2,−2](4,4,4)(4,4,4)(4,4,4)(4,4,4)(4,4,4)(4,4,4)(4,4,4)(4,4,4)(4,4,4)(4,4,4)(5,5,5)(5,5,5) [−1,−1]×[−2,−2]×[−2,−2](4,4,4)(4,4,4)(4,4,4)(4,4,4)(5,4,4)(5,4,4)(5,4,4)(6,4,4)(6,4,4)(6,4,4)(6,5,5)(6,6,6) [−2,−2]×[−1,−1]×[−2,−2](4,4,4)(4,4,4)(4,4,4)(4,4,4)(4,5,4)(4,5,4)(4,5,4)(4,6,4)(4,6,4)(4,6,4)(5,6,5)(6,6,6) [−2,−2]×[−2,−2]×[−1,−1](4,4,4)(4,4,4)(4,4,4)(4,4,4)(4,4,5)(4,4,5)(4,4,5)(4,4,6)(4,4,6)(4,4,6)(5,5,6)(6,6,6) [−2,−2]×[−1,−1]×[−1,−1](4,5,5)(4,5,5)(4,6,6)(4,6,6)(4,6,6)(4,6,6)(4,6,6)(4,6,6)(4,6,6)(4,6,6)(5,7,7)(5,6,6) [−1,−1]×[−2,−2]×[−1,−1](5,4,5)(5,4,5)(6,4,6)(6,4,6)(6,4,6)(6,4,6)(6,4,6)(6,4,6)(6,4,6)(6,4,6)(7,5,7)(7,5,6) [−1,−1]×[−1,−1]×[−2,−2](5,5,4)(5,5,4)(6,6,4)(6,6,4)(6,6,4)(6,6,4)(6,6,4)(6,6,4)(6,6,4)(6,6,4)(7,7,5)(6,6,5) [−1,−1]×[−1,−1]×[−1,−1](5,5,5)(5,5,5)(5,5,5)(5,5,5)(5,5,5)(6,6,6)(6,6,6)(6,6,6)(6,6,6)(7,6,6)(8,7,8)(7,7,7) [0,1]×[−2,−1]×[−2,−1](6,5,5)(6,5,5)(6,5,5)(6,6,6)(6,6,6)(6,6,6)(9,6,6)(8,7,7)(8,7,7)(9,7,7)(9,9,9)(9,9,9) [−2,−1]×[0,1]×[−2,−1](5,6,5)(5,6,5)(5,6,5)(6,6,6)(6,6,6)(6,6,6)(6,9,6)(7,9,7)(7,9,7)(7,9,7)(9,9,9)(9,9,9) [−2,−1]×[−2,−1]×[0,1](5,5,6)(5,5,6)(5,5,6)(6,6,6)(6,6,6)(6,6,6)(6,6,8)(7,7,8)(7,7,8)(7,7,9)(9,9,9)(9,9,9) [−2,−1]×[0,0]×[0,0](4,6,6)(4,6,6)(4,6,6)(4,6,6)(4,6,7)(5,7,7)(6,7,7)(6,8,8)(6,8,8)(6,9,9)(6,9,9)(7,9,9) [0,0]×[−2,−1]×[0,0](6,4,6)(6,4,6)(6,4,6)(6,4,6)(6,4,6)(7,5,7)(7,6,7)(8,6,8)(8,6,7)(9,6,9)(9,6,9)(9,7,9) [0,0]×[0,0]×[−2,−1](5,6,4)(5,4,4)(6,6,4)(6,6,4)(6,6,4)(7,7,5)(7,7,6)(8,8,6)(8,8,6)(9,9,6)(9,9,6)(9,9,7) [0,0]×[0,0]×[0,0](4,4,4)(4,4,4)(4,4,4)(5,5,5)(6,6,6)(6,6,6)(7,7,7)(7,7,7)(7,7,7)(8,8,8)(8,9,9)(8,8,8) [−2,−1]×[0,0]×[1,1](2,3,3)(2,3,3)(3,3,3)(3,4,4)(4,4,4)(4,5,5)(4,5,5)(4,5,5)(4,5,5)(4,5,5)(5,5,5)(5,6,6) [−2,−1]×[1,1]×[0,0](2,3,3)(2,3,3)(3,3,3)(3,4,4)(4,4,4)(4,5,5)(4,5,5)(4,5,5)(4,5,5)(4,5,5)(5,5,5)(5,6,6) [0,0]×[−2,−1]×[1,1](3,2,3)(3,2,3)(3,3,3)(4,3,4)(4,4,4)(5,4,5)(5,4,5)(5,4,5)(5,4,5)(5,4,5)(5,5,5)(6,5,6) [1,1]×[−2,−1]×[0,0](3,2,3)(3,2,3)(3,3,3)(4,3,4)(4,4,4)(5,4,5)(5,4,5)(5,4,5)(5,4,5)(5,4,5)(5,5,5)(6,5,6) [0,0]×[1,1]×[−2,−1](3,3,2)(3,3,2)(3,3,3)(4,4,3)(4,4,4)(5,5,4)(5,5,4)(5,5,4)(5,5,4)(5,5,4)(5,5,5)(6,6,5) Table7.10.Problem(He):modesizesofiteratescorrespondingtoTable7.9. totaliterationnumber202122232425262728293031 outeriterationnumber455566667777 [−2,−2]×[−2,−2]×[−2,−2](245,244,245)(359,362,359)(416,362,430)(265,264,264)(361,358,358)(389,387,391)(392,390,394)(281,281,281)(394,394,393)(431,429,429)(460,452,447)(306,306,306) [−1,−1]×[−2,−2]×[−2,−2](339,234,234)(451,291,291)(514,292,292)(349,244,244)(451,298,298)(512,328,324)(559,340,339)(398,280,280)(534,336,339)(578,359,359)(598,376,375)(428,308,308) [−2,−2]×[−1,−1]×[−2,−2](233,339,234)(291,450,291)(291,492,291)(243,347,243)(296,447,296)(326,512,325)(340,558,340)(280,398,280)(337,535,335)(354,581,355)(375,599,377)(308,428,307) [−2,−2]×[−2,−2]×[−1,−1](234,234,339)(291,291,444)(291,291,502)(244,244,346)(297,296,442)(329,329,509)(340,342,558)(280,280,398)(338,337,534)(354,354,583)(377,377,596)(307,307,430) [−2,−2]×[−1,−1]×[−1,−1](215,312,310)(281,417,415)(281,417,415)(220,330,330)(290,424,424)(310,463,468)(324,511,508)(251,370,371)(331,507,503)(343,540,542)(349,565,562)(276,401,402) [−1,−1]×[−2,−2]×[−1,−1](310,215,310)(417,283,416)(417,283,416)(330,221,330)(424,289,422)(465,309,465)(510,324,507)(370,251,371)(499,334,499)(529,341,540)(562,350,562)(399,276,402) [−1,−1]×[−1,−1]×[−2,−2](312,312,215)(419,420,281)(419,420,281)(328,330,221)(424,422,290)(463,464,309)(507,507,324)(371,371,251)(499,499,332)(523,536,339)(563,562,352)(401,401,276) [−1,−1]×[−1,−1]×[−1,−1](297,297,296)(404,403,399)(404,403,399)(307,307,306)(414,414,413)(447,447,447)(485,488,482)(359,359,361)(493,492,486)(517,519,521)(538,534,532)(383,383,383) [0,1]×[−2,−1]×[−2,−1](767,530,529)(1005,560,554)(1200,622,619)(839,536,536)(1058,603,602)(1215,692,690)(1360,749,745)(1045,629,628)(1321,716,716)(1442,782,775)(1480,797,791)(1170,690,689) [−2,−1]×[0,1]×[−2,−1](530,767,529)(562,1010,555)(621,1200,618)(536,838,536)(605,1061,603)(692,1217,691)(749,1362,747)(631,1043,631)(719,1326,719)(773,1447,765)(798,1482,797)(692,1169,690) [−2,−1]×[−2,−1]×[0,1](530,529,767)(563,560,1016)(624,621,1200)(536,536,841)(600,600,1065)(693,692,1225)(749,750,1356)(629,629,1044)(719,719,1334)(773,769,1457)(799,797,1491)(693,693,1170) [−2,−1]×[0,0]×[0,0](416,374,374)(554,474,474)(554,474,474)(425,387,388)(552,484,482)(589,524,524)(610,570,571)(534,450,450)(659,598,600)(659,613,615)(678,623,625)(561,488,488) [0,0]×[−2,−1]×[0,0](375,416,376)(470,555,469)(470,555,471)(386,426,389)(483,548,485)(523,589,525)(571,612,569)(449,534,449)(596,659,600)(612,659,611)(651,699,638)(490,560,488) [0,0]×[0,0]×[−2,−1](376,375,418)(472,471,559)(472,471,559)(388,385,433)(481,480,542)(524,524,595)(568,569,612)(449,450,533)(600,563,660)(626,623,672)(635,625,672)(490,490,559) [0,0]×[0,0]×[0,0](306,306,307)(480,477,478)(481,478,479)(330,331,331)(461,460,460)(478,474,469)(510,522,521)(405,404,404)(565,558,563)(565,558,563)(568,564,572)(435,434,433) [−2,−1]×[0,0]×[1,1](299,268,273)(331,346,432)(479,398,505)(317,290,314)(353,326,471)(474,428,509)(474,433,527)(413,378,436)(413,414,540)(576,489,651)(576,489,651)(455,408,481) [−2,−1]×[1,1]×[0,0](299,270,268)(330,434,345)(480,502,397)(317,310,289)(353,473,324)(474,517,430)(474,529,435)(410,438,379)(412,534,412)(576,622,463)(576,625,463)(455,487,410) [0,0]×[−2,−1]×[1,1](268,298,272)(348,330,436)(408,485,514)(289,317,315)(377,351,475)(402,463,514)(446,494,530)(380,412,434)(414,412,539)(463,577,598)(463,577,601)(408,455,487) [1,1]×[−2,−1]×[0,0](271,298,268)(436,331,349)(506,481,396)(311,318,290)(475,349,375)(521,463,408)(540,501,447)(437,411,378)(537,412,415)(598,575,462)(601,575,462)(487,455,413) [0,0]×[1,1]×[−2,−1](268,272,299)(350,438,330)(405,519,477)(286,310,317)(377,477,394)(403,522,461)(436,531,471)(378,440,404)(417,547,519)(463,592,526)(466,595,540)(408,491,459)
7 Numerical Realization and Experiments
0 5 10 15 20 25 30
Figure 7.19. Problem (He): maximum ranks on subdivision elements (rows) in dependence on the iteration number (columns). Black corresponds to the maximum value 9.
8 Conclusion and Outlook
We have considered adaptive methods that exploit both low-rank structure of solutions, and their near-sparsity in a wavelet basis. We have studied in particular the approximation of two-electron wave functions, and the approximation of the operators arising in the corresponding eigenvalue problems.
For solutions of model problems, we have obtained approximation complexity estimates. These show that in certain cases of solutions with limited Besov regularity, the nonlinear parametrization of wavelet coefficients yields an improvement, compared with a direct wavelet approximation, in the achievable convergence rate in terms of the total number of parameters.
For the hookium model problem with electron interaction cusp, the results of Section 4.3 show that we can expect almost three times the approximation rate that would be possible with a direct wavelet approximation. The analytical estimate is confirmed by the numerical experiments, and similar numerical observations are made in the case of helium, where no corresponding analytical approximability result is available.
In examples with unlimited Besov regularity, such as hydrogen, we have seen that essentially, the best rate possible in the one-dimensional case for H1–approximation by the wavelet basis is recovered. This can be achieved with standard adaptive wavelet methods based on anisotropic tensor product wavelets as well. For tensor decompositions of wavelet coefficients, however, the constants in the estimates can be expected to be more favorable in higher dimensions. Consider, for instance, the coefficients on an arbitrary fixed wavelet level in an approximation of a separable function in d dimensions with an isolated singularity at zero. If the wavelets have support size 10, we will then typically need at least the order of 10 basis functions in each coordinate direction. In a linearly parametrized wavelet expansion, this leads to 10ddegrees of freedom for this fixed wavelet level alone. In the present approach with a multiplicative parametrization, only 10d coefficients are required. Even in cases in which the same convergence rate is achieved by both constructions, for functions in higher dimensions with suitable structure, the total approximation complexity may thus be substantially more favorable for nonlinearly parametrized wavelet coefficients.
In this work, we have proven the convergence of adaptive low-rank schemes for the computation of such approximations. The total complexity of these methods remains to be investigated in further detail. The numerical experiments support the conjecture that one can expect a computational complexity that is optimal in a similar sense as in the case of adaptive wavelet methods.
In this regard, however, for two-electron problems there remains an issue with the approximation of the electron interaction Coulomb potential. For the resulting lower-dimensional factor matrices, our construction of compressed matrices yields s∗–compressibility with some s∗ < 1 depending on the order of the wavelet basis. However, to make full use of approximability of solutions, we would need s∗ = 32. There is no clear indication in our numerical results that an asymptotically better construction of compressed matrices is possible. In order to circumvent the limitation by the s∗-compressibility of the interaction potential, it may therefore be necessary to make stronger use of structural a priori knowledge on the wavelet coefficients of interaction cusps.
For an improvement in the efficiency of the schemes in general, it may be useful to replace the Daubechies-type wavelets used here by orthonormal spline multiwavelets, which can yield sub-stantially better compressibility of certain operators (e.g., of the Laplacian). By an appropriate modification of the given construction, however, we cannot expect to obtain s∗ ≥ 1 for the lower-dimensional components in the approximate interaction potential in this manner either, which can be seen from an inspection of the proof of Theorem 6.27. Apart from this, there are other possible
8 Conclusion and Outlook
benefits in using such spline basis functions, in particular simpler and potentially more efficient computation of integrals. This can improve the s∗-computability (cf. Definition 3.29) of operators, since the the methods for general wavelets considered in Sections 6.5 and 6.6 have the shortcoming that the number of operations required for the evaluation of each single matrix entry does not remain uniformly bounded.
Another direction for further developments are modifications to the basic algorithms that improve the quantitative behaviour. Algorithms 5.5 and 5.6 in their basic form have the advantage of a relatively transparent basic concept. A first modification, which yields a promising improvement in the numerical experiments of Chapter 7, are additional iterations on fixed index sets. Second, one may use improved approximations for Rayleigh quotients. We have not yet incorporated these options in our convergence analysis here. Moreover, the construction of eigenvalue solvers for nonsymmetric problems, which would allow the treatment of the explicitly correlated formulation of Section 2.2, would be of interest as well.
The tensor representation is used here essentially as a black box that provides the required operations, but one could also consider a combination of the developed wavelet concepts with different iterative schemes that are more directly adapted to the underlying tensor representations (as investigated, e.g., in [82]).
A particularly important point is the application of the iterative methods we have considered to problems in dimensions higher than six. This necessitates the use of alternatives to the Tucker format, which has been sufficient for our purposes in this work. Such alternatives are, for instance, theH-Tucker format or the Tensor Train format. Different variants of tensor preconditioning that are intermediate between a direct application of rescaling operations and the levelwise subdivision considered here may be of interest in higher dimensions as well. The basic iterative schemes given in Chapter 5 can thus in principle be extended to higher-dimensional problems with a suitable tensor structure.
Concerning the application to the Schr¨odinger equation we have considered, the question arises whether there are practically feasible ways of improving the approximation of electron interaction cusps beyond the constraints that one obtains for tensor product bases. There are known methods, e.g., those of Hylleraas type discussed briefly in Section 2.3, that enable a better approximation of electron interaction cusps by suitable coordinate changes. The methods of this type known to date are, however, restricted to special systems with a limited number of electrons.
The basic concepts we have considered here are in principle applicable to systems with more than two electrons, but there are a number of additional difficulties – besides the higher dimensionality – that need to be addressed. This concerns, in particular, the partial antisymmetry conditions that need to be enforced for three or more electrons, and the handling of bivariate tensor factors corresponding to electron pairs. As noted in [154], the efficient application of operators in a direct wavelet discretization with antisymmetry constraints is not straightforward, and this is even less clear for wavelet coefficients represented in a tensor format. The corresponding operations on sums of separable functions, under antisymmetry constraints and combined with a fixed set of electron pair functions, have been considered in [113]. This raises the question whether similar results are possible for tensor formats imposing additional structure, such as the Tucker andH-Tucker format.
Index
explicit correlation, 10, 11, 42, 46, 60 exponential decay, 9, 14Galerkin discretization, 30, 33, 74, 104 Gaussian geminals, 11, 57
Hooke’s law atom, 14, 45, 57, 163 hookium, see Hooke’s law atom hydrogen, 7, 13, 44, 52, 156 inequality norm equivalence, 19, 21, 26, 27, 29 order of polynomial reproduction, 19
Sobolev of mixed smoothness, 27, 42 tensor product Besov, 29
Index
wave function, 7 wavelets
biorthogonal, 18 Daubechies, 22, 96 hyperbolic, 28
Ojanen, 23 orthonormal, 18 weak formulation, 9, 30 weighting factor, 98
List of Symbols
|λ| 19, 26 a(i)n 80 Asq 21 Asq(H) 20 B˜sp(Rd; D) 29 ˇ
vi 69 χd 25 Dd 48 hα 107
Hs,kmix(R3; n) 27 jα 107
Kd(r) 69 k(λ) 19, 26 Λ¯d,n 78 Λ(i)n (·) 80 max|λ| 26 min|λ| 26 N1(·, Dd) 48
∇ 18
∇j 19
∇(d) 25
∇(d)j 25
∇d 26 (∇(d))Dj 26
⊗ 23 Ψλ 25, 26 Sd 76 σ(i)n 80 s(λ) 19, 26 Sσ± 8 Td(r) 79 Vee 8 Vne 8
WD 98, 107 Zj0 18, 98
List of Tables
6.1 Combinations of wavelet indices used for the numerical experiments . . . 145
6.2 Reference values for integrals . . . 145
6.3 Rescaling factors for reference values . . . 146
6.4 Results of dyadic refinement scheme . . . 148
6.5 Timings for evaluation of integrals . . . 149
7.1 Problem (δ3): multilinear ranks of iterates . . . 157
7.2 Problem (δ3): mode sizes of iterates . . . 157
7.3 Problem (H): multilinear ranks of iterates . . . 161
7.4 Problem (H): mode sizes of iterates . . . 161
7.5 Problem (δ6): multilinear ranks of iterates . . . 164
7.6 Problem (δ6): mode sizes of iterates . . . 164
7.7 Problem (Hk): multilinear ranks of iterates . . . 168
7.8 Problem (Hk): mode sizes of iterates . . . 168
7.9 Problem (He): multilinear ranks of iterates . . . 171
7.10 Problem (He): mode sizes of iterates . . . 171
List of Figures
3.1 Relation of Besov spaces to approximation spaces . . . 22
5.1 Structure of the partitionsJ`,{1,2}(2) . . . 75
5.2 Structure of the partitionsJ`,{1,2,3}(3) . . . 78
6.1 Eigenvalue error in Galerkin discretization with approximate potential . . . 106
6.2 Wavelet and scaling function used in the numerical integration tests . . . 113
6.3 Levelwise decay of entries for approximations of one-electron Coulomb potentials . . 114
6.4 Levelwise decay of entries for approximations of two-electron Coulomb potentials . . 122
6.5 Levelwise decay of entries for approximations of two-electron Coulomb potentials . . 122
6.6 Integration error in dependence of given h and Nh, tests 1 and 2 . . . 146
6.7 Integration error in dependence of given h and Nh, tests 3 and 4 . . . 147
6.8 Integration error in dependence of given h and Nh, tests 5 and 6 . . . 147
7.1 Problem (δ3): residual and eigenvalue approximations . . . 154
7.2 Problem (δ3): convergence on one-dimensional section . . . 154
7.3 Problem (δ3): coefficients in mode frames . . . 155
7.4 Problem (δ3): ranks on subdivision elements . . . 155
7.5 Problem (H): number of terms in operator approximation . . . 158
7.6 Problem (H): residual and eigenvalue approximations . . . 158
7.7 Problem (H): convergence on one-dimensional section . . . 159
7.8 Problem (H): coefficients in mode frames . . . 159
7.9 Problem (H): ranks on subdivision elements . . . 160
7.10 Problem (δ6): residual and eigenvalue approximations . . . 162
7.11 Problem (δ6): coefficients in mode frames . . . 163
7.12 Problems (δ3) and (δ6): operations in apply . . . 165
7.13 Problem (Hk): residual and eigenvalue approximations . . . 166
7.14 Problem (Hk): coefficients in mode frames . . . 167
7.15 Problem (Hk): ranks on subdivision elements . . . 167
7.16 Problem (Hk): convergence on one-dimensional section . . . 169
7.17 Problem (He): residual and eigenvalue approximations . . . 170
7.18 Problem (He): coefficients in mode frames . . . 170
7.19 Problem (He): ranks on subdivision elements . . . 172
A Supplementary Proofs
A.1 Anisotropic Besov Regularity for Hooke’s Law Atom
We follow the lines of [51, Lemma 2.1]; since the modification to the argument that we need only involves some detail changes, we adopt the notation from [51] and refer to specific equations in [51] that are changed. Recall that the proof in [51] applies to u0 as in (4.5) without essential changes, but using the superexponential decay of u0 towards infinity instead of the boundedness of the domain.
Proof of Theorem 4.14. We use x, y ∈ R3 as coordinates for R6. For the explicitly correlated eigenfunction w0 as in (4.6), we make use of asymptotic smoothness property
∂
αx∂yβw0(x, y)
≤ cα,β|x − y|3−|α|−|β|. In particular, ∂x31w0 is uniformly bounded.
We assume j1 ≥ j2 without restriction of generality. As in equation (2.8) in [51], we have X
i∈∆0
2−(3+1/2)j12−j12−3j123j2/2
∂x31w0
∞,j1,a1×i . 2−15j1/2+3j2/2
withO(1) summands on the left hand side. Equation (2.9) in [51] becomes, here with p > 6, X
i∈∆\∆0
2−(p+3/2)j1−3j1+3j2/2k∂xp1w0k∞,j1,a1×i
. 2−(p+3/2)j1+3j2/2 X
i∈∆\∆0
2−3j1 sup
(x,y)∈j1,a1×i
|x − y|3−p
. 2−(p+3/2)j1+3j2/2
Z 2−j2+2
2−j1
r2+3−pdr . 2−15j1/2+3j2/2. Equation (2.10) is replaced by
X
j1≥j2≥0
2qj1 X
a1,a2
|hΨj1,j2,a1,a2, w0i|q . X
j1≥j2≥0
2qj123j1(2−15j1/2+3j2/2)q
= X
j1≥0
2−(13q/2−3)j1 X
j2≤j1
23qj2/2∼ X
j1≥0
2−(5q−3)j1,
which requires q > 3/5. We replace (2.11) by X
a1,a2
|hΨj1,j2,a1,a2, w0i|q . 2−(p+3/2)q(j1+j2)
∂xp1∂yp1w0
∞,j1,a1×j2,a2
. X
a1,a2
2−(pq+3q/2−3)j12−(pq+3q/2−3)j22−3(j1+j2) sup
(x,y)∈j1,a1×j2,a2|x − y|(3−2p)q . 2−(pq+3q/2−3)(j1+j2)
Z ∞ 2−j2
r2+(3−2p)qdr
A Supplementary Proofs
where 2 + 3q− 2pq < −1 follows with p > α + 1 and α = 3/q − 3/2, hence . 2−(pq+3q/2−3)(j1+j2)2−(3+(3−2p)q)j2 = 2−(pq+3q/2−3)j12(pq−9q/2)j2, and finally
X
j1≥j2≥0
2qj12−(pq+3q/2−3)j12(pq−9q/2)j2 ∼ X
j1≥0
2−(pq+q/2−3)j12(pq−9q/2)j1 = X
j1≥0
2−(5q−3)j1.
Thus w0 is in the space ˜Bαq(R3; 2) with α = 3/q− 3/2 if q > 3/5, that is, if α < 7/2. Analogously to the argument for u0 in [51], it can be seen that this result is sharp.