Sparse grids: applications to multi-dimensional Schrödinger problems
Abstract: "Sparse grid methods applied to solve partial differential equations allow for a substantial reduction of numerical effort (to obtain equal error magnitudes) compared to conventional finite element methods. A short introduction to this new approach is given. Using a Ritz-Galerkin meth...
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Buch |
| Sprache: | German |
| Veröffentlicht: |
München
1995
|
| Schriftenreihe: | Technische Universität <München>: TUM-I
9507 |
| Schlagworte: | |
| Zusammenfassung: | Abstract: "Sparse grid methods applied to solve partial differential equations allow for a substantial reduction of numerical effort (to obtain equal error magnitudes) compared to conventional finite element methods. A short introduction to this new approach is given. Using a Ritz-Galerkin method on rectangular sparse grids, stationary Schrödinger equations of dimensionality D [> or =] 2 are solved numerically for a number of generic problems and the results are compared to exact values, perturbative results, and numerical computations of other authors. For problems with oscillator potentials (harmonic or anharmonic), the accuracy of eigenvalues for similar numbers of grid points and equal order of basis functions is increased by up to two orders of magnitude with respect to conventional FEM. Good solutions are obtained for singular potentials (hydrogen atom and hydrogen molecular ion), where the sparse grid was automatically refined using a local adaptation strategy. Schrödinger problems of high dimensionality (up to D=8) become tractable with this algorithm, regardless of symmetries or separabilities of the potential functions, i.e. similar accuracies are to be expected for arbitrary potentials. As an example of a physically significant and intrinsically high-dimensional problem, eigenstates of a spin boson coupling model were computed." |
| Beschreibung: | 20 S. graph. Darst. |
Internformat
MARC
| LEADER | 00000nam a2200000 cb4500 | ||
|---|---|---|---|
| 001 | BV010372308 | ||
| 003 | DE-604 | ||
| 005 | 20040416 | ||
| 007 | t | ||
| 008 | 950829s1995 gw d||| t||| 00||| ger d | ||
| 016 | 7 | |a 945208529 |2 DE-101 | |
| 035 | |a (OCoLC)35123072 | ||
| 035 | |a (DE-599)BVBBV010372308 | ||
| 040 | |a DE-604 |b ger |e rakwb | ||
| 041 | 0 | |a ger | |
| 044 | |a gw |c DE | ||
| 049 | |a DE-29T |a DE-12 |a DE-91G | ||
| 088 | |a TUM I 9507 | ||
| 100 | 1 | |a Hilgenfeldt, Sascha |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Sparse grids |b applications to multi-dimensional Schrödinger problems |c Sascha Hilgenfeldt ; Robert Balder ; Christoph Zenger |
| 264 | 1 | |a München |c 1995 | |
| 300 | |a 20 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Technische Universität <München>: TUM-I |v 9507 | |
| 490 | 1 | |a Sonderforschungsbereich Methoden und Werkzeuge für die Nutzung Paralleler Rechnerarchitekturen: SFB-Bericht / A |v 1995,5 | |
| 520 | 3 | |a Abstract: "Sparse grid methods applied to solve partial differential equations allow for a substantial reduction of numerical effort (to obtain equal error magnitudes) compared to conventional finite element methods. A short introduction to this new approach is given. Using a Ritz-Galerkin method on rectangular sparse grids, stationary Schrödinger equations of dimensionality D [> or =] 2 are solved numerically for a number of generic problems and the results are compared to exact values, perturbative results, and numerical computations of other authors. For problems with oscillator potentials (harmonic or anharmonic), the accuracy of eigenvalues for similar numbers of grid points and equal order of basis functions is increased by up to two orders of magnitude with respect to conventional FEM. Good solutions are obtained for singular potentials (hydrogen atom and hydrogen molecular ion), where the sparse grid was automatically refined using a local adaptation strategy. Schrödinger problems of high dimensionality (up to D=8) become tractable with this algorithm, regardless of symmetries or separabilities of the potential functions, i.e. similar accuracies are to be expected for arbitrary potentials. As an example of a physically significant and intrinsically high-dimensional problem, eigenstates of a spin boson coupling model were computed." | |
| 650 | 4 | |a Multigrid methods (Numerical analysis) | |
| 650 | 4 | |a Schrödinger equation | |
| 700 | 1 | |a Balder, Robert |e Verfasser |4 aut | |
| 700 | 1 | |a Zenger, Christoph |e Verfasser |4 aut | |
| 810 | 2 | |a A |t Sonderforschungsbereich Methoden und Werkzeuge für die Nutzung Paralleler Rechnerarchitekturen: SFB-Bericht |v 1995,5 |w (DE-604)BV004627888 |9 1995,5 | |
| 830 | 0 | |a Technische Universität <München>: TUM-I |v 9507 |w (DE-604)BV006185376 |9 9507 | |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-006905241 | ||
Datensatz im Suchindex
| _version_ | 1804124796537012224 |
|---|---|
| any_adam_object | |
| author | Hilgenfeldt, Sascha Balder, Robert Zenger, Christoph |
| author_facet | Hilgenfeldt, Sascha Balder, Robert Zenger, Christoph |
| author_role | aut aut aut |
| author_sort | Hilgenfeldt, Sascha |
| author_variant | s h sh r b rb c z cz |
| building | Verbundindex |
| bvnumber | BV010372308 |
| ctrlnum | (OCoLC)35123072 (DE-599)BVBBV010372308 |
| format | Book |
| fullrecord | <?xml version="1.0" encoding="UTF-8"?><collection xmlns="http://www.loc.gov/MARC21/slim"><record><leader>02845nam a2200385 cb4500</leader><controlfield tag="001">BV010372308</controlfield><controlfield tag="003">DE-604</controlfield><controlfield tag="005">20040416 </controlfield><controlfield tag="007">t</controlfield><controlfield tag="008">950829s1995 gw d||| t||| 00||| ger d</controlfield><datafield tag="016" ind1="7" ind2=" "><subfield code="a">945208529</subfield><subfield code="2">DE-101</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(OCoLC)35123072</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-599)BVBBV010372308</subfield></datafield><datafield tag="040" ind1=" " ind2=" "><subfield code="a">DE-604</subfield><subfield code="b">ger</subfield><subfield code="e">rakwb</subfield></datafield><datafield tag="041" ind1="0" ind2=" "><subfield code="a">ger</subfield></datafield><datafield tag="044" ind1=" " ind2=" "><subfield code="a">gw</subfield><subfield code="c">DE</subfield></datafield><datafield tag="049" ind1=" " ind2=" "><subfield code="a">DE-29T</subfield><subfield code="a">DE-12</subfield><subfield code="a">DE-91G</subfield></datafield><datafield tag="088" ind1=" " ind2=" "><subfield code="a">TUM I 9507</subfield></datafield><datafield tag="100" ind1="1" ind2=" "><subfield code="a">Hilgenfeldt, Sascha</subfield><subfield code="e">Verfasser</subfield><subfield code="4">aut</subfield></datafield><datafield tag="245" ind1="1" ind2="0"><subfield code="a">Sparse grids</subfield><subfield code="b">applications to multi-dimensional Schrödinger problems</subfield><subfield code="c">Sascha Hilgenfeldt ; Robert Balder ; Christoph Zenger</subfield></datafield><datafield tag="264" ind1=" " ind2="1"><subfield code="a">München</subfield><subfield code="c">1995</subfield></datafield><datafield tag="300" ind1=" " ind2=" "><subfield code="a">20 S.</subfield><subfield code="b">graph. Darst.</subfield></datafield><datafield tag="336" ind1=" " ind2=" "><subfield code="b">txt</subfield><subfield code="2">rdacontent</subfield></datafield><datafield tag="337" ind1=" " ind2=" "><subfield code="b">n</subfield><subfield code="2">rdamedia</subfield></datafield><datafield tag="338" ind1=" " ind2=" "><subfield code="b">nc</subfield><subfield code="2">rdacarrier</subfield></datafield><datafield tag="490" ind1="1" ind2=" "><subfield code="a">Technische Universität <München>: TUM-I</subfield><subfield code="v">9507</subfield></datafield><datafield tag="490" ind1="1" ind2=" "><subfield code="a">Sonderforschungsbereich Methoden und Werkzeuge für die Nutzung Paralleler Rechnerarchitekturen: SFB-Bericht / A</subfield><subfield code="v">1995,5</subfield></datafield><datafield tag="520" ind1="3" ind2=" "><subfield code="a">Abstract: "Sparse grid methods applied to solve partial differential equations allow for a substantial reduction of numerical effort (to obtain equal error magnitudes) compared to conventional finite element methods. A short introduction to this new approach is given. Using a Ritz-Galerkin method on rectangular sparse grids, stationary Schrödinger equations of dimensionality D [> or =] 2 are solved numerically for a number of generic problems and the results are compared to exact values, perturbative results, and numerical computations of other authors. For problems with oscillator potentials (harmonic or anharmonic), the accuracy of eigenvalues for similar numbers of grid points and equal order of basis functions is increased by up to two orders of magnitude with respect to conventional FEM. Good solutions are obtained for singular potentials (hydrogen atom and hydrogen molecular ion), where the sparse grid was automatically refined using a local adaptation strategy. Schrödinger problems of high dimensionality (up to D=8) become tractable with this algorithm, regardless of symmetries or separabilities of the potential functions, i.e. similar accuracies are to be expected for arbitrary potentials. As an example of a physically significant and intrinsically high-dimensional problem, eigenstates of a spin boson coupling model were computed."</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Multigrid methods (Numerical analysis)</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Schrödinger equation</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Balder, Robert</subfield><subfield code="e">Verfasser</subfield><subfield code="4">aut</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Zenger, Christoph</subfield><subfield code="e">Verfasser</subfield><subfield code="4">aut</subfield></datafield><datafield tag="810" ind1="2" ind2=" "><subfield code="a">A</subfield><subfield code="t">Sonderforschungsbereich Methoden und Werkzeuge für die Nutzung Paralleler Rechnerarchitekturen: SFB-Bericht</subfield><subfield code="v">1995,5</subfield><subfield code="w">(DE-604)BV004627888</subfield><subfield code="9">1995,5</subfield></datafield><datafield tag="830" ind1=" " ind2="0"><subfield code="a">Technische Universität <München>: TUM-I</subfield><subfield code="v">9507</subfield><subfield code="w">(DE-604)BV006185376</subfield><subfield code="9">9507</subfield></datafield><datafield tag="999" ind1=" " ind2=" "><subfield code="a">oai:aleph.bib-bvb.de:BVB01-006905241</subfield></datafield></record></collection> |
| id | DE-604.BV010372308 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T17:51:22Z |
| institution | BVB |
| language | German |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-006905241 |
| oclc_num | 35123072 |
| open_access_boolean | |
| owner | DE-29T DE-12 DE-91G DE-BY-TUM |
| owner_facet | DE-29T DE-12 DE-91G DE-BY-TUM |
| physical | 20 S. graph. Darst. |
| publishDate | 1995 |
| publishDateSearch | 1995 |
| publishDateSort | 1995 |
| record_format | marc |
| series | Technische Universität <München>: TUM-I |
| series2 | Technische Universität <München>: TUM-I Sonderforschungsbereich Methoden und Werkzeuge für die Nutzung Paralleler Rechnerarchitekturen: SFB-Bericht / A |
| spelling | Hilgenfeldt, Sascha Verfasser aut Sparse grids applications to multi-dimensional Schrödinger problems Sascha Hilgenfeldt ; Robert Balder ; Christoph Zenger München 1995 20 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Technische Universität <München>: TUM-I 9507 Sonderforschungsbereich Methoden und Werkzeuge für die Nutzung Paralleler Rechnerarchitekturen: SFB-Bericht / A 1995,5 Abstract: "Sparse grid methods applied to solve partial differential equations allow for a substantial reduction of numerical effort (to obtain equal error magnitudes) compared to conventional finite element methods. A short introduction to this new approach is given. Using a Ritz-Galerkin method on rectangular sparse grids, stationary Schrödinger equations of dimensionality D [> or =] 2 are solved numerically for a number of generic problems and the results are compared to exact values, perturbative results, and numerical computations of other authors. For problems with oscillator potentials (harmonic or anharmonic), the accuracy of eigenvalues for similar numbers of grid points and equal order of basis functions is increased by up to two orders of magnitude with respect to conventional FEM. Good solutions are obtained for singular potentials (hydrogen atom and hydrogen molecular ion), where the sparse grid was automatically refined using a local adaptation strategy. Schrödinger problems of high dimensionality (up to D=8) become tractable with this algorithm, regardless of symmetries or separabilities of the potential functions, i.e. similar accuracies are to be expected for arbitrary potentials. As an example of a physically significant and intrinsically high-dimensional problem, eigenstates of a spin boson coupling model were computed." Multigrid methods (Numerical analysis) Schrödinger equation Balder, Robert Verfasser aut Zenger, Christoph Verfasser aut A Sonderforschungsbereich Methoden und Werkzeuge für die Nutzung Paralleler Rechnerarchitekturen: SFB-Bericht 1995,5 (DE-604)BV004627888 1995,5 Technische Universität <München>: TUM-I 9507 (DE-604)BV006185376 9507 |
| spellingShingle | Hilgenfeldt, Sascha Balder, Robert Zenger, Christoph Sparse grids applications to multi-dimensional Schrödinger problems Technische Universität <München>: TUM-I Multigrid methods (Numerical analysis) Schrödinger equation |
| title | Sparse grids applications to multi-dimensional Schrödinger problems |
| title_auth | Sparse grids applications to multi-dimensional Schrödinger problems |
| title_exact_search | Sparse grids applications to multi-dimensional Schrödinger problems |
| title_full | Sparse grids applications to multi-dimensional Schrödinger problems Sascha Hilgenfeldt ; Robert Balder ; Christoph Zenger |
| title_fullStr | Sparse grids applications to multi-dimensional Schrödinger problems Sascha Hilgenfeldt ; Robert Balder ; Christoph Zenger |
| title_full_unstemmed | Sparse grids applications to multi-dimensional Schrödinger problems Sascha Hilgenfeldt ; Robert Balder ; Christoph Zenger |
| title_short | Sparse grids |
| title_sort | sparse grids applications to multi dimensional schrodinger problems |
| title_sub | applications to multi-dimensional Schrödinger problems |
| topic | Multigrid methods (Numerical analysis) Schrödinger equation |
| topic_facet | Multigrid methods (Numerical analysis) Schrödinger equation |
| volume_link | (DE-604)BV004627888 (DE-604)BV006185376 |
| work_keys_str_mv | AT hilgenfeldtsascha sparsegridsapplicationstomultidimensionalschrodingerproblems AT balderrobert sparsegridsapplicationstomultidimensionalschrodingerproblems AT zengerchristoph sparsegridsapplicationstomultidimensionalschrodingerproblems |