Order-Theoretic Combination Techniques and the Electronic Schrödinger Equation
Order-Theoretic Combination Techniques and the Electronic Schrödinger Equation
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DissertationDate of Exam
02.02.2024Date of Publication
22.05.2024Advisor
Griebel, MichaelCo-Referee
Garcke, JochenMetadata
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Abstract
Most standard constructions of the combination technique [M. Griebel et al., Iterative Methods in Linear Algebra, Elsevier, North Holland, p. 263] manipulate families of functions organised by downward-closed subsets of Nd. We introduce instead an alternative formulation, with functions indexed from a more general kind of partially ordered set (poset). The combinatorial and order-theoretic machinery of Möbius inversion helps us to construct combination sums of functions organised by order ideals of a poset grid. An adaptive algorithm is given for the quasi-optimal assembly of such an order ideal. This order-theoretic combination technique (OTCT) formalism is applied in the quantum-chemical setting of the high-dimensional electronic Schrödinger equation. Here, the OTCT allows us to connect, understand, and improve on a number of existing approaches.
We consider first a selection of existing extrapolative composite methods. Extending on the idea of the CQML approach [P. Zaspel et al., J. Chem. Theory Comput., 15(3), 2018], an application of basically just the standard version of the combination technique leads to a generalised composite method (GCM). This approach is systematically improvable and appears comparable, if not yet truly competitive with standard composite methods from the perspectives of both accuracy and of cost.
We turn then to energy-based fragmentation methods, which are often founded upon a truncated many-body expansion (MBE). It is well-known that Möbius inversion can provide non-recursive expressions for the individual MBE terms, and so the OTCT delivers by construction a framework for the adaptive truncation of MBE-like formulae. The same also functions for a class of related graph-based decompositions described in the existing literature. We term these in our context as SUPANOVA (SUbgraph Poset ANOVA) decompositions, and motivate them as extensions to the BOSSANOVA decomposition [M. Griebel et al., Extraction of Quantifiable Information from Complex Systems, Springer, Cham, 2014, p. 211; F. Heber, Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn, 2014]. We identify a subtle technical issue that afflicts BOSSANOVA in certain cases, and apply instead an adaptive SUPANOVA decomposition defined for convex subgraphs.
Finally, we combine the GCM and SUPANOVA ideas to obtain a poset grid that recovers many existing multilevel fragmentation methods. We extend the ML-BOSSANOVA method [S. R. Chinnamsetty et al., Multiscale Model. Simul., 16(2), 2018], exploring now also a hierarchy of ab initio theories. Although an initial assessment is inconclusive, this ML-SUPANOVA formulation appears well-founded and paves the way to a number of interesting possible applications in the future.
We consider first a selection of existing extrapolative composite methods. Extending on the idea of the CQML approach [P. Zaspel et al., J. Chem. Theory Comput., 15(3), 2018], an application of basically just the standard version of the combination technique leads to a generalised composite method (GCM). This approach is systematically improvable and appears comparable, if not yet truly competitive with standard composite methods from the perspectives of both accuracy and of cost.
We turn then to energy-based fragmentation methods, which are often founded upon a truncated many-body expansion (MBE). It is well-known that Möbius inversion can provide non-recursive expressions for the individual MBE terms, and so the OTCT delivers by construction a framework for the adaptive truncation of MBE-like formulae. The same also functions for a class of related graph-based decompositions described in the existing literature. We term these in our context as SUPANOVA (SUbgraph Poset ANOVA) decompositions, and motivate them as extensions to the BOSSANOVA decomposition [M. Griebel et al., Extraction of Quantifiable Information from Complex Systems, Springer, Cham, 2014, p. 211; F. Heber, Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn, 2014]. We identify a subtle technical issue that afflicts BOSSANOVA in certain cases, and apply instead an adaptive SUPANOVA decomposition defined for convex subgraphs.
Finally, we combine the GCM and SUPANOVA ideas to obtain a poset grid that recovers many existing multilevel fragmentation methods. We extend the ML-BOSSANOVA method [S. R. Chinnamsetty et al., Multiscale Model. Simul., 16(2), 2018], exploring now also a hierarchy of ab initio theories. Although an initial assessment is inconclusive, this ML-SUPANOVA formulation appears well-founded and paves the way to a number of interesting possible applications in the future.
Subjects
adaptive algorithms, ANOVA-like decompositions, combination technique, composite methods, convex subgraphs, electronic Schrödinger equation, fragmentation methods, graph-based decompositions, many-body expansion, Möbius inversion, multilevel decompositions
Classification (DDC)
510 MathematikBarker, James Nicholas John: Order-Theoretic Combination Techniques and the Electronic Schrödinger Equation. - Bonn, 2024. - Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn.
Online-Ausgabe in bonndoc: https://nbn-resolving.org/urn:nbn:de:hbz:5-76117
Online-Ausgabe in bonndoc: https://nbn-resolving.org/urn:nbn:de:hbz:5-76117
@phdthesis{handle:20.500.11811/11554,
urn: https://nbn-resolving.org/urn:nbn:de:hbz:5-76117,
doi: https://doi.org/10.48565/bonndoc-293,
author = {{James Nicholas John Barker}},
title = {Order-Theoretic Combination Techniques and the Electronic Schrödinger Equation},
school = {Rheinische Friedrich-Wilhelms-Universität Bonn},
year = 2024,
month = may,
note = {Most standard constructions of the combination technique [M. Griebel et al., Iterative Methods in Linear Algebra, Elsevier, North Holland, p. 263] manipulate families of functions organised by downward-closed subsets of Nd. We introduce instead an alternative formulation, with functions indexed from a more general kind of partially ordered set (poset). The combinatorial and order-theoretic machinery of Möbius inversion helps us to construct combination sums of functions organised by order ideals of a poset grid. An adaptive algorithm is given for the quasi-optimal assembly of such an order ideal. This order-theoretic combination technique (OTCT) formalism is applied in the quantum-chemical setting of the high-dimensional electronic Schrödinger equation. Here, the OTCT allows us to connect, understand, and improve on a number of existing approaches.
We consider first a selection of existing extrapolative composite methods. Extending on the idea of the CQML approach [P. Zaspel et al., J. Chem. Theory Comput., 15(3), 2018], an application of basically just the standard version of the combination technique leads to a generalised composite method (GCM). This approach is systematically improvable and appears comparable, if not yet truly competitive with standard composite methods from the perspectives of both accuracy and of cost.
We turn then to energy-based fragmentation methods, which are often founded upon a truncated many-body expansion (MBE). It is well-known that Möbius inversion can provide non-recursive expressions for the individual MBE terms, and so the OTCT delivers by construction a framework for the adaptive truncation of MBE-like formulae. The same also functions for a class of related graph-based decompositions described in the existing literature. We term these in our context as SUPANOVA (SUbgraph Poset ANOVA) decompositions, and motivate them as extensions to the BOSSANOVA decomposition [M. Griebel et al., Extraction of Quantifiable Information from Complex Systems, Springer, Cham, 2014, p. 211; F. Heber, Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn, 2014]. We identify a subtle technical issue that afflicts BOSSANOVA in certain cases, and apply instead an adaptive SUPANOVA decomposition defined for convex subgraphs.
Finally, we combine the GCM and SUPANOVA ideas to obtain a poset grid that recovers many existing multilevel fragmentation methods. We extend the ML-BOSSANOVA method [S. R. Chinnamsetty et al., Multiscale Model. Simul., 16(2), 2018], exploring now also a hierarchy of ab initio theories. Although an initial assessment is inconclusive, this ML-SUPANOVA formulation appears well-founded and paves the way to a number of interesting possible applications in the future.},
url = {https://hdl.handle.net/20.500.11811/11554}
}
urn: https://nbn-resolving.org/urn:nbn:de:hbz:5-76117,
doi: https://doi.org/10.48565/bonndoc-293,
author = {{James Nicholas John Barker}},
title = {Order-Theoretic Combination Techniques and the Electronic Schrödinger Equation},
school = {Rheinische Friedrich-Wilhelms-Universität Bonn},
year = 2024,
month = may,
note = {Most standard constructions of the combination technique [M. Griebel et al., Iterative Methods in Linear Algebra, Elsevier, North Holland, p. 263] manipulate families of functions organised by downward-closed subsets of Nd. We introduce instead an alternative formulation, with functions indexed from a more general kind of partially ordered set (poset). The combinatorial and order-theoretic machinery of Möbius inversion helps us to construct combination sums of functions organised by order ideals of a poset grid. An adaptive algorithm is given for the quasi-optimal assembly of such an order ideal. This order-theoretic combination technique (OTCT) formalism is applied in the quantum-chemical setting of the high-dimensional electronic Schrödinger equation. Here, the OTCT allows us to connect, understand, and improve on a number of existing approaches.
We consider first a selection of existing extrapolative composite methods. Extending on the idea of the CQML approach [P. Zaspel et al., J. Chem. Theory Comput., 15(3), 2018], an application of basically just the standard version of the combination technique leads to a generalised composite method (GCM). This approach is systematically improvable and appears comparable, if not yet truly competitive with standard composite methods from the perspectives of both accuracy and of cost.
We turn then to energy-based fragmentation methods, which are often founded upon a truncated many-body expansion (MBE). It is well-known that Möbius inversion can provide non-recursive expressions for the individual MBE terms, and so the OTCT delivers by construction a framework for the adaptive truncation of MBE-like formulae. The same also functions for a class of related graph-based decompositions described in the existing literature. We term these in our context as SUPANOVA (SUbgraph Poset ANOVA) decompositions, and motivate them as extensions to the BOSSANOVA decomposition [M. Griebel et al., Extraction of Quantifiable Information from Complex Systems, Springer, Cham, 2014, p. 211; F. Heber, Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn, 2014]. We identify a subtle technical issue that afflicts BOSSANOVA in certain cases, and apply instead an adaptive SUPANOVA decomposition defined for convex subgraphs.
Finally, we combine the GCM and SUPANOVA ideas to obtain a poset grid that recovers many existing multilevel fragmentation methods. We extend the ML-BOSSANOVA method [S. R. Chinnamsetty et al., Multiscale Model. Simul., 16(2), 2018], exploring now also a hierarchy of ab initio theories. Although an initial assessment is inconclusive, this ML-SUPANOVA formulation appears well-founded and paves the way to a number of interesting possible applications in the future.},
url = {https://hdl.handle.net/20.500.11811/11554}
}
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