Optimization of mesh hierarchies in multilevel Monte Carlo samplers
We perform a general optimization of the parameters in the multilevel Monte Carlo (MLMC) discretization hierarchy based on uniform discretization methods with general approximation orders and computational costs. We optimize hierarchies with geometric and non-geometric sequences of mesh sizes and sh...
| Main Authors: | , , , |
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| 格式: | Journal article |
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Springer
2015
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| _version_ | 1826295498069245952 |
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| author | Haji-Ali, A Nobile, F von Schwerin, E Tempone, R |
| author_facet | Haji-Ali, A Nobile, F von Schwerin, E Tempone, R |
| author_sort | Haji-Ali, A |
| collection | OXFORD |
| description | We perform a general optimization of the parameters in the multilevel Monte Carlo (MLMC) discretization hierarchy based on uniform discretization methods with general approximation orders and computational costs. We optimize hierarchies with geometric and non-geometric sequences of mesh sizes and show that geometric hierarchies, when optimized, are nearly optimal and have the same asymptotic computational complexity as non-geometric optimal hierarchies. We discuss how enforcing constraints on parameters of MLMC hierarchies affects the optimality of these hierarchies. These constraints include an upper and a lower bound on the mesh size or enforcing that the number of samples and the number of discretization elements are integers. We also discuss the optimal tolerance splitting between the bias and the statistical error contributions and its asymptotic behavior. To provide numerical grounds for our theoretical results, we apply these optimized hierarchies together with the Continuation MLMC Algorithm (Collier et al., BIT Numer Math 55(2):399–432, 2015). The first example considers a three-dimensional elliptic partial differential equation with random inputs. Its space discretization is based on continuous piecewise trilinear finite elements and the corresponding linear system is solved by either a direct or an iterative solver. The second example considers a one-dimensional Itô stochastic differential equation discretized by a Milstein scheme. |
| first_indexed | 2024-03-07T04:01:57Z |
| format | Journal article |
| id | oxford-uuid:c4e37a05-01e2-4e2c-b58b-cef4bd1c1583 |
| institution | University of Oxford |
| last_indexed | 2024-03-07T04:01:57Z |
| publishDate | 2015 |
| publisher | Springer |
| record_format | dspace |
| spelling | oxford-uuid:c4e37a05-01e2-4e2c-b58b-cef4bd1c15832022-03-27T06:26:52ZOptimization of mesh hierarchies in multilevel Monte Carlo samplersJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:c4e37a05-01e2-4e2c-b58b-cef4bd1c1583Symplectic Elements at OxfordSpringer2015Haji-Ali, ANobile, Fvon Schwerin, ETempone, RWe perform a general optimization of the parameters in the multilevel Monte Carlo (MLMC) discretization hierarchy based on uniform discretization methods with general approximation orders and computational costs. We optimize hierarchies with geometric and non-geometric sequences of mesh sizes and show that geometric hierarchies, when optimized, are nearly optimal and have the same asymptotic computational complexity as non-geometric optimal hierarchies. We discuss how enforcing constraints on parameters of MLMC hierarchies affects the optimality of these hierarchies. These constraints include an upper and a lower bound on the mesh size or enforcing that the number of samples and the number of discretization elements are integers. We also discuss the optimal tolerance splitting between the bias and the statistical error contributions and its asymptotic behavior. To provide numerical grounds for our theoretical results, we apply these optimized hierarchies together with the Continuation MLMC Algorithm (Collier et al., BIT Numer Math 55(2):399–432, 2015). The first example considers a three-dimensional elliptic partial differential equation with random inputs. Its space discretization is based on continuous piecewise trilinear finite elements and the corresponding linear system is solved by either a direct or an iterative solver. The second example considers a one-dimensional Itô stochastic differential equation discretized by a Milstein scheme. |
| spellingShingle | Haji-Ali, A Nobile, F von Schwerin, E Tempone, R Optimization of mesh hierarchies in multilevel Monte Carlo samplers |
| title | Optimization of mesh hierarchies in multilevel Monte Carlo samplers |
| title_full | Optimization of mesh hierarchies in multilevel Monte Carlo samplers |
| title_fullStr | Optimization of mesh hierarchies in multilevel Monte Carlo samplers |
| title_full_unstemmed | Optimization of mesh hierarchies in multilevel Monte Carlo samplers |
| title_short | Optimization of mesh hierarchies in multilevel Monte Carlo samplers |
| title_sort | optimization of mesh hierarchies in multilevel monte carlo samplers |
| work_keys_str_mv | AT hajialia optimizationofmeshhierarchiesinmultilevelmontecarlosamplers AT nobilef optimizationofmeshhierarchiesinmultilevelmontecarlosamplers AT vonschwerine optimizationofmeshhierarchiesinmultilevelmontecarlosamplers AT temponer optimizationofmeshhierarchiesinmultilevelmontecarlosamplers |