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#### Open Access in DiVA

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Grandin, Magnus
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Grandin, Magnus
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Division of Scientific ComputingComputational Science
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Adaptive Solvers for High-Dimensional PDE Problems on Clusters of Multicore ProcessorsPrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
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2014 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Abstract [en]

##### Place, publisher, year, edition, pages

Uppsala: Acta Universitatis Upsaliensis, 2014. , p. 34
##### Series

Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology, ISSN 1651-6214 ; 1199
##### Keywords [en]

adaptive mesh refinement, anisotropic refinement, exponential integrators, Lanczos' algorithm, hybrid parallelization, time-dependent Schrödinger equation
##### National Category

Computational Mathematics
##### Research subject

Scientific Computing
##### Identifiers

URN: urn:nbn:se:uu:diva-234984ISBN: 978-91-554-9095-9 (print)OAI: oai:DiVA.org:uu-234984DiVA, id: diva2:758606
##### Public defence

2014-12-12, Room 2446, Polacksbacken, Lägerhyddsvägen 2, Uppsala, 10:15 (English)
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##### Projects

eSSENCEUPMARCAvailable from: 2014-11-21 Created: 2014-10-27 Last updated: 2019-02-25Bibliographically approved
##### List of papers

Accurate numerical solution of time-dependent, high-dimensional partial differential equations (PDEs) usually requires efficient numerical techniques and massive-scale parallel computing. In this thesis, we implement and evaluate discretization schemes suited for PDEs of higher dimensionality, focusing on high order of accuracy and low computational cost.

Spatial discretization is particularly challenging in higher dimensions. The memory requirements for uniform grids quickly grow out of reach even on large-scale parallel computers. We utilize high-order discretization schemes and implement adaptive mesh refinement on structured hyperrectangular domains in order to reduce the required number of grid points and computational work. We allow for anisotropic (non-uniform) refinement by recursive bisection and show how to construct, manage and load balance such grids efficiently. In our numerical examples, we use finite difference schemes to discretize the PDEs. In the adaptive case we show how a stable discretization can be constructed using SBP-SAT operators. However, our adaptive mesh framework is general and other methods of discretization are viable.

For integration in time, we implement exponential integrators based on the Lanczos/Arnoldi iterative schemes for eigenvalue approximations. Using adaptive time stepping and a truncated Magnus expansion, we attain high levels of accuracy in the solution at low computational cost. We further investigate alternative implementations of the Lanczos algorithm with reduced communication costs.

As an example application problem, we have considered the time-dependent Schrödinger equation (TDSE). We present solvers and results for the solution of the TDSE on equidistant as well as adaptively refined Cartesian grids.

1. An implementation framework for solving high-dimensional PDEs on massively parallel computers$(function(){PrimeFaces.cw("OverlayPanel","overlay359735",{id:"formSmash:j_idt516:0:j_idt520",widgetVar:"overlay359735",target:"formSmash:j_idt516:0:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

2. Communication-efficient algorithms for numerical quantum dynamics$(function(){PrimeFaces.cw("OverlayPanel","overlay375913",{id:"formSmash:j_idt516:1:j_idt520",widgetVar:"overlay375913",target:"formSmash:j_idt516:1:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

3. Numerical evaluation of the Communication-Avoiding Lanczos algorithm$(function(){PrimeFaces.cw("OverlayPanel","overlay505799",{id:"formSmash:j_idt516:2:j_idt520",widgetVar:"overlay505799",target:"formSmash:j_idt516:2:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

4. Stable difference methods for block-oriented adaptive grids$(function(){PrimeFaces.cw("OverlayPanel","overlay758564",{id:"formSmash:j_idt516:3:j_idt520",widgetVar:"overlay758564",target:"formSmash:j_idt516:3:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

5. Data structures and algorithms for high-dimensional structured adaptive mesh refinement$(function(){PrimeFaces.cw("OverlayPanel","overlay758570",{id:"formSmash:j_idt516:4:j_idt520",widgetVar:"overlay758570",target:"formSmash:j_idt516:4:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

6. Parallel data structures and algorithms for high-dimensional structured adaptive mesh refinement$(function(){PrimeFaces.cw("OverlayPanel","overlay758573",{id:"formSmash:j_idt516:5:j_idt520",widgetVar:"overlay758573",target:"formSmash:j_idt516:5:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

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