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Towards an adaptive solver for high-dimensional PDE problems on clusters of multicore processors
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Institutionen för informationsteknologi, Avdelningen för beräkningsvetenskap. Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Institutionen för informationsteknologi, Tillämpad beräkningsvetenskap.
2012 (engelsk)Licentiatavhandling, med artikler (Annet vitenskapelig)
Abstract [en]

Accurate numerical simulation of time-dependent phenomena in many spatial dimensions is a challenging computational task apparent in a vast range of application areas, for instance quantum dynamics, financial mathematics, systems biology and plasma physics. Particularly problematic is that the number of unknowns in the governing equations (the number of grid points) grows exponentially with the number of spatial dimensions introduced, often referred to as the curse of dimensionality. This limits the range of problems that we can solve, since the computational effort and requirements on memory storage directly depend on the number of unknowns for which to solve the equations.

In order to push the limit of tractable problems, we are developing an implementation framework, HAParaNDA, for high-dimensional PDE-problems. By using high-order accurate schemes and adaptive mesh refinement (AMR) in space, we aim at reducing the number of grid points used in the discretization, thereby enabling the solution of larger and higher-dimensional problems. Within the framework, we use structured grids for spatial discretization and a block-decomposition of the spatial domain for parallelization and load balancing. For integration in time, we use exponential integration, although the framework allows the flexibility of other integrators to be implemented as well. Exponential integrators using the Lanzcos or the Arnoldi algorithm has proven a succesful and efficient approach for large problems. Using a truncation of the Magnus expansion, we can attain high levels of accuracy in the solution.

As an example application, we have implemented a solver for the time-dependent Schrödinger equation using this framework. We provide scaling results for small and medium sized clusters of multicore nodes, and show that the solver fulfills the expected rate of convergence.

sted, utgiver, år, opplag, sider
Uppsala University, 2012.
Serie
IT licentiate theses / Uppsala University, Department of Information Technology, ISSN 1404-5117 ; 2012-003
HSV kategori
Forskningsprogram
Beräkningsvetenskap
Identifikatorer
URN: urn:nbn:se:uu:diva-169259OAI: oai:DiVA.org:uu-169259DiVA, id: diva2:505808
Veileder
Prosjekter
eSSENCEUPMARCTilgjengelig fra: 2012-03-09 Laget: 2012-02-25 Sist oppdatert: 2019-02-25bibliografisk kontrollert
Delarbeid
1. An implementation framework for solving high-dimensional PDEs on massively parallel computers
Åpne denne publikasjonen i ny fane eller vindu >>An implementation framework for solving high-dimensional PDEs on massively parallel computers
2010 (engelsk)Inngår i: Numerical Mathematics and Advanced Applications: 2009, Berlin: Springer-Verlag , 2010, s. 417-424Konferansepaper, Publicerat paper (Fagfellevurdert)
sted, utgiver, år, opplag, sider
Berlin: Springer-Verlag, 2010
HSV kategori
Identifikatorer
urn:nbn:se:uu:diva-132927 (URN)10.1007/978-3-642-11795-4_44 (DOI)000395207900044 ()978-3-642-11794-7 (ISBN)
Prosjekter
eSSENCEUPMARC
Tilgjengelig fra: 2010-10-29 Laget: 2010-10-29 Sist oppdatert: 2018-06-16bibliografisk kontrollert
2. Communication-efficient algorithms for numerical quantum dynamics
Åpne denne publikasjonen i ny fane eller vindu >>Communication-efficient algorithms for numerical quantum dynamics
2012 (engelsk)Inngår i: Applied Parallel and Scientific Computing: Part II, Berlin: Springer-Verlag , 2012, s. 368-378Konferansepaper, Publicerat paper (Fagfellevurdert)
sted, utgiver, år, opplag, sider
Berlin: Springer-Verlag, 2012
Serie
Lecture Notes in Computer Science ; 7134
HSV kategori
Identifikatorer
urn:nbn:se:uu:diva-135980 (URN)10.1007/978-3-642-28145-7_36 (DOI)000309716000036 ()978-3-642-28144-0 (ISBN)
Konferanse
PARA 2010: State of the Art in Scientific and Parallel Computing
Prosjekter
eSSENCEUPMARC
Tilgjengelig fra: 2012-02-16 Laget: 2010-12-09 Sist oppdatert: 2018-01-12bibliografisk kontrollert
3. Stable difference methods for block-structured adaptive grids
Åpne denne publikasjonen i ny fane eller vindu >>Stable difference methods for block-structured adaptive grids
2011 (engelsk)Rapport (Annet vitenskapelig)
Abstract [en]

The time-dependent Schrödinger equation describes quantum dynamical phenomena. Solving it numerically, the small-scale interactions that are modeled require very fine spatial resolution. At the same time, the solutions are localized and confined to small regions in space. Using the required resolution over the entire high-dimensional domain often makes the model problems intractable due to the prohibitively large grids that result from such a discretization. In this paper, we present a block-structured adaptive mesh refinement scheme, aiming at efficient adaptive discretization of high-dimensional partial differential equations such as the time-dependent Schrödinger equation. Our framework allows for anisotropic grid refinement in order to avoid unnecessary refinement. For spatial discretization, we use standard finite difference stencils together with summation-by-parts operators and simultaneous-approximation-term interface treatment. We propagate in time using exponential integration with the Lanczos method. Our theoretical and numerical results show that our adaptive scheme is stable for long time integrations. We also show that the discretizations meet the expected convergence rates.

Serie
Technical report / Department of Information Technology, Uppsala University, ISSN 1404-3203 ; 2011-022
HSV kategori
Identifikatorer
urn:nbn:se:uu:diva-159854 (URN)
Prosjekter
eSSENCE
Tilgjengelig fra: 2011-10-11 Laget: 2011-10-11 Sist oppdatert: 2024-05-30bibliografisk kontrollert
4. Numerical evaluation of the Communication-Avoiding Lanczos algorithm
Åpne denne publikasjonen i ny fane eller vindu >>Numerical evaluation of the Communication-Avoiding Lanczos algorithm
2012 (engelsk)Rapport (Annet vitenskapelig)
Abstract [en]

The Lanczos algorithm is widely used for solving large sparse symmetric eigenvalue problems when only a few eigenvalues from the spectrum are needed. Due to sparse matrix-vector multiplications and frequent synchronization, the algorithm is communication intensive leading to poor performance on parallel computers and modern cache-based processors. The Communication-Avoiding Lanczos algorithm [Hoemmen; 2010] attempts to improve performance by taking the equivalence of s steps of the original algorithm at a time. The scheme is equivalent to the original algorithm in exact arithmetic but as the value of s grows larger, numerical roundoff errors are expected to have a greater impact. In this paper, we investigate the numerical properties of the Communication-Avoiding Lanczos (CA-Lanczos) algorithm and how well it works in practical computations. Apart from the algorithm itself, we have implemented techniques that are commonly used with the Lanczos algorithm to improve its numerical performance, such as semi-orthogonal schemes and restarting. We present results that show that CA-Lanczos is often as accurate as the original algorithm. In many cases, if the parameters of the s-step basis are chosen appropriately, the numerical behaviour of CA-Lanczos is close to the standard algorithm even though it is somewhat more sensitive to loosing mutual orthogonality among the basis vectors.

Serie
Technical report / Department of Information Technology, Uppsala University, ISSN 1404-3203 ; 2012-001
HSV kategori
Identifikatorer
urn:nbn:se:uu:diva-169257 (URN)
Prosjekter
eSSENCE
Tilgjengelig fra: 2012-01-22 Laget: 2012-02-25 Sist oppdatert: 2024-05-30bibliografisk kontrollert

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