08:30
Computing and Model Order Reduction (Part 1)
Chair: Matthias Möller
08:30
25 mins
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Optimal Control on a Model for Cervical Cancer
Tri Sri Noor Asih, Widodo Widodo, Dwi Rizkiana Dewi
Abstract: The optimum control can be applied into biology problems. In cervical cancer mathematical model, the optimum control representating treatment that applied to the cervical cancer patient. If we devided the cervical cells into sub population of normal cells, infected cells, pre-cancer cells and cancer cells, the control function can be applied into sub population of pre-cancer cells. This is according to the medical research that pre-cancer stage can be regress into infected stage by effective treatment. So we include the control function in pre-cancer stage, find the optimum control by Pontryagin’s Maximum Principle, and made simulation of it. Then we compare the solution of the system between system with optimum control and system without control.
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08:55
25 mins
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Reproducible Interval Matrix Multiplication
Katsuhisa Ozaki
Abstract: This talk is concerned with interval arithmetic, with a particular focus on interval matrix multiplication. There are several interval methods for matrix multiplication; for example, see [1,2,3]. Methods in [1] and [2] involve four and two matrix multiplications, respectively. The main advantage of the methods offered in [1,2] is that they exploit high-performance routines in Basic Linear Algebra Subprograms.
Computational environments can vary. Differences are found in the number of cores in central processing unit, the numerical libraries, support of fused multiply-add (a floating-point multiply–add operation performed in one step), and so forth. We may obtain different computed results using each computational environment. The main reason for these differing results is a rounding error of floating-point arithmetic. Reproducibility is the ability to obtain bitwise identical numerical results from run to run on the same input data on the different architectures. Our goal is to obtain the midpoint-radius interval matrices for interval matrix multiplication that are identical bitwise in different computational environments. We extend and apply error-free transformation of matrix multiplication [4] to interval matrix multiplication. We assume that a divide-and-conquer method is not applied to matrix multiplication. Then, the reproducible result can be obtained.
Moreover, we discuss acceleration of the performance using low-precision arithmetic as in [3]. Finally, we show numerical examples to illustrate the efficiency of the proposed methods.
References
[1] S.M. Rump, Fast and parallel interval arithmetic. BIT Numerical Mathematics, 39:3 (1999), 539–560.
[2] T. Ogita, S. Oishi, Fast Inclusion of Interval Matrix Multiplication, Reliable Computing, 11:3 (2005), 191–205.
[3] K. Ozaki, T. Ogita, F. Bunger, S. Oishi: Accelerating interval matrix multiplication by mixed precision arithmetic, Nonlinear Theory and its Applications, IEICE, 6:3 (2015), 364-376.
[4] K. Ozaki, T. Ogita, S. Oishi, S. M. Rump: Error-Free Transformation of Matrix Multiplication by Using Fast Routines of Matrix Multiplication and its Applications, Numerical Algorithms, 59:1 (2012), 95-118.
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09:20
25 mins
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Parallel algebraic linear solvers for high-order discontinuous Galerkin methods for compressible turbulent flows
Emeric Martin, Florent Renac
Abstract: Abstract: Discontinuous Galerkin (DG) methods are high-order discretizations based on piecewise polynomial approximations that are of main interest for the solution of nonlinear convection dominated flow problems. In the present work, we propose efficient and robust strategies for an inexact Newton-Krylov implicit time integration in the context of steady-state solutions of transonic turbulent flows at high Reynolds number. Matrices arising from DG methods are real, non-symmetric, sparse with a block structure and a symmetric pattern, often large, and very often ill-conditioned. We investigate parallel GMRES and flexible-GMRES iterative algorithms, with possible deflated restarting approaches [1,2], combined with local block ILU(k) preconditioners [3]. Numerical experiments with the Aghora code [4] on stiff problems with large CFL numbers will be presented to assess the overall performance of these strategies in terms of convergence rate and of CPU and memory costs related to the space discretization order of the DG method.
References: [1] R. B. Morgan, GMRES with Deflated Restarting, SIAM J. Sci. Comput., 24(1), pp. 20-37, 2002. [2] L. Giraud, S. Gratton, X. Pinel, and X. Vasseur, Flexible GMRES with deflated restarting, SIAM J. Sci. Comput., 32(4), pp. 1858-1878, 2010. [3] A. Chapman, Y. Saad, and L. Wigton, High-order ILU preconditioners for CFD problems, Int. J. Numer. Meth. Fluids, 33(6), pp. 767-788, 2000. [4] F. Renac, M. de la Llave Plata, E. Martin, J.-B. Chapelier and V. Couaillier, Aghora: A high-order DG solver for turbulent flow simulations, in: IDIHOM: Industrialization of High-Order Methods - A Top-Down Approach, Notes on Numerical Fluid Mechanics and Multidisciplinary Design, 128, Springer, 2015.
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09:45
25 mins
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3D Lattice-Boltzmann Simulations Using GPGPU and an Adaptive Grid
Radek Fučík, Tomáš Oberhuber, Pavel Eichler, Jakub Klinkovský, Aleš Wodecki
Abstract: This talk is concerned with the development of an adaptive grid structure ap- plied to simulations using the Lattice-Boltzmann method. The implementation is based on a problem specific multilevel octree structure that is pregenerated based on a preview simulation. The applications that we are targeting include but are not limited to three-dimensional turbulent flow of fluid around obstacles, the motion of curves and the phase field simulation of crystal growth (of pure substances and alloys). Some of these simulations have been validated against experimental data provided by collaborating institutions: IKEM Praha, Czech republic and CESEP, Colorado School of Mines, Golden, USA. A range of sim- ulation results will be presented and an outlook for future development will be given.
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