scholarly journals Hierarchical Gradient-Based Optimization with B-Splines on Sparse Grids

2016 ◽  
pp. 315-336 ◽  
Author(s):  
Julian Valentin ◽  
Dirk Pflüger
Keyword(s):  
Sparse Grids ◽  
B Splines ◽  
Gradient Based

Gradient-Based Two-Scale Topology Optimization With B-Splines on Sparse Grids

2020 ◽  
Vol 42 (4) ◽  
pp. B1092-B1114
Author(s):  
Julian Valentin ◽  
Daniel Hübner ◽  
Michael Stingl ◽  
Dirk Pflüger
Keyword(s):  
Topology Optimization ◽  
Sparse Grids ◽  
B Splines ◽  
Gradient Based

scholarly journals Solving High-Dimensional Dynamic Portfolio Choice Models with Hierarchical B-Splines on Sparse Grids

2021 ◽  
Author(s):  
Peter Schober ◽  
Julian Valentin ◽  
Dirk Pflüger
Keyword(s):  
Portfolio Choice ◽  
Value Function ◽  
Choice Models ◽  
Sparse Grids ◽  
Continuous Control ◽  
B Splines ◽  
Gradient Based ◽  
Dynamic Portfolio ◽  
Dynamic Portfolio Choice ◽  
The Value Function

AbstractDiscrete time dynamic programming to solve dynamic portfolio choice models has three immanent issues: firstly, the curse of dimensionality prohibits more than a handful of continuous states. Secondly, in higher dimensions, even regular sparse grid discretizations need too many grid points for sufficiently accurate approximations of the value function. Thirdly, the models usually require continuous control variables, and hence gradient-based optimization with smooth approximations of the value function is necessary to obtain accurate solutions to the optimization problem. For the first time, we enable accurate and fast numerical solutions with gradient-based optimization while still allowing for spatial adaptivity using hierarchical B-splines on sparse grids. When compared to the standard linear bases on sparse grids or finite difference approximations of the gradient, our approach saves an order of magnitude in total computational complexity for a representative dynamic portfolio choice model with varying state space dimensionality, stochastic sample space, and choice variables.

B-splines on sparse grids for surrogates in uncertainty quantification

2021 ◽  
Vol 209 ◽  
pp. 107430
Author(s):  
Michael F. Rehme ◽  
Fabian Franzelin ◽  
Dirk Pflüger
Keyword(s):  
Uncertainty Quantification ◽  
Sparse Grids ◽  
B Splines

Automated Optimisation of T-Root Rotor Grooves With B-Splines and Finite Element Analysis

2015 ◽  
Author(s):  
Jonathan Henson ◽  
Richard Dolan ◽  
Gareth Thomas ◽  
Christos Georgakis
Keyword(s):  
Finite Element ◽  
Steam Turbine ◽  
Peak Stress ◽  
Turbine Rotor ◽  
Element Analysis ◽  
B Splines ◽  
Steam Turbine Rotor ◽  
Spline Curves ◽  
Python Scripting ◽  
Gradient Based

An Alstom tool is described for the automated and simultaneous design optimisation of 2 and 4-hook T-root grooving of multiple steam turbine rotor stages in order to minimise the peak stress. The finite element axisymmetric thermal-stress calculation is performed with Abaqus in a few hours on modest hardware. The tool embeds Python scripting to facilitate the rotor groove model definition and meshing within Abaqus/CAE, with emphasis placed on minimising the effort for the initial setup. Rotor groove shapes are described with B-splines, maintained and modified within the in-house tool. Their shape is progressively refined as directed by a hybrid evolutionary-gradient based optimisation engine in order to achieve the minimum stress objective. In the region of highest stress, the groove boundary shape adjusts as the optimisation proceeds to conform to the local contours of stress. Application to a low pressure steam turbine rotor demonstrates comparable or lower stresses with this tool compared to those from manual expert optimisation. The method can be readily extended to other geometric entities on the rotor described with B-spline curves, e.g. cavities, seals.

scholarly journals Uncertainty quantification for the Hokkaido Nansei-Oki tsunami using B-splines on adaptive sparse grids

2021 ◽  
Vol 62 ◽  
pp. C30-C44
Author(s):  
Michael Rehme ◽  
Stephen Roberts ◽  
Dirk Pflüger
Keyword(s):  
Uncertainty Quantification ◽  
Surrogate Model ◽  
Polynomial Chaos ◽  
Spline Interpolation ◽  
Sparse Grids ◽  
Stochastic Collocation ◽  
B Splines ◽  
Spatially Adaptive ◽  
Modeling Uncertainties ◽  
Run Up

Modeling uncertainties in the input parameters of computer simulations is an established way to account for inevitably limited knowledge. To overcome long run-times and high demand for computational resources, a surrogate model can replace the original simulation. We use spatially adaptive sparse grids for the creation of this surrogate model. Sparse grids are a discretization scheme designed to mitigate the curse of dimensionality, and spatial adaptivity further decreases the necessary number of expensive simulations. We combine this with B-spline basis functions which provide gradients and are exactly integrable. We demonstrate the capability of this uncertainty quantification approach for a simulation of the Hokkaido Nansei–Oki Tsunami with anuga. We develop a better understanding of the tsunami behavior by calculating key quantities such as mean, percentiles and maximum run-up. We compare our approach to the popular Dakota toolbox and reach slightly better results for all quantities of interest.  References B. M. Adams, M. S. Ebeida, et al. Dakota. Sandia Technical Report, SAND2014-4633, Version 6.11 User’s Manual, July 2014. 2019. https://dakota.sandia.gov/content/manuals. J. H. S. de Baar and S. G. Roberts. Multifidelity sparse-grid-based uncertainty quantification for the Hokkaido Nansei–Oki tsunami. Pure Appl. Geophys. 174 (2017), pp. 3107–3121. doi: 10.1007/s00024-017-1606-y. H.-J. Bungartz and M. Griebel. Sparse grids. Acta Numer. 13 (2004), pp. 147–269. doi: 10.1017/S0962492904000182. M. Eldred and J. Burkardt. Comparison of non-intrusive polynomial chaos and stochastic collocation methods for uncertainty quantification. 47th AIAA. 2009. doi: 10.2514/6.2009-976. K. Höllig and J. Hörner. Approximation and modeling with B-splines. Philadelphia: SIAM, 2013. doi: 10.1137/1.9781611972955. M. Matsuyama and H. Tanaka. An experimental study of the highest run-up height in the 1993 Hokkaido Nansei–Oki earthquake tsunami. National Tsunami Hazard Mitigation Program Review and International Tsunami Symposium (ITS). 2001. O. Nielsen, S. Roberts, D. Gray, A. McPherson, and A. Hitchman. Hydrodymamic modelling of coastal inundation. MODSIM 2005. 2005, pp. 518–523. https://www.mssanz.org.au/modsim05/papers/nielsen.pdf. J. Nocedal and S. J. Wright. Numerical optimization. Springer, 2006. doi: 10.1007/978-0-387-40065-5. D. Pflüger. Spatially Adaptive Sparse Grids for High-Dimensional Problems. Dr. rer. nat., Technische Universität München, Aug. 2010. https://www5.in.tum.de/pub/pflueger10spatially.pdf. M. F. Rehme, F. Franzelin, and D. Pflüger. B-splines on sparse grids for surrogates in uncertainty quantification. Reliab. Eng. Sys. Saf. 209 (2021), p. 107430. doi: 10.1016/j.ress.2021.107430. M. F. Rehme and D. Pflüger. Stochastic collocation with hierarchical extended B-splines on Sparse Grids. Approximation Theory XVI, AT 2019. Springer Proc. Math. Stats. Vol. 336. Springer, 2020. doi: 10.1007/978-3-030-57464-2_12. S Roberts, O. Nielsen, D. Gray, J. Sexton, and G. Davies. ANUGA. Geoscience Australia. 2015. doi: 10.13140/RG.2.2.12401.99686. I. J. Schoenberg and A. Whitney. On Pólya frequence functions. III. The positivity of translation determinants with an application to the interpolation problem by spline curves. Trans. Am. Math. Soc. 74.2 (1953), pp. 246–259. doi: 10.2307/1990881. W. Sickel and T. Ullrich. Spline interpolation on sparse grids. Appl. Anal. 90.3–4 (2011), pp. 337–383. doi: 10.1080/00036811.2010.495336. C. E. Synolakis, E. N. Bernard, V. V. Titov, U. Kânoğlu, and F. I. González. Standards, criteria, and procedures for NOAA evaluation of tsunami numerical models. NOAA/Pacific Marine Environmental Laboratory. 2007. https://nctr.pmel.noaa.gov/benchmark/. J. Valentin and D. Pflüger. Hierarchical gradient-based optimization with B-splines on sparse grids. Sparse Grids and Applications—Stuttgart 2014. Lecture Notes in Computational Science and Engineering. Vol. 109. Springer, 2016, pp. 315–336. doi: 10.1007/978-3-319-28262-6_13. D. Xiu and G. E. Karniadakis. The Wiener–Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24.2 (2002), pp. 619–644. doi: 10.1137/S1064827501387826.

Free-Form Optimal Conjugation Design

2012 ◽  
Author(s):  
Bowen Yu ◽  
Kwun-lon Ting
Keyword(s):  
Optimization Methods ◽  
Free Form ◽  
Control Points ◽  
Contact Ratio ◽  
B Splines ◽  
Geometry Design ◽  
Gear Efficiency ◽  
Gradient Based ◽  
Optimization Of Geometry ◽  
Geometric Relationship

The paper presents the first optimal conjugation design methodology based on free-form conjugation modeling theory. The methodology is implemented for planar circular gearing and general for any planar gearing. According to the previous research of free-form conjugation, conjugate profiles are modeled by contact path geometries or cutter geometries which are represented by NURBS (non-uniform rational B-splines). The interchangeability between control points and interpolation points of NURBS are introduced in general to offer reasonable constraints of special interpolation conditions in conjugation design. To adapt to the flexibility brought by free-form techniques, the determination of an important conjugate property—contact ratio is carried out through geometric relationship. To make use of NURBS for optimal design, conjugate properties and their differentiations are represented by important parameters and then by control points and interpolation points of NURBS. The interested properties are relative curvature, specific sliding ratio and nominal Hertz contact stress, which are the main factors of gear efficiency and wear. The properties are well-known for their difficulties in optimization. The paper shows that with appropriate manipulations in mathematics and programming, it is feasible for gradient based optimization methods, which are accurate and fast in convergence. The methodology is consistent with the regular optimization of geometry design, and can be integrated into geometry design systems. The examples show the effectiveness of the proposed optimization framework.

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