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Control and Optimization with PDE Constraints pp 117–136Cite as

Birkhäuser

Gradient Computation for Model Calibration with Pointwise Observations

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Part of the book series: International Series of Numerical Mathematics ((ISNM,volume 164))

Abstract

Mathematical models for option pricing often result in partial differential equations of parabolic type. The calibration of these models leads to an optimization problem with PDE constraints and usually pointwise observations in the objective function. Thus, the adjoint equation of this problem involves Dirac delta functions and needs a special treatment from a numerical point of view. We show by means of numerical results that also the order of discretizing and optimizing plays an important role.

Support acknowledged by DFG Special Priority Program 1253 ‘Optimization with Partial Differential Equations’.

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Notes

  1. 1.

    Note the different scaling of Figs. 3a and 3b, causing a cut of the peaks at T=1 and T=2 in Fig. 3b.

  2. 2.

    Note that the relative difference between reference gradient (ΔT=3.125e-4, Δx=0.0025) based on first optimize-Rannacher and first discretize-Rannacher is 7.47e-009.

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Authors and Affiliations

  1. Fachbereich IV, Abteilung Mathematik, Universität Trier, 54286, Trier, Germany

    Ekkehard W. Sachs & Matthias Schu

Authors
  1. Ekkehard W. Sachs
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  2. Matthias Schu
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Correspondence to Ekkehard W. Sachs .

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Editors and Affiliations

  1. , Institute for Mathematics and Scientific, University of Graz, Heinrichstr. 36, Graz, 8010, Austria

    Kristian Bredies

  2. , Institute for Mathematics and Scientific, University of Graz, Heinrichstr. 36, Graz, 8010, Austria

    Christian Clason

  3. , Institute for Mathematics and Scientific, University of Graz, Heinrichstr. 36, Graz, 8010, Austria

    Karl Kunisch  & Gregory von Winckel  & 

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Sachs, E.W., Schu, M. (2013). Gradient Computation for Model Calibration with Pointwise Observations. In: Bredies, K., Clason, C., Kunisch, K., von Winckel, G. (eds) Control and Optimization with PDE Constraints. International Series of Numerical Mathematics, vol 164. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0631-2_7

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