290 research outputs found
Reduced basis methods for pricing options with the Black-Scholes and Heston model
In this paper, we present a reduced basis method for pricing European and
American options based on the Black-Scholes and Heston model. To tackle each
model numerically, we formulate the problem in terms of a time dependent
variational equality or inequality. We apply a suitable reduced basis approach
for both types of options. The characteristic ingredients used in the method
are a combined POD-Greedy and Angle-Greedy procedure for the construction of
the primal and dual reduced spaces. Analytically, we prove the reproduction
property of the reduced scheme and derive a posteriori error estimators.
Numerical examples are provided, illustrating the approximation quality and
convergence of our approach for the different option pricing models. Also, we
investigate the reliability and effectivity of the error estimators.Comment: 25 pages, 27 figure
Efficient parametric analysis of the chemical master equation through model order reduction
Background: Stochastic biochemical reaction networks are commonly modelled by
the chemical master equation, and can be simulated as first order linear
differential equations through a finite state projection. Due to the very high
state space dimension of these equations, numerical simulations are
computationally expensive. This is a particular problem for analysis tasks
requiring repeated simulations for different parameter values. Such tasks are
computationally expensive to the point of infeasibility with the chemical
master equation. Results: In this article, we apply parametric model order
reduction techniques in order to construct accurate low-dimensional parametric
models of the chemical master equation. These surrogate models can be used in
various parametric analysis task such as identifiability analysis, parameter
estimation, or sensitivity analysis. As biological examples, we consider two
models for gene regulation networks, a bistable switch and a network displaying
stochastic oscillations. Conclusions: The results show that the parametric
model reduction yields efficient models of stochastic biochemical reaction
networks, and that these models can be useful for systems biology applications
involving parametric analysis problems such as parameter exploration,
optimization, estimation or sensitivity analysis.Comment: 23 pages, 8 figures, 2 table
Kernel Methods for Surrogate Modeling
This chapter deals with kernel methods as a special class of techniques for
surrogate modeling. Kernel methods have proven to be efficient in machine
learning, pattern recognition and signal analysis due to their flexibility,
excellent experimental performance and elegant functional analytic background.
These data-based techniques provide so called kernel expansions, i.e., linear
combinations of kernel functions which are generated from given input-output
point samples that may be arbitrarily scattered. In particular, these
techniques are meshless, do not require or depend on a grid, hence are less
prone to the curse of dimensionality, even for high-dimensional problems.
In contrast to projection-based model reduction, we do not necessarily assume
a high-dimensional model, but a general function that models input-output
behavior within some simulation context. This could be some micro-model in a
multiscale-simulation, some submodel in a coupled system, some initialization
function for solvers, coefficient function in PDEs, etc.
First, kernel surrogates can be useful if the input-output function is
expensive to evaluate, e.g. is a result of a finite element simulation. Here,
acceleration can be obtained by sparse kernel expansions. Second, if a function
is available only via measurements or a few function evaluation samples, kernel
approximation techniques can provide function surrogates that allow global
evaluation.
We present some important kernel approximation techniques, which are kernel
interpolation, greedy kernel approximation and support vector regression.
Pseudo-code is provided for ease of reproducibility. In order to illustrate the
main features, commonalities and differences, we compare these techniques on a
real-world application. The experiments clearly indicate the enormous
acceleration potentia
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