In this paper a new method for parameter estimation for elliptic partial differential equations is introduced. Parameter estimation includes minimizing an objective function, which is a measure for the difference between the parameter-dependent solution of the differential equation and some given data. We assume, that the given data results in a good approximation of the state of the system.In order to evaluate the objective function the solution of a differential equation has to be computed and hence, a large system of linear equations has to be solved. Minimization methods involve many evaluations of the objective function and therefore, the differential equation has to be solved multiple times. Thus, the computing time for parameter estimation can be large. Model order reduction was developed in order to reduce the computational effort of solving these differential equations multiple times. We use the given approximation of the state of the system as reduced basis and omit computing any snapshots. Therefore, our approach decreases the effort of the offline phase drastically. Furthermore, the dimension of the reduced system is one and thus, is much smaller than the dimension of other approaches. However, the obtained reduced model is a good approximation only close to the given data. Hence, the reduced system can lead to large errors for parameter sets, which correspond to solutions far away from the given approximation of the state of the system. In order to prevent convergence of the parameter estimator to such a local minimizer we penalise the approximation error between the full and the reduced system. ARTICLE HISTORY

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The timescales of chemical reactions range from nanoseconds to minutes. Hence, chemical reaction systems result in stiff systems of differential equations. Usually, implicit integration schemes are used in order to solve these stiff systems of differential equations. Thus, Newton's method is used in order to solve a nonlinear equation system in each time step. Thereby the evaluation of the chemical source term requests subtraction of very large numbers, and round‐off errors by cancellation occur. This can cause severe convergence problems within Newton's method, resulting in step size reductions. The system of differential equations is replaced by a less stiff modified system in order to reduce the round‐off errors and the computing time. Thereby the approximation error between the given system of differential equations and the modified system of differential equations is smaller than the given tolerance.

We present a new method for parameter estimation for elliptic partial differential equations. Parameter estimation requires the evaluation of the partial differential equation for many different parameter sets. Therefore, model order reduction is reasonable. Model order reduction is composed of an offline phase and an online phase. In the offline phase the reduced model is constructed using snapshots. In this paper we use the given measurement as only snapshot. Hence, the computational costs of the offline phase are reduced.

Chemical reaction systems often result in high-dimensional, stiff differential equations. Solving these differential equations is computational demanding. The computational effort can be reduced by the usage of an operator splitting scheme. The most frequently used splitting schemes are the Lie-Trotter splitting [1] and the Strang splitting [2]. However, the usage of a splitting scheme results in an additional splitting error. The Lie-Trotter splitting is a first order scheme, and the Strang splitting is a second order scheme. If the timescale of the fastest chemical reaction is smaller than the splitting time step, the Strang splitting suffers from order reduction. Therefore, the Lie-Trotter splitting and the Strang splitting are first order schemes for reasonable step sizes. The Richardson extrapolation of the Lie-Trotter splitting is a second order scheme. We show that the Richardson extrapolation does not suffer from order reduction for a splitting between a fast chemical source term and a slow transport term.