Computation of Reachable Sets Based on Hamilton-Jacobi-Bellman Equation with Running Cost Function

A novel method for computing reachable sets is proposed in this paper. In the proposed method, a Hamilton-Jacobi-Bellman equation with running cost function is numerically solved and the reachable sets of different time horizons are characterized by a family of non-zero level sets of the solution of the Hamilton-Jacobi-Bellman equation. In addition to the classical reachable set, by setting different running cost functions and terminal conditions of the Hamilton-Jacobi-Bellman equation, the proposed method allows to compute more generalized reachable sets, which are referred to as cost-limited reachable sets. In order to overcome the difficulty of solving the Hamilton-Jacobi-Bellman equation caused by the discontinuity of the solution, a method based on recursion and grid interpolation is employed. At the end of this paper, some examples are taken to illustrate the validity and generality of the proposed method.

Computation of Reachable Sets Based on Hamilton-Jacobi-Bellman Equation with Running Cost Function

A novel method for computing reachable sets is proposed in this paper. In the proposed method, a Hamilton-Jacobi-Bellman equation with running cost function is numerically solved and the reachable sets of different time horizons are characterized by a family of non-zero level sets of the solution of the Hamilton-Jacobi-Bellman equation. In addition to the classical reachable set, by setting different running cost functions and terminal conditions of the Hamilton-Jacobi-Bellman equation, the proposed method allows to compute more generalized reachable sets, which are referred to as cost-limited reachable sets. In order to overcome the difficulty of solving the Hamilton-Jacobi-Bellman equation caused by the discontinuity of the solution, a method based on recursion and grid interpolation is employed. At the end of this paper, some examples are taken to illustrate the validity and generality of the proposed method.

On the uncertainty of height anomaly differences predicted by least-squares collocation

A network of pointwise available height anomalies, derived from levelling and GPS observations, can be densified by adjusting a gravimetric quasigeoid using least-squares collocation. The resulting type of Corrector Surface Model (CSM) is applied by Norwegian surveyors to convert ellipsoidal heights to normal heights expressed in the official height system NN2000. In this work, the uncertainty related to the use of a CSM to predict differences in height anomaly was sought. As previously, the application of variograms to determine the local statistical properties of the adopted collocation model led to predictions that were consistent with their computed uncertainties. For the purpose of predicting height anomaly differences, the effect of collocation was seen to be moderate in general for the small spatial separations considered (< 10 km). However, the relative impact of collocation could be appreciable, and increasing with distance, near the network. At last, it was argued that conservative uncertainties of height anomaly differences may be obtained by rescaling output of a grid interpolation by \sqrt \Delta, where Δ is the spatial separation of the two locations for which the difference is sought.

Mathematical Modeling, Laboratory Experiments, and Sensitivity Analysis of Bioplug Technology at Darcy Scale

Summary In this paper, we study a Darcy-scale mathematical model for biofilm formation in porous media. The pores in the core are divided into three phases: water, oil, and biofilm. The water and oil flow are modeled by a generalized version of Darcy's law, and the substrate is transported by mechanical dispersion, diffusion, and convection in the water phase. Initially, there is biofilm on the pore walls. The biofilm consumes substrate for production of biomass and modifies the pore space, which changes the rock permeability. The model includes detachment of biomass caused by water flux and death of bacteria, and it is implemented in the MATLAB Reservoir Simulation Toolbox (MRST). We discuss the capability of the numerical simulator to capture results from laboratory experiments. We perform a novel sensitivity analysis based on sparse-grid interpolation and multiwavelet expansion to identify the critical model parameters. Numerical experiments using diverse injection strategies are performed to study the impact of different porosity/permeability relationships in a core saturated with water and oil.

A Class of Explicit Integrators with o-grid Interpolation for Solving Non-linear Systems of First Order ODEs

This research work is aimed at constructing a class of explicit integrators with improved stability and accuracy by incorporating an off-gird interpolation point for the purpose of making them effcient for solving stiff initial value problems. Accordingly, continuous formulations of a class of hybrid explicit integrators are derived using multi-step collocation method through matrix inversion technique, for step numbers k = 2; 3; 4: The discrete schemes were deduced from their respective continuous formulations. The stability and convergence analysis were carried out and shown to be A(α)-stable and convergent respectively. The discrete schemes when implemented as block integrators to solve some non-linear problems, it was observed that the results obtained compete favorably with the MATLAB ode23 solver.

Irregular Grid Interpolation using Radial Basis Function for Large Cylindrical Volume

Irregular grid interpolation is one of the numerical functions that often used to approximate value on an arbitrary location in the area closed by non-regular grid pivot points. In this paper, we propose method for achieving efficient computation time of radial basis function-based non-regular grid interpolation on a cylindrical coordinate. Our method consist of two stages. The first stage is the computation of weights from solving linear RBF systems constructed by known pivot points. We divide the volume into many subvolumes. At second stages, interpolation on an arbitrary point could be done using weights calculated on the first stage. At first, we find the nearest point with the query point by structuring pivot points in a K-D tree structure. After that, using the closest pivot point, we could compute the interpolated value with RBF functions. We present the performance of our method based on computation time on two stages and its precision by calculating the mean square error between the interpolated values and analytic functions. Based on the performance evaluation, our method is acceptable.