Three algorithms for solving a simplified 3-D advection-diffusion equation were compared as to their accuracy and speed in the context of insect and spore dispersal. The algorithms tested were the explicit central dif...
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Three algorithms for solving a simplified 3-D advection-diffusion equation were compared as to their accuracy and speed in the context of insect and spore dispersal. The algorithms tested were the explicit central difference (ECD) method, the implicit Crank-Nicholson (ICN) method, and the implicit Chapeau function (ICF) method. The three algorithms were used only to simulate the diffusion process. A hold-release wind shifting method was developed to simulate the wind advection process, which shifts the concentration an integer number of grids and accumulates the remaining wind travel distance (which is less than the grid spacing) to the next time step. The test problem was the dispersal of a cloud of particles (originally in only one grid cell) in a 3-D space. The major criterion for testing the accuracy was R-2, which represents the proportion of the total variation in particle distribution in all grid cells that is accounted for by the particle distribution through numerical solutions. Other criteria included total remaining mass, peak positive density, and largest negative density. High R-2 values were obtained for the ECD method with (Delta tK(z))/(Delta z)(2) less than or equal to 0.5 (Delta t = time step;K-z = vertical eddy diffusion coefficient;Delta z = vertical grid spacing), and for the two implicit methods with Delta tK(z)/(Delta z)(2) less than or equal to 5. The ICN method gave higher R-2 values than the ICF method when the concentration gradients were high, but its accuracy decreased more rapidly with the progress of time than the ICF method with a combination of a large grid spacing and a large time step. With very steep concentration gradients, the ICF method generated huge negative values, the ICN method generated negative values to a lesser extent, and the ECD method generated only small negative values. It was also found that good mass and/or peak preservation did not necessarily correspond to a higher R-2 value. Based on the R-2 value and the req
A numerical algorithm to obtain the consistent conditions satisfied by singular arcs for singular linear-quadratic optimal control problems is presented. The algorithm is based on the Presymplectic Constraint Algorith...
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A numerical algorithm to obtain the consistent conditions satisfied by singular arcs for singular linear-quadratic optimal control problems is presented. The algorithm is based on the Presymplectic Constraint Algorithm (PCA) by Gotay-Nester (Gotay et al., J Math Phys 19:2388-2399, 1978;Volckaert and Aeyels 1999) that allows to solve presymplectic Hamiltonian systems and that provides a geometrical framework to the Dirac-Bergmann theory of constraints for singular Lagrangian systems (Dirac, Can J Math 2:129-148, 1950). The numerical implementation of the algorithm is based on the singular value decomposition that, on each step, allows to construct a semi-explicit system. Several examples and experiments are discussed, among them a family of arbitrary large singular LQ systems with index 2 and a family of examples of arbitrary large index, all of them exhibiting stable behaviour.
We present a novel reformulation of nonsmooth differential equations with state jumps enabling their easier simulation and use in optimal control problems without the need for integer variables. The main idea is to in...
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We present a novel reformulation of nonsmooth differential equations with state jumps enabling their easier simulation and use in optimal control problems without the need for integer variables. The main idea is to introduce an auxiliary differential equation to mimic the state jump map. Thereby, a clock state is introduced which does not evolve during the runtime of the auxiliary system. The pieces of the trajectory that correspond to the parts when the clock state was evolving recover the solution of the original system with jumps. Our reformulation results in nonsmooth ordinary differential equations where the discontinuity is in the first time derivative of the trajectory, rather than in the trajectory itself. This class of systems is easier to handle both theoretically and numerically. The reformulation is suitable for partially elastic mechanical impact problems. We provide numerical examples demonstrating the ease of use of this reformulation in both simulation and optimal control. In the optimal control example, we solve a sequence of nonlinear programming problems (NLPs) in a homotopy penalization approach and recover a time-optimal trajectory with state jumps.
The intracellular diffusive movement of molecular substances that are buffered by diffusing chelators is often modeled as movement between compartments with constant concentrations within which the buffering occurs. H...
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The intracellular diffusive movement of molecular substances that are buffered by diffusing chelators is often modeled as movement between compartments with constant concentrations within which the buffering occurs. Here, an algorithm to solve such a system of time-dependent differential equations is presented. This Dynamic and Balanced Operator Splitting Scheme (DABOSS) combines dynamic time stepping and operator splitting techniques. The time stepping minimizes the number of time steps while bounding local errors. The balanced operator splitting separates the diffusion and reaction components (each of which is solved efficiently) in a way that preserves the correct steady-state behavior. Analysis shows that DABOSS scales almost linearly in the number of compartments and remains accurate over very long simulations. Moreover, DABOSS works efficiently for nanometer-sized compartments with sources of flux, showing that it is applicable to situations where more spatial resolution is desired.
A numerical method and the theory leading to its success are developed in this letter to solve nonstandard optimal control problems involving sweeping processes, in which the sweeping set C is non-convex and coincides...
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A numerical method and the theory leading to its success are developed in this letter to solve nonstandard optimal control problems involving sweeping processes, in which the sweeping set C is non-convex and coincides with the zero-sublevel set of a smooth function having a Lipschitz gradient, and the fixed initial state is allowed to be any point of C. This numerical method was introduced by Pinho et al. (2020) for a special form of our problem in which the function whose zero-sublevel set defines C is restricted to be twice differentiable and convex, and the initial state is confined in the interior of their convex set C. The remarkable feature of this method is manifested in approximating the sweeping process by a sequence of standard control systems invoking an innovative exponential penalty term in lieu of the normal cone, whose presence in the sweeping process renders most standard methods inapplicable. For a general setting, we prove that the optimal solution of the approximating standard optimal control problem converges uniformly to an optimal solution of the original problem (see Remark 3). This numerical method is shown to be efficient through an example in which C is not convex and the initial state is on its boundary.
We propose new easily computable bounds for different quantities which are solutions of Markov renewal equations linked to some continuous-time semi-Markov process (SMP). The idea is to construct two new discrete-time...
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We propose new easily computable bounds for different quantities which are solutions of Markov renewal equations linked to some continuous-time semi-Markov process (SMP). The idea is to construct two new discrete-time SMP which bound the initial SMP in some sense. The solution of a Markov renewal equation linked to the initial SMP is then shown to be bounded by solutions of Markov renewal equations linked to the two discrete time SMP. Also, the bounds are proved to converge. To illustrate the results, numerical bounds are provided for two quantities from the reliability field: mean sojourn times and probability transitions.
We tackle a nonlinear optimal control problem for a stochastic differential equation in Euclidean space and its state-linear counterpart for the Fokker-Planck-Kolmogorov equation in the space of probabilities. Our app...
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We tackle a nonlinear optimal control problem for a stochastic differential equation in Euclidean space and its state-linear counterpart for the Fokker-Planck-Kolmogorov equation in the space of probabilities. Our approach is founded on a novel concept of local optimality, stronger than conventional Pontryagin's minimum and originally crafted for deterministic optimal ensemble control problems. A key practical outcome is a rapidly converging numerical algorithm, which proves its feasibility for problems involving Markovian and open-loop strategies.
In this paper, we solve a complex partial differential equation motivated by applications in finance where the solution of the system gives the price of European options, including transaction costs and stochastic vol...
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In this paper, we solve a complex partial differential equation motivated by applications in finance where the solution of the system gives the price of European options, including transaction costs and stochastic volatility. The model is based on theoretical analysis, and the resulting differential equation is solved using PDE2D software. The stability analysis agrees well with experimental results.
The singular and singularly perturbed boundary value problems (SBVPs and SPBVPs) arise in the modeling of various chemical processes such as the isothermal gas sphere, electroactive polymer film, thermal explosion, an...
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The singular and singularly perturbed boundary value problems (SBVPs and SPBVPs) arise in the modeling of various chemical processes such as the isothermal gas sphere, electroactive polymer film, thermal explosion, and chemical reactor theory. Efficient numerical methods are desirable for solving such problems with a wide scope of influence. Here we derive the implicit-explicit local differential transform method (IELDTM) based on the Taylor series to solve chemical SBVPs and SPBVPs. The differential equations are directly utilized to determine the local Taylor coefficients and the entire system of algebraic equations is assembled using explicit/implicit continuity relations regarding the direction parameter. The IELDTM has an effective h - p refinement property and increasing the order of the method does not affect the degrees of freedom. We have validated the theoretical convergence results of the IELDTM with various numerical experiments and provided detailed discussions. It has been proven that the IELDTM yields more accurate solutions with fewer CPU times than various recent numerical methods for solving chemical BVPs.
Specialized numerical schemes for calculation of high order field derivatives are considered. The input data are a regular rectangular mesh with potential values at its nodes, the output data are high order derivative...
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Specialized numerical schemes for calculation of high order field derivatives are considered. The input data are a regular rectangular mesh with potential values at its nodes, the output data are high order derivative values at grid nodes and associated expressions which enable the calculation of potentials, field values or high order derivatives inside a grid cell with high accuracy. An important feature is that differentiation and approximation techniques are based on a priori knowledge about the Laplace equation which allow the production of numerical schemes with higher order properties. (C) 2011 Elsevier B.V. All rights reserved.
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