Imre Fekete

Sidafa Conde, Imre Fekete, John N. Shadid

We construct a family of embedded pairs for optimal strong stability preserving explicit Runge-Kutta methods of order $2 \leq p \leq 4$ to be used to obtain numerical solution of spatially discretized hyperbolic PDEs. In this construction, the goals include non-defective methods, large region of absolute stability, and optimal error measurement as defined in [5,19]. The new family of embedded pairs offer the ability for strong stability preserving (SSP) methods to adapt by varying the step-size based on the local error estimation while maintaining their inherent nonlinear stability properties. Through several numerical experiments, we assess the overall effectiveness in terms of precision versus work while also taking into consideration accuracy and stability.

Gustaf Söderlind, Imre Fekete, István Faragó

In order to be convergent, linear multistep methods must be zero stable. While constant step size theory was established in the 1950's, zero stability on nonuniform grids is less well understood. Here we investigate zero stability on compact intervals and smooth nonuniform grids. In practical computations, step size control can be implemented using smooth (small) step size changes. The resulting grid $\{t_n\}_{n=0}^N$ can be modeled as the image of an equidistant grid under a smooth deformation map, i.e., $t_n = Φ(τ_n)$, where $τ_n = n/N$ and the map $Φ$ is monotonically increasing with $Φ(0)=0$ and $Φ(1)=1$. The model is justified for any fixed order method operating in its asymptotic regime when applied to smooth problems, since the step size is then determined by the (smooth) principal error function which determines $Φ$, and a tolerance requirement which determines $N$. Given any strongly stable multistep method, there is an $N^*$ such that the method is zero stable for $N>N^*$, provided that $Φ\in C^2[0,1]$. Thus zero stability holds on all nonuniform grids such that adjacent step sizes satisfy $h_n/h_{n-1} = 1 + \mathrm O(N^{-1})$ as $N\rightarrow\infty$. The results are exemplified for BDF-type methods.

Imre Fekete, David I. Ketcheson, Lajos Lóczi

We propose a technique for investigating stability properties like positivity and forward invariance of an interval for method-of-lines discretizations, and apply the technique to study positivity preservation for a class of TVD semi-discretizations of 1D scalar hyperbolic conservation laws. This technique is a generalization of the approach suggested in ref. 12. We give more relaxed conditions on the time-step for positivity preservation for slope-limited semi-discretizations integrated in time with explicit Runge-Kutta methods. We show that the step-size restrictions derived are sharp in a certain sense, and that many higher-order explicit Runge-Kutta methods, including the classical 4th-order method and all non-confluent methods with a negative Butcher coefficient, cannot generally maintain positivity for these semi-discretizations under any positive step size. We also apply the proposed technique to centered finite difference discretizations of scalar hyperbolic and parabolic problems.

Imre Fekete, András Molnár, Péter L. Simon

The paper contributes to strengthening the relation between machine learning and the theory of differential equations. In this context, the inverse problem of fitting the parameters, and the initial condition of a differential equation to some measurements constitutes a key issue. The paper explores an abstraction that can be used to construct a family of loss functions with the aim of fitting the solution of an initial value problem to a set of discrete or continuous measurements. It is shown, that an extension of the adjoint equation can be used to derive the gradient of the loss function as a continuous analogue of backpropagation in machine learning. Numerical evidence is presented that under reasonably controlled circumstances the gradients obtained this way can be used in a gradient descent to fit the solution of an initial value problem to a set of continuous noisy measurements, and a set of discrete noisy measurements that are recorded at uncertain times.