arXiv subjects

Bernhard A. Schmitt

Publications and source records attributed to Bernhard A. Schmitt.

9 recordsLinked to original sources

arXiv2026

Stiff Norm Estimates for Variable-Stepsize Peer Two-Step Methods

Peer two-step methods are attractive for the numerical solution of stiff initial value problems because they combine favorable features of Runge-Kutta and multi-step methods, including high-order accuracy, strong stability properties, and the avoidance of order reduction. This paper develops rigorous norm estimates for the propagation operators of variable-stepsize diagonally-implicit $L(α)$-stable Peer methods applied to stiff semi-linear systems. The analysis relies on a class of Peer methods whose propagation matrices possess a grid-independent left eigenvector, enabling the construction of suitable matrix norms for entire coefficient families. We show that this class contains $L(α)$-stable methods of stage order $q=s$ and super-convergence of order $s+1$ with $s$ stages, exceeding the classical stage-order barrier $q\le 2$ of irreducible $s$-stage diagonally-implicit Runge-Kutta methods, and construct such methods up to order 5. Achieving high stage order is one of the principal advantages of Peer methods, especially for stiff differential equations where it helps mitigate order reduction. The resulting theory provides uniform stability bounds for stiff semi-linear problems on variable grids and establishes a foundation for further applications to optimal control and related boundary value problems. Comparative numerical results are presented for three classical stiff benchmark problems, including three novel Peer two-step methods of order 3, 4, 5, and four singly diagonally implicit Runge-Kutta methods with a first explicit step of stage order 2 and order 3, 4, 5, and 6.

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arXiv2026

Implicit Third-Order Peer Triplets with Variable Stepsizes for Gradient-Based Solutions in Large-Scale ODE-Constrained Optimal Control

This paper is concerned with the theory, construction and application of variable-stepsize implicit Peer two-step methods that are super-convergent for variable stepsizes, i.e., preserve their classical order achieved for uniform stepsizes when applied in a gradient-based solution algorithm to solve ODE-constrained optimal control problems in a first-discretize-then-optimize setting. Gradients of the objective function can be computed most efficiently using approximate adjoint variables. High accuracy with moderate computational effort can be achieved through time integration methods that satisfy a sufficiently large number of adjoint order conditions for variable stepsizes and provide gradients with higher-order consistency. In this paper, we enhance our previously developed variable implicit two-step Peer triplets constructed in [J. Comput. Appl. Math. 460, 2025] to get ready for large-scale dynamical systems with varying time scales without losing efficiency. A key advantage of Peer methods is their use of multiple stages with the same high stage order, which prevents order reduction - an issue commonly encountered in semi discretized PDE problems with boundary control. Two third-order methods with four stages, good stability properties, small error constants, and a grid adaptation by equi-distributing global errors are constructed and tested for a 1D boundary heat control problem and an optimal control of cytotoxic therapies in the treatment of prostate cancer.

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arXiv2024

Implicit Peer Triplets in Gradient-Based Solution Algorithms for ODE Constrained Optimal Control

It is common practice to apply gradient-based optimization algorithms to numerically solve large-scale ODE constrained optimal control problems. Gradients of the objective function are most efficiently computed by approximate adjoint variables. High accuracy with moderate computing time can be achieved by such time integration methods that satisfy a sufficiently large number of adjoint order conditions and supply gradients with higher orders of consistency. In this paper, we upgrade our former implicit two-step Peer triplets constructed in [Algorithms, 15:310, 2022] to meet those new requirements. Since Peer methods use several stages of the same high stage order, a decisive advantage is their lack of order reduction as for semi-discretized PDE problems with boundary control. Additional order conditions for the control and certain positivity requirements now intensify the demands on the Peer triplet. We discuss the construction of $4$-stage methods with order pairs $(3,3)$ and $(4,3)$ in detail and provide three Peer triplets of practical interest. We prove convergence of order $s-1$, at least, for $s$-stage methods if state, adjoint and control satisfy the corresponding order conditions. Numerical tests show the expected order of convergence for the new Peer triplets.

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arXiv2024

Variable-Stepsize Implicit Peer Triplets in ODE Constrained Optimal Control

This paper is concerned with the theory, construction and application of implicit Peer two-step methods that are super-convergent for variable stepsizes, i.e., preserve their classical order achieved for uniform stepsizes when applied to ODE constrained optimal control problems in a first-discretize-then-optimize setting. We upgrade our former implicit two-step Peer triplets constructed in [Algorithms, 15:310, 2022] to get ready for dynamical systems with varying time scales without loosing efficiency. Peer triplets consist of a standard Peer method for interior time steps supplemented by matching methods for the starting and end steps. A decisive advantage of Peer methods is their absence of order reduction since they use stages of the same high stage order. The consistency analysis of variable-stepsize implicit Peer methods results in additional order conditions and severe new difficulties for uniform zero-stability, which intensifies the demands on the Peer triplet. Further, we discuss the construction of 4-stage methods with order pairs (4,3) and (3,3) for state and adjoint variables in detail and provide four Peer triplets of practical interest. We rigorously prove convergence of order $s-1$ for $s$-stage Peer methods applied on grids with bounded or smoothly changing stepsize ratios. Numerical tests show the expected order of convergence for the new variable-stepsize Peer triplets.

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arXiv2022

A Stiff MOL Boundary Control Problem for the 1D Heat Equation with Exact Discrete Solution

Method-of-lines discretizations are demanding test problems for stiff integration methods. However, for PDE problems with known analytic solution the presence of space discretization errors or the need to use codes to compute reference solutions may limit the validity of numerical test results. To overcome these drawbacks we present in this short note a simple test problem with boundary control, a situation where one-step methods may suffer from order reduction. We derive exact formulas for the solution of an optimal boundary control problem governed by a one dimensional discrete heat equation and an objective function that measures the distance of the final state from the target and the control costs. This analytical setting is used to compare the numerically observed convergence orders for selected implicit Runge-Kutta and Peer two-step methods of classical order four which are suitable for optimal control problems.

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arXiv2022

Krylov Methods for Adjoint-Free Singular Vector Based Perturbations in Dynamical Systems

The estimation of weather forecast uncertainty with ensemble systems requires a careful selection of perturbations to establish a reliable sampling of the error growth potential in the phase space of the model. Usually, the singular vectors of the tangent linear model propagator are used to identify the fastest growing modes (classical singular vector perturbation (SV) method). In this paper we present an efficient matrix-free block Krylov method for generating fast growing perturbations in high dimensional dynamical systems. A specific matrix containing the non-linear evolution of perturbations is introduced, which we call Evolved Increment Matrix (EIM). Instead of solving an equivalent eigenvalue problem, we use the Arnoldi method for a direct approximation of the leading singular vectors of this matrix, which however is never computed explicitly. This avoids linear and adjoint models but requires forecasts with the full non-linear system. The performance of the approximated perturbations is compared with singular vectors of a full EIM (not with the classical SV method). We show promising results for the Lorenz96 differential equations and a shallow water model, where we obtain good approximations of the fastest growing perturbations by using only a small number of Arnoldi iterations.

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arXiv2022

Implicit A-Stable Peer Triplets for ODE Constrained Optimal Control Problems

This paper is concerned with the construction and convergence analysis of novel implicit Peer triplets of two-step nature with four stages for nonlinear ODE constrained optimal control problems. We combine the property of superconvergence of some standard Peer method for inner grid points with carefully designed starting and end methods to achieve order four for the state variables and order three for the adjoint variables in a first-discretize-then-optimize approach together with A-stability. The notion triplets emphasizes that these three different Peer methods have to satisfy additional matching conditions. Four such Peer triplets of practical interest are constructed. Also as a benchmark method, the well-known backward differentiation formula BDF4, which is only $A(73.35^o)$-stable, is extended to a special Peer triplet to supply an adjoint consistent method of higher order and BDF type with equidistant nodes. Within the class of Peer triplets, we found a diagonally implicit $A(84^o)$-stable method with nodes symmetric in $[0,1]$ to a common center that performs equally well. Numerical tests with three well established optimal control problems confirm the theoretical findings also concerning A-stability.

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arXiv2020

Discrete Adjoint Implicit Peer Methods in Optimal Control

It is well known that in the first-discretize-then-optimize approach in the control of ordinary differential equations the adjoint method may converge under additional order conditions only. For Peer two-step methods we derive such adjoint order conditions and pay special attention to the boundary steps. For $s$-stage methods, we prove convergence of order $s$ for the state variables if the adjoint method satisfies the conditions for order $s\!-\!1$, at least. We remove some bottlenecks at the boundaries encountered in an earlier paper of the first author et al. [J. Comput. Appl. Math., 262:73-86, 2014] and discuss the construction of 3-stage methods for the order pair (3,2) in detail including some matrix background for the combined forward and adjoint order conditions. The impact of nodes having equal differences is highlighted. It turns out that the most attractive methods are related to BDF. Three 3-stage methods are constructed which show the expected orders in numerical tests.

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arXiv2015

Algebraic criteria for A-stability of peer two-step methods

A new criterion for A-stability of peer two-step methods is presented which is verifiable exactly in exact arithmetic by checking semi-definiteness of a certain test matrix. It depends on the existence of two positive definite weight matrices for a given method. Although the initial approach is different using properties of the numerical radius the criterion itself resembles the one from algebraic stability of General Linear Methods. Known numerical algorithms for the computation of the unknown weight matrices suffer from rank deficiencies of the test matrix. For $s$-stage peer methods of order $s-1$ this rank defect is identified with an explicit block diagonal decomposition of the test matrix in trivial and definite blocks. In the design of methods its coefficients are unknown and an explicit parametrization of A-stable peer methods of order $s-1$ is presented with a weight matrix as parameter. This leads to a general existence result for any number of stages. The restrictions for efficient L-stable peer methods like diagonally-implicit and parallel ones are also discussed and such methods with $3$ and $4$ stages are constructed.

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