H2\mathcal{H}_2 Pseudo-Optimal Reduction of Structured DAEs by Rational Interpolation

In this contribution, we extend the concept of $\mathcal{H}_2$ inner product and $\mathcal{H}_2$ pseudo-optimality to dynamical systems modeled by differential-algebraic equations (DAEs). To this end, we derive projected Sylvester equations that characterize the $\mathcal{H}_2$ inner product in terms of the matrices of the DAE realization. Using this result, we extend the $\mathcal{H}_2$ pseudo-optimal rational Krylov algorithm for ordinary differential equations to the DAE case. This algorithm computes the globally optimal reduced-order model for a given subspace of $\mathcal{H}_2$ defined by poles and input residual directions. Necessary and sufficient conditions for $\mathcal{H}_2$ pseudo-optimality are derived using the new formulation of the $\mathcal{H}_2$ inner product in terms of tangential interpolation conditions. Based on these conditions, the cumulative reduction procedure combined with the adaptive rational Krylov algorithm, known as CUREd SPARK, is extended to DAEs. Important properties of this procedure are that it guarantees stability preservation and adaptively selects interpolation frequencies and reduced order. Numerical examples are used to illustrate the theoretical discussion. Even though the results apply in theory to general DAEs, special structures will be exploited for numerically efficient implementations

Variational collision integrator for polymer chains

The numerical simulation of many-particle systems (e.g., in molecular dynamics) often involves constraints of various forms. We present a symplectic integrator for mechanical systems with holonomic (bilateral) and unilateral contact constraints, the latter being in the form of a nonpenetration condition. The scheme is based on a discrete variant of Hamilton’s principle in which both the discrete trajectory and the unknown collision time are varied (cf. [Fetecau et al., 2003, SIAM J. Applied Dynamical Systems, 2, pp. 381–416]). As a consequence, the collision event enters the discrete equations of motion as an unknown that has to be computed on-the-fly whenever a collision is imminent. The additional bilateral constraints are e ciently dealt with employing a discrete null space reduction (including a projection and a local reparametrisation step) which considerably reduces the number of unknowns and improves the condition number during each time-step as compared to a standard treatment with Lagrange multipliers. We illustrate the numerical scheme with a simple example from polymer dynamics, a linear chain of beads, and test it against other standard numerical schemes for collision problems

Multimethod optimization in the cloud: A case‐study in systems biology modelling

[Abstract] Optimization problems appear in many different applications in science and engineering. A large number of different algorithms have been proposed for solving them; however, there is no unique general optimization method that performs efficiently across a diverse set of problems. Thus, a multimethod optimization, in which different algorithms cooperate to outperform the results obtained by any of them in isolation, is a very appealing alternative. Besides, as real‐life optimization problems are becoming more and more challenging, the use of HPC techniques to implement these algorithms represents an effective strategy to speed up the time‐to‐solution. In addition, a parallel multimethod approach can benefit from the effortless access to q large number of distributed resources facilitated by cloud computing. In this paper, we propose a self‐adaptive cooperative parallel multimethod for global optimization. This proposal aims to perform a thorough exploration of the solution space by means of multiple concurrent executions of a broad range of search strategies. For its evaluation, we consider an extremely challenging case‐study from the field of computational systems biology. We also assess the performance of the proposal on a public cloud, demonstrating both the potential of the multimethod approach and the opportunity that the cloud provides for these problems.Gobierno de España; DPI2014‐55276‐C5‐2‐RGobierno de España; DPI2017‐82896‐C2‐2‐RGobierno de España; TIN2016‐75845‐PXunta de Galicia; R2016/045Xunta de Galicia; ED431C 2017/0