Reduced-order Computation Research Articles

A game-theoretic approach to deciphering the dynamics of amyloid-β aggregation along competing pathways.

Aggregation of amyloid-β (Aβ) peptides is a significant event that underpins Alzheimer's disease (AD). Aβ aggregates, especially the low-molecular weight oligomers, are the primary toxic agents in AD pathogenesis. Therefore, there is increasing interest in understanding their formation and behaviour. In this paper, we use our previously established results on heterotypic interactions between Aβ and fatty acids (FAs) to investigate off-pathway aggregation under the control of FA concentrations to develop a mathematical framework that captures the mechanism. Our framework to define and simulate the competing on- and off-pathways of Aβ aggregation is based on the principles of game theory. Together with detailed simulations and biophysical experiments, our models describe the dynamics involved in the mechanisms of Aβ aggregation in the presence of FAs to adopt multiple pathways. Specifically, our reduced-order computations indicate that the emergence of off- or on-pathway aggregates are tightly controlled by a narrow set of rate constants, and one could alter such parameters to populate a particular oligomeric species. These models agree with the detailed simulations and experimental data on using FA as a heterotypic partner to modulate the temporal parameters. Predicting spatio-temporal landscape along competing pathways for a given heterotypic partner such as lipids is a first step towards simulating scenarios in which the generation of specific ‘conformer strains’ of Aβ could be predicted. This approach could be significant in deciphering the mechanisms of amyloid aggregation and strain generation, which are ubiquitously observed in many neurodegenerative diseases.

A new approach for the calculation of the cut-off resolution parameter in bridging methods for turbulent flow simulation

The bridging HRL (hybrid RANS/LES) method is being increasingly used to solve complex and high-Reynolds number industrial flows. The increase in popularity is equally due to the conceptual simplicity and the potential ability to affect cut-off at an optimal length scale depending upon the problem on hand. In the bridging-type of HRL methods, the ratios of resolved-to-total kinetic energy and dissipation serve to partition the flow field into resolved and computational fields for reduced-order computations. Modeled transport equations for unresolved kinetic energy and dissipation equations are solved. In this paper, we develop an additional model transport equation to determine the resolved turbulent kinetic energy equation which enables an accurate and computational means of specifying the resolution control parameter for optimal computations. The proposed approach obviates the need for expensive averaging operations currently employed to compute the partition between resolved and modeled fields. This development will expedite the bridging HRL computations for flows with transient boundaries and moving geometries. The development is in the context of Partially-Averaged Navier–Stokes (PANS) model, but the conclusions are broadly applicable to other bridging HRL approaches. The advantage of the new approach is demonstrated for the flow past a square cylinder.

A harmonic balance technique for the reduced-order computation of vortex-induced vibration

We present a harmonic balance (HB) method to model frequency lock-in effect during vortex-induced vibration (VIV) of elastically mounted circular cylinder and a flexible riser section in a freestream uniform flow. The fluid flow and structure are coupled by a fixed-point iteration process through a frequency updating algorithm. By minimizing the structural residual in the standard least-square norm, the convergence of HB-based fixed-point algorithm is achieved for a range of reduced velocity. To begin with, the HB solver is first assessed for a periodic unsteady flow around a stationary circular cylinder. A freely vibrating circular cylinder is then adopted for the reduced-order computation of VIV at low Reynolds numbers of Re=100 and 180 with one- and two-degrees-of-freedom. The coupled VIV dynamics and the frequency lock-in phenomenon are accurately captured. The results show that the HB solver is able to predict the amplitude of vibration, frequency and forces comparable to its time domain counterpart, while providing a significant reduction with regard to overall computational cost. The proposed new scheme is then demonstrated for a fully-coupled three dimensional (3D) analysis of a linear-elastic riser section undergoing vortex-induced vibration in the lock-in range. The results reveal the 3D effects through isosurfaces of streamwise vorticity blobs distributed over the span of flexible riser section. In comparison to time domain results, the 3D flow-structure interactions are accurately predicted while providing a similar speed up rate that of 2D simulations. This further corroborates that the HB solver can be extended to 3D flow-structure dynamics without compromising efficiency and accuracy.

Near-Optimal Control for Singularly Perturbed Stochastic Systems

This paper addresses linear quadratic control with state-dependent noise for singularly perturbed stochastic systems (SPSS). First, the asymptotic structure of the stochastic algebraic Riccati equation (SARE) is established for two cases. Second, a new iterative algorithm that combines Newton's method with the fixed point algorithm is established. As a result, the quadratic convergence and the reduced-order computation in the same dimension of the subsystem are attained. As another important feature, a high-order state feedback controller that uses the obtained iterative solution is given and the degradation of the cost performance is investigated for the stochastic case for the first time. Furthermore, the parameter independent controller is also given in case the singular perturbation is unknown. Finally, in order to demonstrate the efficiency of the proposed algorithm, a numerical example is given for the practical megawatt-frequency control problem.

Recursive Computation of Static Output Feedback Stochastic Nash Games for Weakly-Coupled Large-Scale Systems

This paper discusses the infinite horizon static output feedback stochastic Nash games involving state-dependent noise in weakly coupled large-scale systems. In order to construct the strategy, the conditions for the existence of equilibria have been derived from the solutions of the sets of cross-coupled stochastic algebraic Riccati equations (CSAREs). After establishing the asymptotic structure along with the positive semidefiniteness for the solutions of CSAREs, recursive algorithm for solving CSAREs is derived. As a result, it is shown that the proposed algorithm attains the reduced-order computations and the reduction of the CPU time. As another important contribution, the uniqueness of the strategy set is proved for the sufficiently small parameter e. Finally, in order to demonstrate the efficiency of the proposed algorithm, numerical example is given.

Numerical solution of stochastic Nash games with state-dependent noise for weakly coupled large-scale systems

This paper discusses the infinite horizon stochastic Nash games with state-dependent noise. After establishing the asymptotic structure along with the positive semidefiniteness for the solutions of the cross-coupled stochastic algebraic Riccati equations (CSAREs), a new algorithm that combines Newton’s method with two fixed point algorithms for solving the CSAREs is derived. As a result, it is shown that the proposed algorithm attains quadratic convergence and the reduced-order computations for sufficiently small parameter ε. As another important feature, the high-order approximate strategy that is based on the iterative solutions is proposed. Using such strategy, the degradation of the cost functional is investigated. Finally, in order to demonstrate the efficiency of the proposed algorithms, computational examples are provided.

Numerical computation for H∞ output feedback control for strongly coupled large-scale systems

In this paper, H ∞ output feedback control for strongly coupled large-scale systems are discussed. When the positive coupling parameter ε which connect the other subsystems is large, a successive algorithm for solving the algebraic Riccati equations (ARE) is developed for the first time. Since the proposed algorithm is derived using Newton’s method, it is noteworthy that the quadratic convergence and uniqueness of the obtained solution are both guaranteed for strongly coupled parameter ε. Moreover, in order to reduce the computation in the resulting Newton’s method, the gradient-based iterative (GI) algorithm is combined. As a result, it is shown that the reduced-order computation is attained. Finally, in order to demonstrate the efficiency of the proposed algorithms, computational examples are provided.