Time-invariant BAS-decentralized large-scale linear regulator problem

We shall investigate randomized algorithms for solving large-scale linear inverse problems with general Tikhonov regularizations. Our first approach transforms general form inverse problems into standard form, then we apply randomized algorithms to reduce large-scale systems of standard form to much smaller-scale systems and seek their regularized solutions in combination with some popular choice rules for regularization parameters. Our second approach involves a new random generalized SVD algorithm that can essentially reduce the sizes of the original large-scale ill-posed systems. The reduced systems can provide approximate regularized solutions with about the same accuracy as the ones by the classical generalized SVD, but they are much more stable and much less expensive as they need only to work on problems of much smaller sizes. Numerical results are presented to demonstrate the efficiency and accuracy of the algorithms.

This paper provides an alternating optimization algorithm for large-scale matrix rank minimization problems and its parallel implementation on GPU. The matrix rank minimization problem has a lot of important applications in signal processing, and several useful algorithms have been proposed. However most algorithms cannot be applied to a large-scale problem because of high computational cost. This paper proposes a null space based algorithm, which provides a low-rank solution without computing inverse matrix nor singular value decomposition. The algorithm can be parallelized easily without any approximation and can be applied to a large-scale problem. Numerical examples show that the algorithm provides a low-rank solution efficiently and can be speed up by parallel GPU computing.

The problem of the robust stability of large-scale dynamical systems including delayed states and parameter perturbations in interconnections is considered. By using algebraic Riccati equations and some analytical methods, some sufficient conditions on linear decentralized state feedback controllers are derived so that the systems remain stable in the presence of delayed states and parameter perturbations. Such conditions give some bounds on the perturbations of interconnections with delayed states and uncertain parameters, and result in a quantitative measures of robustness for large-scale dynamical systems including delayed states and uncertain parameters in interconnections. The results obtained in this paper are applicable not only to large-scale systems with multiple time-varying delays, but also to large-scale systems without exact knowledge of the delays, i.e., large-scale systems with uncertain delays.

Efficient optimization of a large-scale biorefinery system using a novel decomposition based approach

We provide experimental evidence that current desktop computers feature enough computational power to solve large-scale dense linear algebra problems. While the high computational cost of the numerical methods for solving these problems can be tackled by the multiple cores of current processors, we propose to use the disk to store the large data structures associated with these applications. Our results also show that the limited amount of RAM and the comparatively slow disk of the system pose no problem for the solution of very large dense linear systems and linear least-squares problems. Thus, current desktop computers are revealed as an appealing, cost-effective platform for research groups that have to deal with large dense linear algebra problems but have no direct access to large computing facilities.

: Controls research under this program concentrated on the development of a network thermodynamic stabilization framework for hybrid control design of large-scale and multiagent aerospace systems. In particular, we concentrate on hybrid control, hierarchical control, impulsive dynamical systems, nonnegative dynamical systems, compartmental systems, large-scale systems, nonlinear switching control, cooperative control, and adaptive control. Application areas include large flexible interconnected space structures, spacecraft stabilization, cooperative control of unmanned air vehicles, network systems, swarms of air and space vehicle formations, and pharmacological systems.

This work presents the development of a 3D hybrid coupled dual continuum and discrete fracture model for simulating coupled flow, reaction and deformation processes relevant to fractured reservoirs with multiscale fracture system, e.g., coal and shale, efficiently and accurately. In this hybrid model, the natural fracture network and coal matrix are described together by a dual continuum approach and the large fractures are represented explicitly by the discrete fracture approach. A combination of different types of elements is used for spatial discretization. Large fractures are discretised with lower-dimensional interface elements and continuum domains with higher-dimensional elements. The coupling between the two models is achieved via the principle of superposition. To reduce computational time of simulations for complex and large-scale problems, a hybrid MPI/OpenMP parallel scheme is implemented in this work. The developed model is applied to investigate coupled thermal, hydraulic, and mechanical processes associated with CO2 sequestration and enhanced coalbed methane recovery. The results demonstrate capabilities of the model to adequately capture the effects of multiscale fracture system and their coupled behaviour during CO2 injection and methane recovery from coal reservoirs. Performance of the proposed parallelisation scheme was tested by comparing computation times of serial and parallel implementations. A good performance improvement was achieved, the speedup using parallelized scheme reaches up to about 10 times along with satisfactory scalability for considered application example. The findings of this work support developments and improvements of efficient advanced numerical models to study coupled THCM behaviour in fractured porous geomaterials.

This paper addresses the development of a dynamic programming decomposition-coordination algorithm for the optimal control of discrete-time large-scale systems with multiple time delays. The systems under study can be converted into the corresponding augmented systems satisfying the Markov processes denning the augmented state vectors. The resulting dynamic programming can then be further decomposed into a series of subproblems which might be solved in parallel by combining the dynamic programming techniques with approaches to matrix partition. The entire system's solution is finally given by the iterative computation between the subproblems being the local control level and the coordinator. The major advantages of the proposed algorithm over others are to avoid two-point boundary-value problems of the systems with multiple time delays and to significantly reduce the computer memory and time required for such large-scale systems optimal control problems.

The authors present an approach to the linear programming state estimation problem that is suitable for large-scale interconnected power systems. It uses the Dantzig-Wolfe decomposition algorithm. The performance of the method is enhanced by using a multiple column strategy and several other modifications to the original algorithm. Scaling techniques are used to guarantee a reliable solution. Test results on standard test systems are used to illustrate the usefulness of the method and its viability for parallel processor implementation.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Stochastic optimal control for non-linear dynamical systems under noisy observations

An MPI-CUDA Implementation and Optimization for Parallel Sparse Equations and Least Squares (LSQR)

Applying Linear Regression to systems with a massive amount of observations, a scenario which is becoming increasingly common in the era of Big Data, poses major algorithmic and computational challenges. This paper proposes a novel high-performance FPGA-based architecture for large-scale Linear Regression problems as well as a heterogeneous system comprising the custom FPGA architecture, an enhanced GPU module and a multi-core CPU for addressing the aforementioned problem. The system adaptively assigns Linear Regression workloads to the three computing devices to minimise runtime. The device with the highest performance is chosen based on an analytical framework, as well as the workload's size and structure. A quantitative comparison with existing FPGA, GPU and multi-core CPU designs yields speed-ups of up to 18.07×, 32.67× and 25.84× respectively.

Stochastic inverse modeling deals with the estimation of functions from sparse data, which is a problem with a nonunique solution, with the objective to evaluate best estimates, measures of uncertainty, and sets of solutions that are consistent with the data. As finer resolutions become desirable, the computational requirements increase dramatically when using conventional solvers. A method is developed in this paper to solve large-scale stochastic linear inverse problems, based on the hierarchical matrix (or ℋ 2 matrix) approach. The proposed approach can also exploit the sparsity of the underlying measurement operator, which relates observations to unknowns. Conventional direct algorithms for solving large-scale linear inverse problems, using stochastic linear inversion techniques, typically scale as 𝒪(n 2 m+nm 2), where n is the number of measurements and m is the number of unknowns. We typically have n ≪ m. In contrast, the algorithm presented here scales as 𝒪(n 2 m), i.e., it scales linearly with the larger problem dimension m. The algorithm also allows quantification of uncertainty in the solution at a computational cost that also grows only linearly in the number of unknowns. The speedup gained is significant since the number of unknowns m is often large. The effectiveness of the algorithm is demonstrated by solving a realistic crosswell tomography problem by formulating it as a stochastic linear inverse problem. In the case of the crosswell tomography problem, the sparsity of the measurement operator allows us to further reduce the cost of our proposed algorithm from 𝒪(n 2 m) to $\mathcal {O}(n^{2} \sqrt {m} + nm)$ . The computational speedup gained by using the new algorithm makes it easier, among other things, to optimize the location of sources and receivers, by minimizing the mean square error of the estimation. Without this fast algorithm, this optimization would be computationally impractical using conventional methods.

We develop a framework for goal-oriented optimal design of experiments (GOODE) for large-scale Bayesian linear inverse problems governed by PDEs. This framework differs from classical Bayesian optimal design of experiments (ODE) in the following sense: we seek experimental designs that minimize the posterior uncertainty in the experiment end-goal, e.g. a quantity of interest (QoI), rather than the estimated parameter itself. This is suitable for scenarios in which the solution of an inverse problem is an intermediate step and the estimated parameter is then used to compute a QoI. In such problems, a GOODE approach has two benefits: the designs can avoid wastage of experimental resources by a targeted collection of data, and the resulting design criteria are computationally easier to evaluate due to the often low-dimensionality of the QoIs. We present two modified design criteria, A-GOODE and D-GOODE, which are natural analogues of classical Bayesian A- and D-optimal criteria. We analyze the connections to other ODE criteria, and provide interpretations for the GOODE criteria by using tools from information theory. Then, we develop an efficient gradient-based optimization framework for solving the GOODE optimization problems. Additionally, we present comprehensive numerical experiments testing the various aspects of the presented approach. The driving application is the optimal placement of sensors to identify the source of contaminants in a diffusion and transport problem. We enforce sparsity of the sensor placements using an -norm penalty approach, and propose a practical strategy for specifying the associated penalty parameter.

The paper discusses an iterative scheme for solving large-scale three-dimensional linear elasticity problems, discretized on a tensor product of two-dimensional and one-dimensional meshes. A framework is chosen of the additive AMLI method to develop a preconditioner of a black-box' type which is robust with respect to discontinuities of the problem coefficients and imposes only weak (and acceptable in practice) restrictions on the choice of the meshing procedure. The preconditioner works on a hierarchical sequence of nested finite element spaces to solve the problem with arithmetic cost, nearly proportional to the number of degrees of freedom on the finest mesh. It is particularly well suited for the case when the solution is known to be strongly varying in certain subregions of the domain and the mesh is locally prerefined there to reduce the discretization error.