Improving Efficiency of Rational Krylov Subspace Methods

This thesis studies two classes of numerical linear algebra problems, approximating the product of a function of a matrix with a vector, and solving the linear eigenvalue problem $Av=\lambda Bv$ for a small number of eigenvalues. These problems are solved by rational Krylov subspace methods (RKSM). We present several improvements in two directions: pole selection and applying inexact methods. In Chapter 3, a flexible extended Krylov subspace method ($\mathcal{F}$-EKSM) is considered for numerical approximation of the action of a matrix function $f(A)$ to a vector $b$, where the function $f$ is of Markov type. $\mathcal{F}$-EKSM has the same framework as the extended Krylov subspace method (EKSM), replacing the zero pole in EKSM with a properly chosen fixed nonzero poles. For symmetric positive definite matrices, the optimal fixed pole is derived for $\mathcal{F}$-EKSM to achieve the lowest possible upper bound on the asymptotic convergence factor, which is lower than that of EKSM. The analysis is based on properties of Faber polynomials of $A$ and $(I-A/s)^{-1}$. For large and sparse matrices that can be handled efficiently by LU factorizations, numerical experiments show that $\mathcal{F}$-EKSM and a variant of RKSM based on a small number of fixed poles outperform EKSM in both storage and runtime, and they usually have advantage over adaptive RKSM in runtime. Chapter 4 concerns the theory and development of inexact RKSM for approximating the action of a function of matrix $f(A)$ to a column vector $b$. At each step of RKSM, a shifted linear system of equations needs to be solved to enlarge the subspace. For large-scale problems, arising from discretizations of PDEs in 3D domains, such a linear system is usually solved by an iterative method approximately. The main question is how to relax the accuracy of these linear solves without negatively affecting the convergence for approximating $f(A)b$. Our insight into this issue is obtained by exploring the residual bounds on the rational Krylov subspace approximations to $f(A)b$, based on the decaying behavior of the entries in the first column of the matrix function of the block Rayleigh quotient of $A$ with respect to the rational Krylov subspaces. The decay bounds on these entries for both analytic functions and Markov functions can be efficiently and accurately evaluated by appropriate quadrature rules. A heuristic based on these bounds is proposed to relax the tolerances of the linear solves arising from each step of RKSM. As the algorithm progresses toward convergence, the linear solves can be performed with increasingly lower accuracy and computational cost. Numerical experiments for large nonsymmetric matrices show the effectiveness of the tolerance relaxation strategy for the inexact linear solves of RKSM. In Chapter 5, inexact RKSM are studied to solve large-scale nonsymmetric eigenvalue problems. Similar to the problem setting in Chapter 4, each iteration (outer step) of RKSM requires solution to a shifted linear system to enlarge the subspace, but these linear solves by direct methods are prohibitive due to the problem scale. Errors are introduced at each outer step if these linear systems are solved approximately by iterative methods (inner step), and these errors accumulate in the rational Krylov subspace. In this thesis, we derive an upper bound on the errors that can be introduced at each outer step to maintain the same convergence as exact RKSM for approximating an invariant subspace. Since this bound is inversely proportional to the current eigenresidual norm of the desired invariant subspace, the tolerance of iterative linear solves at each outer step can be relaxed with the outer iteration progress. A restarted variant of the inexact RKSM is also proposed. Numerical experiments show the effectiveness of relaxing the inner tolerance to save computational cost

Indefinite linearized augmented Lagrangian method for convex programming with linear inequality constraints

The augmented Lagrangian method (ALM) is a benchmark for tackling the convex optimization problem with linear constraints; ALM and its variants for linearly equality-constrained convex minimization models have been well studied in the literatures. However, much less attention has been paid to ALM for efficiently solving the linearly inequality-constrained convex minimization model. In this paper, we exploit an enlightening reformulation of the most recent indefinite linearized (equality-constrained) ALM, and present a novel indefinite linearized ALM scheme for efficiently solving the convex optimization problem with linear inequality constraints. The proposed method enjoys great advantages, especially for large-scale optimization cases, in two folds mainly: first, it significantly simplifies the optimization of the challenging key subproblem of the classical ALM by employing its linearized reformulation, while keeping low complexity in computation; second, we prove that a smaller proximity regularization term is needed for convergence guarantee, which allows a bigger step-size and can largely reduce required iterations for convergence. Moreover, we establish an elegant global convergence theory of the proposed scheme upon its equivalent compact expression of prediction-correction, along with a worst-case $\mathcal{O}(1/N)$ convergence rate. Numerical results demonstrate that the proposed method can reach a faster converge rate for a higher numerical efficiency as the regularization term turns smaller, which confirms the theoretical results presented in this study

On Reliability of Smart Grid Neighborhood Area Networks

With the integration of the advanced computing and communication technologies, smart grid system is dedicated to enhance the efficiency and the reliability of future power systems greatly through renewable energy resources, as well as distributed communication intelligence and demand response. Along with advanced features of smart grid, the reliability of smart grid communication system emerges to be a critical issue, since millions of smart devices are interconnected through communication networks throughout critical power facilities, which has an immediate and direct impact on the reliability of the entire power infrastructure. In this paper, we present a comprehensive survey of reliability issues posted by the smart grid with a focus on communications in support of neighborhood area networks (NAN). Specifically, we focus on network architecture, reliability requirements and challenges of both communication networks and systems, secure countermeasures, and case studies in smart grid NAN. We aim to provide a deep understanding of reliability challenges and effective solutions toward reliability issues in smart grid NAN

Calculation Method for Uplift Capacity of Suction Caisson in Sand Considering Different Drainage Conditions

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Reconstruction of mandibular defects using vascularized fibular osteomyocutaneous flap combined with nonvascularized fibular flap

The height of single-layer fibular flap is not long enough to return to the ideal height of the mandible. While the double-layer vascularized fibular osteomyocutaneous flap(VFF) is more complicated in shaping and fixation, along with a longer operation time. The aim of this study was to investigate the clinical effect of VFF combined with nonvascularized fibular flap(NVFF) in the reconstruction of mandibular defect. From September 2016 to June 2018, 15 patients with benign mandibular tumors underwent reconstruction with VFF and NVFF. SimPlant Pro ? software (version 11.04) was used to simulate reconstruction of the mandible preoperatively. All patients were followed up for 8-23 month, with an average of 11.7 months. 15 VFFs survived well. Among the 15 NVFFs, one was almost completely absorbed, two with partial absorption, and the remaining survived regardless of the small amount of absorption. The postoperative absorption of the whole fibula was 7.53±6.362%, a favorable facial contour and speech function were attained. The VFF combined with NVFF to reconstruct the mandibular defect can restore the vertical height of the mandible and achieve satisfactory clinical results

Efficient Exploration Using Extra Safety Budget in Constrained Policy Optimization

Reinforcement learning (RL) has achieved promising results on most robotic control tasks. Safety of learning-based controllers is an essential notion of ensuring the effectiveness of the controllers. Current methods adopt whole consistency constraints during the training, thus resulting in inefficient exploration in the early stage. In this paper, we propose an algorithm named Constrained Policy Optimization with Extra Safety Budget (ESB-CPO) to strike a balance between the exploration efficiency and the constraints satisfaction. In the early stage, our method loosens the practical constraints of unsafe transitions (adding extra safety budget) with the aid of a new metric we propose. With the training process, the constraints in our optimization problem become tighter. Meanwhile, theoretical analysis and practical experiments demonstrate that our method gradually meets the cost limit's demand in the final training stage. When evaluated on Safety-Gym and Bullet-Safety-Gym benchmarks, our method has shown its advantages over baseline algorithms in terms of safety and optimality. Remarkably, our method gains remarkable performance improvement under the same cost limit compared with baselines.Comment: 7 pages, 8 figure

DexCatch: Learning to Catch Arbitrary Objects with Dexterous Hands

Achieving human-like dexterous manipulation remains a crucial area of research in robotics. Current research focuses on improving the success rate of pick-and-place tasks. Compared with pick-and-place, throw-catching behavior has the potential to increase picking speed without transporting objects to their destination. However, dynamic dexterous manipulation poses a major challenge for stable control due to a large number of dynamic contacts. In this paper, we propose a Stability-Constrained Reinforcement Learning (SCRL) algorithm to learn to catch diverse objects with dexterous hands. The SCRL algorithm outperforms baselines by a large margin, and the learned policies show strong zero-shot transfer performance on unseen objects. Remarkably, even though the object in a hand facing sideward is extremely unstable due to the lack of support from the palm, our method can still achieve a high level of success in the most challenging task. Video demonstrations of learned behaviors and the code can be found on the supplementary website