Explore the vast range of titles available.

Direct search methods are best known as unconstrained optimization techniques that do not explicitly use derivatives. Direct search methods were formally proposed and widely applied in the 1960s but fell out of favor with the mathematical optimization community by the early 1970s because they lacked coherent mathematical analysis. Nonetheless, users remained loyal to these methods, most of which were easy to program, some of which were reliable. In the past fifteen years, these methods have seen a revival due, in part, to the appearance of mathematical analysis, as well as to interest in parallel and distributed computing. This review begins by briefly summarizing the history of direct search methods and considering the special properties of problems for which they are well suited. Our focus then turns to a broad class of methods for which we provide a unifying framework that lends itself to a variety of convergence results. The underlying principles allow generalization to handle bound constraints and linear constraints. We also discuss extensions to problems with nonlinear constraints.

A key feature of federated learning (FL) is that not all clients participate in every communication epoch of each global model update. The rationality for such partial client selection is largely to reduce the communication overhead. However, in many cases, the unselected clients are still able to compute their local model updates, but are not “authorized” to upload the updates in this round, which is a waste of computation capacity. In this work, we propose an algorithm FedUmf—Federated Learning with Unauthorized Model Fusion that utilizes the model updates from the unselected clients. More specifically, a client computes the stochastic gradient descent (SGD) even if it is not selected to upload in the current communication epoch. Then, if this client is selected in the next round, it non-trivially merges the outdated SGD stored in the previous round with the current global model before it starts to compute the new local model. A rigorous convergence analysis is established for FedUmf, which shows a faster convergence rate than the vanilla FedAvg. Comprehensive numerical experiments on several standard classification tasks demonstrate its advantages, which corroborate the theoretical results.

We examine the local convergence properties of pattern search methods, complementing the previously established global convergence properties for this class of algorithms. We show that the step-length control parameter which appears in the definition of pattern search algorithms provides a reliable asymptotic measure of first-order stationarity. This gives an analytical justification for a traditional stopping criterion for pattern search methods. Using this measure of first-order stationarity, we both revisit the global convergence properties of pattern search and analyze the behavior of pattern search in the neighborhood of an isolated local minimizer.

The alternating direction method of multipliers (ADMM) is now widely used in many fields, and its convergence was proved when two blocks of variables are alternatively updated. It is strongly desirable and practically valuable to extend the ADMM directly to the case of a multi-block convex minimization problem where its objective function is the sum of more than two separable convex functions. However, the convergence of this extension has been missing for a long time—neither an affirmative convergence proof nor an example showing its divergence is known in the literature. In this paper we give a negative answer to this long-standing open question: The direct extension of ADMM is not necessarily convergent. We present a sufficient condition to ensure the convergence of the direct extension of ADMM, and give an example to show its divergence.

We present and analyze a second order in time variable step BDF2 numerical scheme for the Cahn-Hilliard equation. The construction relies on a second order backward difference, convex-splitting technique and viscous regularizing at the discrete level. We show that the scheme is unconditionally stable and uniquely solvable. In addition, under mild restriction on the ratio of adjacent time-steps, an optimal second order in time convergence rate is established. The proof involves a novel generalized discrete Gronwall-type inequality. As far as we know, this is the first rigorous proof of second order convergence for a variable step BDF2 scheme, even in the linear case, without severe restriction on the ratio of adjacent time-steps. Results of our numerical experiments corroborate our theoretical analysis.

Based on the Caputo fractional-order derivative, this work investigates the dynamics of a newly developed co-infection model of Human immunodeficiency virus (HIV) and Hepatitis C virus (HCV). Due to their ability to take into account memory history and heritability, Caputo fractional-order derivatives are a natural candidate to study the HIV/HCV co-infection where these two properties are critical to study how infections spread. Furthermore, applying the Caputo fractional-derivative to the co-infection model helps forecast disease progression and offers optimal treatment strategies for understanding complex HIV/HCV interactions and co-evolutionary dynamics. Mathematical analysis of the co-infection model reveals two equilibria, one without sickness and the other with sickness. The next-generation matrix approach is employed to calculate the basic reproduction number for the cases of HIV and HCV only respectively, and the co-infection model of HIV and HCV, jointly that demonstrates the mutual influence of the two diseases. Using the reproduction numbers, the Lyapunov functional method, and the Routh-Hurwitz criterion, we establish the global dynamics of the model. To validate theoretical predictions, the fractional Adams Method (FAM), a popular numerical technique with a predictor-corrector structure, is utilized to compute the model’s numerical solutions. Finally, numerical simulations confirm the theoretical findings, elucidating the high degree of agreement between the theoretical analysis and the numerical results. Different from the existing literature using the L1 scheme, we incorporated a memory trace (MT) procedure in our paper that captures and amalgamates the historical dynamics of the system to evoke the memory effect in detail. One of the novel results obtained from this study is that the memory trace starts to come into existence once fractional power ζ starts to increase from 0 to 1 and completely disappears when ζ becomes 1. Upon increasing the fractional-order ζ from 0, the memory effect exploits a nonlinear proliferation starting from zero. This observed memory effect emphasizes the difference between the integer and non-integer order derivatives and thus claims the existence of memory effects of fractional-order derivatives. The findings of the paper will contribute to a better understanding of the disease outbreak, as well as aid in the development of future predictions and control strategies.

In this paper we propose and analyze a (temporally) third order accurate exponential time differencing (ETD) numerical scheme for the no-slope-selection (NSS) equation of the epitaxial thin film growth model, with Fourier pseudo-spectral discretization in space. A linear splitting is applied to the physical model, and an ETD-based multistep approximation is used for time integration of the corresponding equation. In addition, a third order accurate Douglas-Dupont regularization term, in the form of - A Δ t 2 ϕ 0 ( L N ) Δ N 2 ( u n + 1 - u n ) , is added in the numerical scheme. A careful Fourier eigenvalue analysis results in the energy stability in a modified version, and a theoretical justification of the coefficient A becomes available. As a result of this energy stability analysis, a uniform in time bound of the numerical energy is obtained. And also, the optimal rate convergence analysis and error estimate are derived in details, in the ℓ ∞ ( 0 , T ; H h 1 ) ∩ ℓ 2 ( 0 , T ; H h 3 ) norm, with the help of a careful eigenvalue bound estimate, combined with the nonlinear analysis for the NSS model. This convergence estimate is the first such result for a third order accurate scheme for a gradient flow. Some numerical simulation results are presented to demonstrate the efficiency of the numerical scheme and the third order convergence. The long time simulation results for ε = 0.02 (up to T = 3 × 10 5 ) have indicated a logarithm law for the energy decay, as well as the power laws for growth of the surface roughness and the mound width. In particular, the power index for the surface roughness and the mound width growth, created by the third order numerical scheme, is more accurate than those produced by certain second order energy stable schemes in the existing literature.

We present global convergence rates for a line-search method which is based on random first-order models and directions whose quality is ensured only with certain probability. We show that in terms of the order of the accuracy, the evaluation complexity of such a method is the same as its counterparts that use deterministic accurate models; the use of probabilistic models only increases the complexity by a constant, which depends on the probability of the models being good. We particularize and improve these results in the convex and strongly convex case. We also analyze a probabilistic cubic regularization variant that allows approximate probabilistic second-order models and show improved complexity bounds compared to probabilistic first-order methods; again, as a function of the accuracy, the probabilistic cubic regularization bounds are of the same (optimal) order as for the deterministic case.

Here we have studied the fingering phenomena in fluid flow through fracture porous media with inclination and gravitational effect and investigate the applicability of Adomian decomposition method to the nonlinear partial differential equation arising in this phenomena and prove the convergence of Adomian decomposition scheme, which leads to an abstract result and an analytical approximate solution to the equation. Finally developed a simulation result of saturation of wetting phase with and without considering the inclination effect for some interesting choices of parametric data value and studied the recovery rate of the oil reservoir with dimensionless time.