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In this article, we show that the collective light–matter strong coupling regime where N molecular emitters couple to the photon mode of an optical cavity can be mapped to a quantum impurity model where the photon is the impurity that is coupled to a bath of anharmonic transitions. In the thermodynamic limit where N ≫ 1, we argue that the bath can be replaced with an effective harmonic bath, leading to a dramatic simplification of the problem into one of the coupled harmonic oscillators. We derive simple analytical expressions for linear optical spectra (transmission, reflection, and absorption) where the only molecular input required is the molecular linear susceptibility. This formalism is applied to a series of illustrative examples, showing the role of temperature, disorder, vibronic coupling, and optical saturation of the molecular ensemble, explaining that it is useful even when describing an important class of nonlinear optical experiments. For completeness, we provide Appendixes A–C that include a self-contained derivation of the relevant spectroscopic observables for arbitrary anharmonic systems (for both large and small N) within the rotating-wave approximation. While some of the presented results herein have already been reported in the literature, we provide a unified presentation of the results as well as new interpretations that connect powerful concepts in open quantum systems and linear response theory with molecular polaritonics. [ABSTRACT FROM AUTHOR]

Synchronization is common in both nature and human physiology, often used to illustrate nonlinear dynamics. Interestingly, one can initiate their comprehension of this phenomenon from pure linear systems. In this Tutorial, we begin with a theoretical exploration of coupled oscillators' dynamic behavior, enabling us to discern and contrast the unique attributes of dissipative coupling as opposed to commonly observed coherent coupling. We then examine synchronization in two dissipative coupled linear systems: one with two pendulums mutually linked via the Lenz effect and the other with two RLC oscillators coupled via a resistor. This Tutorial is designed to serve as a concise starting point for researchers interested in exploring synchronization phenomena using a simplified model driven solely by dissipative coupling. [ABSTRACT FROM AUTHOR]

To investigate the impact of non-linear interactions on dynamic coarse graining, we study a simplified model system featuring a tracer particle in a complex environment. Using a projection operator formalism and computer simulations, we systematically derive generalized Langevin equations (GLEs) describing the dynamics of this particle. We compare different kinds of linear and non-linear coarse-graining procedures to understand how non-linearities enter reconstructed GLEs and how they influence the coarse-grained dynamics. For non-linear external potentials, we show analytically and numerically that the non-Gaussian parameter and the incoherent intermediate scattering function will not be correctly reproduced by the GLE if a linear projection is applied. This, however, can be overcome by using non-linear projection operators. We also study anharmonic coupling between the tracer and the environment and demonstrate that the reconstructed memory kernel develops an additional trap-dependent contribution. Our study highlights some open challenges and possible solutions in dynamic coarse graining. [ABSTRACT FROM AUTHOR]

Explicit and implicit size effects in computer simulations result from considering systems with a fixed number of particles and periodic boundary conditions, respectively. We investigate these effects in the relation D*(L) = A(L) exp(α(L)s2(L)) between reduced self-diffusion coefficient D*(L) and two-body excess entropy s2(L) for prototypical simple-liquid systems of linear size L. To this aim, we introduce and validate a finite-size two-body excess entropy integral equation. Our analytical arguments and simulation results show that s2(L) exhibits a linear scaling with 1/L. Since D*(L) displays a similar behavior, we show that the parameters A(L) and α(L) are also linearly proportional to 1/L. By extrapolating to the thermodynamic limit, we report the coefficients A∞ = 0.048 ± 0.001 and α∞ = 1.000 ± 0.013 that agree well with the universal values available in the literature [M. Dzugutov, Nature 381, 137–139 (1996)]. Finally, we find a power law relation between the scaling coefficients for D*(L) and s2(L), suggesting a constant viscosity-to-entropy ratio. [ABSTRACT FROM AUTHOR]

Using computer simulations, we have studied the percolation and the electrical conductance of two-dimensional, random percolating networks of curved, zero-width metallic nanowires. We mimicked the curved nanowires using circular arcs. The percolation threshold decreased as the aspect ratio of the arcs increased. Comparison with published data on the percolation threshold of symmetric quadratic Bézier curves suggests that when the percolation of slightly curved wires is simulated, the particular choice of curve to mimic the shape of real-world wires is of little importance. Considering the electrical properties, we took into account both the nanowire resistance per unit length and the junction (nanowire/nanowire contact) resistance. Using mean-field approximation (MFA), we derived the total electrical conductance of nanowire-based networks as a function of their geometrical and physical parameters. The MFA predictions have been confirmed by our Monte Carlo numerical simulations. For our random homogeneous and isotropic systems of conductive curved wires, the electric conductance decreased as the wire shape changed from a stick to a ring when the wire length remained fixed. [ABSTRACT FROM AUTHOR]

This paper concern to determine a local energy decay estimate for synchronizable solutions of a coupled system of linear wave equations. Here, it is shown that the energy for such solutions decays at a rate of t−n$$ {t}^{-n} $$ (n$$ n $$ is the space dimension) when the coupling matrix satisfy the positive C1$$ {C}_1 $$‐compatibility condition. [ABSTRACT FROM AUTHOR]

Summary: In this work, two state estimation methods are proposed for a class of discrete‐time descriptor linear systems subject to bounded uncertainties. First, we propose set‐valued estimator algorithm using symmetric boxes to compute rigorous bounds of the system states. The observer gains are calculated using L∞$$ {L}_{\infty } $$ norm to attenuate the effects of the uncertainties and to improve the accuracy of the proposed estimator. Second, to obtain tighter state enclosures, zonotopic set computation are developed instead of interval analysis to design a new set‐valued state estimation algorithm. The performances of the proposed state estimation approaches are highlighted on different illustrative examples. [ABSTRACT FROM AUTHOR]

In this work we study the design of H2$$ {H}_2 $$ filters for phase‐type (PH) semi‐Markov jump linear systems, considering partial information on the system's operating mode and possible parameter uncertainties. With respect to the mode of operation, it is assumed that the state space of the semi‐Markov chain can be written as the union of disjoint sets, called clusters, and the only information available to the filter is which cluster the state of the semi‐Markov chain belongs to. A new linear matrix inequality (LMI) parameterization employing the slack variable technique is introduced for the design of switching full‐order filters according to the cluster observations so that suitable bounds on the H2$$ {H}_2 $$ norm of the estimation error are guaranteed for the uncertain system. If the system is restricted to be a Markov jump linear system and the Markov chain is assumed to be perfectly measured, the design conditions are shown to be also necessary leading to the optimal full‐order filter. Furthermore, the general filter can be particularized into an observer form for the case in which the system matrices are the same within each cluster and there are no parameter uncertainties except for possible uncertainties affecting the transition rates of the jumping process. The paper concludes with an illustrative example in the context of networked control systems. [ABSTRACT FROM AUTHOR]

In this paper we study the unique weighted measurable shadowing property for weighted pseudo-orbits of random systems on Banach spaces with the property that linear part of random system admits a tempered exponential dichotomy. We also prove for linear random systems that the tempered exponential dichotomy is necessary for the unique weighted measurable shadowing property to hold. [ABSTRACT FROM AUTHOR]

This paper is devoted to analyzing a kind of fractional stochastic neutral system (FSNS). Firstly, by introducing the notion of newly defined two-parameter Mittag-Leffler matrix function, we derive the solution of the corresponding linear stochastic system. Subsequently, for the linear case, by virtue of the Grammian matrix, we give a suffcient and necessary condition to guarantee the relatively exact controllability for the addressed case. Furthermore, for the nonlinear one, the relatively exact controllability is obtained by fixed point and explore it via Banach contraction principle. Finally, two examples are provided to intensify our theoretical conclusions. [ABSTRACT FROM AUTHOR]

The randomized sparse Kaczmarz (RSK) method is an algorithm used to calculate sparse solutions for the basis pursuit problem. In this paper, we propose an algorithm framework for computing sparse solutions of linear systems, which includes the sparse Kaczmarz and sparse block Kaczmarz algorithms. In order to overcome the limitations of the l1 penalty, we design an effective and new randomized sparse Kaczmarz algorithm (RSK-MCP) based on the non-convex minimax-concave penalty (MCP) in sparse signal reconstruction. Additionally, we prove that the RSK-MCP algorithm is equivalent to the randomized coordinate descent method for the corresponding dual problem. Based on this result, we demonstrate that the RSK-MCP algorithm exhibits linear convergence, meaning it converges to a sparse solution of the MCP model when the regularization of MCP is a strongly convex function. Numerical experiments indicate that the RSK-MCP algorithm outperforms RSK-L1 in terms of both efficiency and accuracy [ABSTRACT FROM AUTHOR]

This paper is concerned with the exact delay range making input-delay systems unstabilizable. The exact range means that the systems are unstabilizable if and only if the delay is within this range. Contributions of this paper are to characterize the exact range and to present a computation method to derive this range. It is shown that the above range is related to unstable eigenvalues of the system matrix. In the discrete-time case, if none of the eigenvalues of the system matrix is a unit root, then the above range is a finite set. If there exist some eigenvalues which are unit roots, this range may be a finite set or may be composed of several arithmetic progressions. When this range contains finite elements, the number of these elements is bounded by the geometric multiplicities of eigenvalues. When this range contains arithmetic progressions, the number of such progressions is bounded by the above multiplicities. On the other hand, our results can provide an upper bound for the well-known delay margin, which is the maximal delay value achievable by a robust controller to stabilize systems. [ABSTRACT FROM AUTHOR]

In this paper, we study a linear elliptic system involving Navier-type boundary conditions or tangential components in a three-dimensional bounded domain, possibly multiply connected and with boundary possibly nonconnected. We establish an appropriate Inf-Sup condition and then we prove the existence and uniqueness of the generalized solutions in Lp-theory. We also investigate the case of strong solutions. [ABSTRACT FROM AUTHOR]

This article investigates the policy iteration (PI) method for the discounted optimal control (DOC) problem of continuous‐time linear systems. We show the properties and convergence of the PI method. The theory analysis shows that the convergence of PI can be ensured without requiring the initial admissible control gain. The convergence rate of the PI method is provided. An iteration‐termination criterion is established for detecting the stability of the closed‐loop system under the control gain obtained by executing PI. Two kinds of data‐driven implementations are constructed without using prior information of the system dynamics. A simulation example is presented to validated the properties of the PI method. [ABSTRACT FROM AUTHOR]

The structured singular value (SSV), or μ$$ \mu $$, is used to assess the robust stability and performance of an uncertain linear time‐invariant system. Existing algorithms compute upper and lower bounds on the SSV for structured uncertainties that contain repeated (real or complex) scalars and/or nonrepeated complex full‐blocks. This paper presents algorithms to compute bounds on the SSV for the case of repeated complex full‐blocks. This specific class of uncertainty is relevant for the input‐output analysis of many convective systems, such as fluid flows. Specifically, we present a power iteration to compute the SSV lower bound for the case of repeated complex full‐blocks. This generalizes existing power iterations for repeated complex scalars and nonrepeated complex full‐blocks. The upper bound can be formulated as a semi‐definite program (SDP), which we solve using a standard interior‐point method to compute optimal scaling matrices associated with the repeated full‐blocks. Our implementation of the method only requires gradient information, which improves the computational efficiency of the method. Finally, we test our proposed algorithms on an example model of incompressible fluid flow. The proposed methods provide less conservative bounds as compared to prior results, which ignore the repeated full‐block structure. [ABSTRACT FROM AUTHOR]

This article studies the stability problem of discrete‐time positive switched linear systems by introducing a nonzero elements chain method. Compared to existing works, the highlight of this article is reflected in the inclusiveness of the subsystem, that is, each subsystem is allowed to be stable, marginally stable or even unstable. Some sufficient conditions depending on the characteristics of system matrices and the time‐dependent switching signals are established, so as to guarantee the exponential stability of positive switched linear systems. In addition, the main result is extended to the case of time‐varying delay. At last, the superiority of the obtained results is well verified by numerical examples. [ABSTRACT FROM AUTHOR]

This article proposes a two‐step interval estimation method for continuous‐time linear time‐invariant systems with unknown but bounded disturbances and measurement noise. In the first step, H∞$$ {H}_{\infty } $$ method is used to design a robust reduced‐order observer that attenuates the impact of the system uncertainties on the estimation error. In the second step, a non‐conservative interval prediction of the estimation error is achieved by applying a symmetric set‐integration approach. The resulting tight set‐valued state estimation is obtained by combining the point estimation from the reduced‐order observer with the predicted estimation error set provided by the set‐integration method. Compared to the conventional methods, the proposed method is built on less‐restrictive design conditions, that allow one to deal with a large range of applications. On the other hand, numerical simulations show that the proposed method offers more precise interval estimation compared to an optimal reduced‐order interval observer selected from recent literature. [ABSTRACT FROM AUTHOR]

This paper proposes a novel Difference-of-Convex (DC) decomposition for polynomials using a power-sum representation, achieved by solving a sparse linear system. We introduce the Boosted DCA with Exact Line Search ( BDCA e ) for addressing linearly constrained polynomial programs within the DC framework. Notably, we demonstrate that the exact line search equates to determining the roots of a univariate polynomial in an interval, with coefficients being computed explicitly based on the power-sum DC decompositions. The subsequential convergence of BDCA e to critical points is proven, and its convergence rate under the Kurdyka–Łojasiewicz property is established. To efficiently tackle the convex subproblems, we integrate the Fast Dual Proximal Gradient method by exploiting the separable block structure of the power-sum DC decompositions. We validate our approach through numerical experiments on the Mean–Variance–Skewness–Kurtosis portfolio optimization model and box-constrained polynomial optimization problems. Comparative analysis of BDCA e against DCA, BDCA with Armijo line search, UDCA, and UBDCA with projective DC decomposition, alongside standard nonlinear optimization solvers FMINCON and FILTERSD, substantiates the efficiency of our proposed approach. [ABSTRACT FROM AUTHOR]

We extend the Duffin–Schaeffer conjecture to the setting of systems of m$m$ linear forms in n$n$ variables. That is, we establish a criterion to determine whether, for a given rate of approximation, almost all or almost no n$n$‐by‐m$m$ systems of linear forms are approximable at that rate using integer vectors satisfying a natural coprimality condition. When m=n=1$m=n=1$, this is the classical 1941 Duffin–Schaeffer conjecture, which was proved in 2020 by Koukoulopoulos and Maynard. Pollington and Vaughan proved the higher dimensional version, where m>1$m>1$ and n=1$n=1$, in 1990. The general statement we prove here was conjectured in 2009 by Beresnevich, Bernik, Dodson, and Velani. For approximations with no coprimality requirement, they also conjectured a generalized version of Catlin's conjecture, and in 2010, Beresnevich and Velani proved the m>1$m>1$ cases of that. Catlin's classical conjecture, where m=n=1$m=n=1$, follows from the classical Duffin–Schaeffer conjecture. The remaining cases of the generalized version, where m=1$m=1$ and n>1$n>1$, follow from our main result. Finally, through the mass transference principle, our main results imply their Hausdorff measure analogues, which were also conjectured by Beresnevich et al. [ABSTRACT FROM AUTHOR]

The usual process for solving a nonhomogeneous system of linear differential equations is to find the general complementary homogeneous solution first and then construct a particular solution from it. When can we flip the script on this process and compute a particular solution first? This paper argues this is possible when the nonhomogeneous forcing function is analytic. It presents two different series expansions for this particular particular solution using power series methods, and reveals a surprising connection to the more familiar variation of parameters formula. [ABSTRACT FROM AUTHOR]

This paper is devoted to study the symmetry and monotonicity of positive solutions for linear coupling elliptic systems in a ball in $ \mathbf{R}^N $ R N . Using the Alexandrov–Serrin method of moving planes combined with the strong maximum principle, we prove that the solutions of elliptic systems with linear couplings in a ball are symmetric w.r.t. 0 and radially decreasing. For our problems, the tangential gradient of solutions and the coupling conditions play important roles in using the moving plane method. Our results on the symmetry of solutions are further research based on the existence of solutions in [1]. [ABSTRACT FROM AUTHOR]

This paper proposes a novel free-weighting matrix (FWM) approach for the event-triggered (ET) consensus and formation control of multi-agent systems over a directed graph. A new FWM equality for the consensus error is proposed. This FWM is applied to attain coherent behavior of systems via a fully distributed ET approach. The proposed approach is also extended for ET formation control to attain a specific pattern of agents. Furthermore, an FWM-based consensus controller design with practical ET cooperative control is considered to address consensus and formation control problems by using the in-degree information of a node rather than the algebraic connectivity of a graph or adaptation of parameters. Additionally, the controller gain is formulated using a more general transformation compared to that used in classical approaches. Moreover, the proposed ET scheme for the designed controller is applied on the transmission side to preserve computational resources by eliminating the Zeno behavior. In contrast to the existing methods, the proposed approach (i) provides a fully distributed protocol without using central information of network or adaptive gain; (ii) provides a practical ET mechanism to assure effective bandwidth and fewer energy consumption; and (iii) can be applied to the ET formation problem for mobile agents. Simulations involving six aerial vehicles are performed to demonstrate the efficacy of the proposed cooperative control strategy. [ABSTRACT FROM AUTHOR]

In this work, preconditioning strategies are developed in the context of generalized continuum formulations used to regularize multifield models for simulating localized failure of quasi‐brittle materials. Specifically, a micropolar continuum extended by a nonlocal damage formulation is considered for regularizing both, shear dominated failure and tensile cracking. For such models, additional microrotation and nonlocal damage fields, and their interactions, increase the complexity and size of the arising linear systems. This increases the demand for specialized preconditioning strategies when iterative solvers are adopted. Herein, a block preconditioning strategy, employing algebraic multigrid methods (AMG) for approximating the application of sub‐block inverses, is developed and tested in three steps. First, a block preconditioner is introduced for linear systems resulting from micropolar models. For this case, a simple sparse Schur complement approximation, which is practical to compute, is proposed and analyzed. It is tested for three different discretizations. Second, the developed preconditioner is extended to reflect the additional nonlocal damage field. This extended three‐field preconditioner is tested on the simulation of a compression test on a sandstone sample. All numerical tests show an improved performance of the block preconditioning approach in comparison to a black‐box monolithic AMG approach. Finally, a problem‐adapted preconditioner setup strategy is proposed, which involves a reuse of the multigrid hierarchy during nonlinear iterations, and additionally accounts for the different stages occurring in the simulation of localized failure. The problem‐adapted preconditioning strategy has the potential to further reduce the total computation time. [ABSTRACT FROM AUTHOR]

The advent of easy access to large amount of data has sparked interest in directly developing the relationships between input and output of dynamic systems. A challenge is that in addition to the applied input and the measured output, the dynamics can also depend on hidden states that are not directly measured. In general, it is unclear what type of data, such as past input and or past output is needed, to learn inverse operators (that predict the input needed to track a desired output for control purposes) with a desired precision. The main contribution of this work is to show that, irrespective of the selected model, removing the hidden-state dependence and achieving a desired precision of inverse operators require (i) a sufficiently-long past history of the output and (ii) sufficiently-precise estimates of the output's instantaneous time derivatives that are necessary and sufficient for linear systems, and under some conditions, for nonlinear systems. This insight, about the required observables (output history and derivative) for removing the hidden-state dependence and achieving precision, is used to develop a data-enabled algorithm to learn the inverse operator for multi-input multi-output square systems. Simulation examples are used to illustrate that neural nets (with universal approximation property) can learn the inverse operator with sufficient precision only if the required observables, identified in this work, are included in training. [ABSTRACT FROM AUTHOR]

This work examines a class of distributed linear systems that fit a tree type of hierarchical structure. Therefore, unidirectional information flow from a subsystem higher in the hierarchy impacts subsystems lower in the hierarchy. While prior efforts at control focused on feedback solutions to these systems, this effort introduces Iterative Learning Control as a feedforward controller. This is termed hierarchical iterative learning control (HILC). This HILC can be implemented in parallel with feedback algorithms or can be used in a series feedforward manner assuming stable, or stabilized, subsystems. An augmentation of the standard learning update operators provides stability and monotonic convergence for the proposed approach. A simulation case study highlights the performance of the proposed design approach. [ABSTRACT FROM AUTHOR]

In this paper, we present a comprehensive framework for studying Krylov subspace methods used to solve the linear system A x = f . These methods aim to achieve convergence within a specified number of iterations, denoted by m, given a particular initial estimate vector x 0 and its corresponding residual r 0 = f - A x 0 . Our analysis focuses on the minimal polynomial Φ m of degree m of A for the vector r 0 . We establish that these methods encompass Petrov-Galerkin methods and minimal seminorm methods as special cases. Additionally, we demonstrate that minimal seminorm methods satisfy implicit Petrov-Galerkin conditions. We provide a general formulation for the iterates based on generalized inverses. The choice of a specific left inverse and the method of constructing the Krylov basis are crucial distinguishing factors among different Krylov subspace methods. We describe and analyze the mathematical properties of these methods, emphasizing their dependency on two matrices. Notably, we prove that CMRH and QMR, as specific instances, also satisfy implicit Petrov-Galerkin orthogonality conditions. Furthermore, we explore techniques to improve the convergence behavior of these methods by carefully selecting vectors in their implementations. Through our investigation, we aim to deepen the understanding of Krylov subspace methods, provide insights into their convergence properties, and identify potential enhancements. We also consider some Krylov subspace methods, which are of product-type methods. In this case, the kth residual r k associated with the approximation x k of the exact solution is given by r k = Ψ k (A) Φ k (A) r 0 , and Ψ k is a polynomial of fixed or variable degree. We will examine particular choices of Ψ k involving local convergence, smoothing, fixed memory, and cost for each iteration. We will also give an enhancement of some product-type methods such as CGS. To illustrate the performance of the derived algorithms, we provide some numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we consider the forward rounding errors of the partial pivoting quaternion LU decomposition. To carry out error analysis of algorithms in quaternion arithmetic, we first propose the accuracy model for the basic quaternion arithmetic operations by the real structure-preserving method. And then, we study the forward rounding errors of the partial pivoting quaternion LU decomposition. Finally, by using the results of the forward rounding errors of the partial pivoting quaternion LU decomposition, we analyze the forward rounding errors of the nonsingular quaternion linear system. [ABSTRACT FROM AUTHOR]

Coupled multi-physics problems are encountered in countless applications and pose significant numerical challenges. In a broad sense, one can categorise the numerical solution strategies for coupled problems into two classes: monolithic approaches and sequential (also known as split, decoupled, partitioned or segregated) approaches. The monolithic approaches treat the entire problem as one, whereas the sequential approaches are iterative decoupling techniques where the different sub-problems are treated separately. Although the monolithic approaches often offer the most robust solution strategies, they tend to require ad-hoc preconditioners and numerical implementations. Sequential methods, on the other hand, offer the possibility to add and remove equations from the model flexibly and rely on existing black-box solvers for each specific equation. Furthermore, when problems are non-linear, inner iterations need to be performed even in monolithic solvers, making the sequential approaches an even more viable alternative. The cost of running inner iterations to recover the multi-physics coupling could, however, easily become prohibitive. Moreover, the sequential approaches might not converge at all. In this work, we present a general formulation of splitting schemes for continuous operators with arbitrary implicit/explicit splitting, like in standard iterative methods for linear systems. By introducing a generic relaxation operator, we find the conditions for the convergence of the iterative schemes. We show how the relaxation operator can be thought of as a preconditioner and constructed based on an approximate Schur complement. We propose a Schur-based Partial Jacobi relaxation operator to stabilise the coupling and show its effectiveness. Although we mainly focus on scalar-scalar linear problems, most results are easily extended to non-linear and higher-dimensional problems. The schemes presented are not explicitly dependent on any particular discretisation methodologies. Numerical tests (1D and 2D) for two PDE systems, namely the Dual-Porosity model and a Quad-Laplacian operator, are carried out to investigate the practical implications of the theoretical results. [ABSTRACT FROM AUTHOR]

Algebraic multigrid (AMG) methods are often robust and effective solvers for solving the large and sparse linear systems that arise from discretized PDEs and other problems, relying on heuristic graph algorithms to achieve their performance. Reduction‐based AMG (AMGr) algorithms attempt to formalize these heuristics by providing two‐level convergence bounds that depend concretely on properties of the partitioning of the given matrix into its fine‐ and coarse‐grid degrees of freedom. MacLachlan and Saad (SISC 2007) proved that the AMGr method yields provably robust two‐level convergence for symmetric and positive‐definite matrices that are diagonally dominant, with a convergence factor bounded as a function of a coarsening parameter. However, when applying AMGr algorithms to matrices that are not diagonally dominant, not only do the convergence factor bounds not hold, but measured performance is notably degraded. Here, we present modifications to the classical AMGr algorithm that improve its performance on matrices that are not diagonally dominant, making use of strength of connection, sparse approximate inverse (SPAI) techniques, and interpolation truncation and rescaling, to improve robustness while maintaining control of the algorithmic costs. We present numerical results demonstrating the robustness of this approach for both classical isotropic diffusion problems and for non‐diagonally dominant systems coming from anisotropic diffusion. [ABSTRACT FROM AUTHOR]

The PageRank model, which was first proposed by Google for its web search engine application, has since become a popular computational tool in a wide range of scientific fields, including chemistry, bioinformatics, neuroscience, bibliometrics, social networks, and others. PageRank calculations necessitate the use of fast computational techniques with low algorithmic and memory complexity. In recent years, much attention has been paid to Krylov subspace algorithms for solving difficult PageRank linear systems, such as those with large damping parameters close to one. In this article, we examine the full orthogonalization method (FOM). We present a convergence study of the method that extends and clarifies part of the conclusions reached in Zhang et al. (J Comput Appl Math. 2016; 296:397–409.). Furthermore, we demonstrate that FOM is breakdown free when solving singular PageRank linear systems with index one and we investigate the effect of using weighted inner‐products instead of conventional inner‐products in the orthonormalization procedure on FOM convergence. Finally, we develop a shifted polynomial preconditioner that takes advantage of the special structure of the PageRank linear system and has a good ability to cluster most of the eigenvalues, making it a good choice for an iterative method like FOM or GMRES. Numerical experiments are presented to support the theoretical findings and to evaluate the performance of the new weighted preconditioned FOM PageRank solver in comparison to other established solvers for this class of problem, including conventional stationary methods, hybrid combinations of stationary and Krylov subspace methods, and multi‐step splitting strategies. [ABSTRACT FROM AUTHOR]

For solving large sparse systems of linear equations, we construct a paradigm of two‐step matrix splitting iteration methods and analyze its convergence property for the nonsingular and the positive‐definite matrix class. This two‐step matrix splitting iteration paradigm adopts only one single splitting of the coefficient matrix, together with several arbitrary iteration parameters. Hence, it can be constructed easily in actual applications, and can also recover a number of representatives of the existing two‐step matrix splitting iteration methods. This result provides systematic treatment for the two‐step matrix splitting iteration methods, establishes rigorous theory for their asymptotic convergence, and enriches algorithmic family of the linear iteration solvers, for the iterative solutions of large sparse linear systems. [ABSTRACT FROM AUTHOR]

State estimation is an essential issue in modern industry, which is used in a wide range of fields. Measurements corrupted by outliers have become an inevitable phenomenon leading to a degradation in the performance of estimators. To mitigate the negative impact of measurement outliers, the development of estimators with outlier suppression capabilities is crucial. In this paper, we propose a simple yet effective outlier suppression technique for discrete-time linear systems in the framework of moving horizon estimation (MHE) combined with the event-triggered mechanism. This is a novel attempt to integrate event-triggered outlier detection and correction mechanisms with the MHE approach. Specifically, we propose two event-based outlier detection methods that can effectively identify measurement outliers. Subsequently, two outlier correction techniques are designed according to the different detection schemes. Lastly, two MHE algorithms are proposed based on the corrected measurements. The proposed algorithms are applied to a target tracking simulation and compared with existing advanced outlier-robust estimators, demonstrating the effectiveness and superiority of the proposed algorithms. [ABSTRACT FROM AUTHOR]

For solving a common fixed point of a finite family of G-nonexpansive mappings, we propose a parallel inertial SP-iteration monotone hybrid algorithm (PISPMHA). Weak convergence theorem is established for PISPMHA in Hilbert spaces endowed with graphs. Convergence behavior of PISPMHA is analyzed and discussed. As applications, we apply PISPMHA to solve linear system, differential, and signal recovery problems. [ABSTRACT FROM AUTHOR]

A general multilayer diffusion problem subject to inhomogeneous boundary conditions with time-dependent boundaries is considered. The embedding method is used to find the analytical solution, which is expressed in terms of time-dependent functions that satisfy a system of linear Volterra integral equations of the first kind. A boundary element method is developed to numerically solve the integral equations and results of numerical simulations are presented. The proposed approach is applicable to multilayer diffusion problems defined on bounded and unbounded spatial domains. • This article considers a unified approach using the embedding method to find the solution of multilayer diffusion problems. • The boundaries may depend on time, and both bounded and unbounded spatial domains can be handled using the same technique. • The solution is in terms of functions that satisfy a system of linear Volterra integral equations of the first kind. • Examples are given for the two-layer diffusion problem on a semi-infinite domain. • A new three-parameter special function in series form is introduced to obtain explicit analytical solutions. [ABSTRACT FROM AUTHOR]

For numerical approximation the reformulation of a PDE as a residual minimisation problem has the advantages that the resulting linear system is symmetric positive definite, and that the norm of the residual provides an a posteriori error estimator. Furthermore, it allows for the treatment of general inhomogeneous boundary conditions. In many minimal residual formulations, however, one or more terms of the residual are measured in negative or fractional Sobolev norms. In this work, we provide a general approach to replace those norms by efficiently evaluable expressions without sacrificing quasi-optimality of the resulting numerical solution. We exemplify our approach by verifying the necessary inf-sup conditions for four formulations of a model second order elliptic equation with inhomogeneous Dirichlet and/or Neumann boundary conditions. We report on numerical experiments for the Poisson problem with mixed inhomogeneous Dirichlet and Neumann boundary conditions in an ultra-weak first order system formulation. [ABSTRACT FROM AUTHOR]

Purpose: The quality of on‐board imaging systems, including cone‐beam computed tomography (CBCT), plays a vital role in image‐guided radiation therapy (IGRT) and adaptive radiotherapy. Recently, there has been an upgrade of the CBCT systems fused in the O‐ring linear accelerators called HyperSight, featuring a high imaging performance. As the characterization of a new imaging system is essential, we evaluated the image quality of the HyperSight system by comparing it with Halcyon 3.0 CBCT and providing benchmark data for routine imaging quality assurance. Methods: The HyperSight features ultra‐fast scan time, a larger kilovoltage (kV) detector, a more substantial kV tube, and an advanced reconstruction algorithm. Imaging protocols in the two modes of operation, treatment mode with IGRT and the CBCT for planning (CBCTp) mode were evaluated and compared with Halcyon 3.0 CBCT. Image quality metrics, including spatial resolution, contrast resolution, uniformity, noise, computed tomography (CT) number linearity, and calibration error, were assessed using a Catphan and an electron density phantom and analyzed with TotalQA software. Results: HyperSight demonstrated substantial improvements in contrast‐to‐noise ratio and noise in both IGRT and CBCTp modes compared to Halcyon 3.0 CBCT. CT number calibration error of HyperSight CBCTp mode (1.06%) closely matches that of a full CT scanner (0.72%), making it suitable for adaptive planning. In addition, the advanced hardware of HyperSight, such as ultra‐fast scan time (5.9 s) or 2.5 times larger heat unit capacity, enhanced the clinical efficiency in our experience. Conclusions: HyperSight represented a significant advancement in CBCT imaging. With its image quality, CT number accuracy, and ultra‐fast scans, HyperSight has a potential to transform patient care and treatment outcomes. The enhanced scan speed and image quality of HyperSight are expected to significantly improve the quality and efficiency of treatment, particularly benefiting patients. [ABSTRACT FROM AUTHOR]

This study presents novel formulations for ℋ∞$$ {\mathscr{H}}_{\infty } $$ robust state‐feedback control synthesis for discrete‐time linear time‐invariant systems based on linear matrix inequalities. The proposed formulations require searching for two adjustment matrices. The synthesis formulations include other formulations from the literature as particular cases according to specific values of the adjustment matrices. We propose the application of evolutionary optimization to determine the optimal values of these two adjustment matrices. One approach to obtaining a single‐step formulation for static output‐feedback control synthesis is to transform a state‐feedback control synthesis formulation via the simple change of variables. The alteration of variables considered in this study requires an adjustment matrix. The adjustment matrix influences the performance of the resulting controller or the existence of a feasible solution to the problem. Here, we also propose the application of evolutionary optimization to determine the optimal value of this adjustment matrix and the two adjustment matrices of the proposed formulations to obtain the optimal ℋ∞$$ {\mathscr{H}}_{\infty } $$ robust control system. Case studies verify that despite the increased complexity, the proposed formulations and the method required to tune them may be indispensable in achieving a robustly stable control system or an enhanced ℋ∞$$ {\mathscr{H}}_{\infty } $$ performance for more intricate problems. [ABSTRACT FROM AUTHOR]