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This article describes a projection method based on a combination of rational and polynomial approximations for efficiently solving large nonlinear eigenvalue problems. In a first stage the nonlinear matrix function T(λ)$$ T\left(\lambda \right) $$ under consideration is approximated by a matrix polynomial in λ$$ \lambda $$. The error resulting from this polynomial approximation is in turn approximated by rational functions with the help of the Cauchy integral formula. The two approximations are combined and a linearization is performed. A key ingredient of the proposed approach is a projection method that uses subspaces spanned by vectors of the same dimension as that of the original problem instead of that of the linearized problem. A procedure is also presented to automatically select shifts and to partition the region of interest into a few subregions. This allows to subdivide the problem into smaller subproblems that are solved independently. The accuracy of the proposed method is theoretically analyzed and its performance is illustrated with a few test problems that have been discussed in the literature. [ABSTRACT FROM AUTHOR]

Interpolation-based methods are well-established and effective approaches for the efficient generation of accurate reduced-order surrogate models. Common challenges for such methods are the automatic selection of good or even optimal interpolation points and the appropriate size of the reduced-order model. An approach that addresses the first problem for linear, unstructured systems is the iterative rational Krylov algorithm (IRKA), which computes optimal interpolation points through iterative updates by solving linear eigenvalue problems. However, in the case of preserving internal system structures, optimal interpolation points are unknown, and heuristics based on nonlinear eigenvalue problems result in numbers of potential interpolation points that typically exceed the reasonable size of reduced-order systems. In our work, we propose a projection-based iterative interpolation method inspired by IRKA for generally structured systems to adaptively compute near-optimal interpolation points as well as an appropriate size for the reduced-order system. Additionally, the iterative updates of the interpolation points can be chosen such that the reduced-order model provides an accurate approximation in specified frequency ranges of interest. For such applications, our new approach outperforms the established methods in terms of accuracy and computational effort. We show this in numerical examples with different structures. [ABSTRACT FROM AUTHOR]

In this paper, we present a variant of the circumcenter method for the convex feasibility problem (CFP), ensuring finite convergence under a Slater assumption. The method replaces exact projections onto the convex sets with projections onto separating half-spaces, perturbed by positive exogenous parameters that decrease to zero along the iterations. If the perturbation parameters decrease slowly enough, such as the terms of a diverging series, finite convergence is achieved. To the best of our knowledge, this is the first circumcenter method for CFP that guarantees finite convergence. [ABSTRACT FROM AUTHOR]

We propose and analyze a novel approach to construct structure preserving approximations for the Poisson-Nernst-Planck equations, focusing on the positivity preserving and mass conservation properties. The strategy consists of a standard time marching step with a projection (or correction) step to satisfy the desired physical constraints (positivity and mass conservation). Based on the L² projection, we construct a second order Crank-Nicolson type finite difference scheme, which is linear (exclude the very efficient L² projection part), positivity preserving, and mass conserving. Rigorous error estimates in the L² norm are established, which are both second order accurate in space and time. The other choice of projection, e.g., H¹ projection, is discussed. Numerical examples are presented to verify the theoretical results and demonstrate the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

This paper surveys an important class of methods that combine iterative projection methods and variational regularization methods for large-scale inverse problems. Iterative methods such as Krylov subspace methods are invaluable in the numerical linear algebra community and have proved important in solving inverse problems due to their inherent regularizing properties and their ability to handle large-scale problems. Variational regularization describes a broad and important class of methods that are used to obtain reliable solutions to inverse problems, whereby one solves a modified problem that incorporates prior knowledge. Hybrid projection methods combine iterative projection methods with variational regularization techniques in a synergistic way, providing researchers with a powerful computational framework for solving very large inverse problems. Although the idea of a hybrid Krylov method for linear inverse problems goes back to the 1980s, several recent advances on new regularization frameworks and methodologies have made this field ripe for extensions, further analyses, and new applications. In this paper, we provide a practical and accessible introduction to hybrid projection methods in the context of solving large (linear) inverse problems. [ABSTRACT FROM AUTHOR]

This paper presents a projection and contraction method with a double inertial extrapolation step and self-adaptive step sizes to solve variational inequalities with quasi-monotonicity in real Hilbert spaces. Weak and strong convergence results are obtained under some mild conditions. We also give linear convergence results under a special case of our proposed method. Preliminary numerical results show that our proposed method is competitive with other related methods in the literature. [ABSTRACT FROM AUTHOR]

The total variation regularization method was introduced by Rudin, Osher, and Fatemi as an efficient technique for regularizing grayscale images. In this work, we aimed to generalize the total variation method to regularize multidimensional problems such as colour image and video restoration. A degradation model in a tensor format is proposed to recover blurred and noisy colour images and videos. The alternating direction method for multipliers (ADMM) and an optimized form of projection methods have been employed to solve the tensorial total variation minimization problem. The structure of the developed approach allows the selection of the optimal parameter. We use the truncated SVD (TSVD) to reduce the size of the problem and to accelerate the convergence of the algorithm. The convergence analysis of the proposed method is proved using convex optimization. Numerical tests for image and video restoration are given showing the effectiveness of the proposed approaches. [ABSTRACT FROM AUTHOR]

In this paper, we present a successive centralization process for the circumcentered-reflection method with several control sequences for solving the convex feasibility problem in Euclidean space. Assuming that a standard error bound holds, we prove the linear convergence of the method with the most violated constraint control sequence. Moreover, under additional smoothness assumptions on the target sets, we establish the superlinear convergence. Numerical experiments confirm the efficiency of our method. [ABSTRACT FROM AUTHOR]

We use supervised machine learning to approximate the expectations typically contained in the optimality conditions of an economic model in the spirit of the parameterized expectations algorithm (PEA) with stochastic simulation. When the set of state variables is generated by a stochastic simulation, it is likely to suffer from multicollinearity. We show that a neural network‐based expectations algorithm can deal efficiently with multicollinearity by extending the optimal debt management problem studied by Faraglia, Marcet, Oikonomou, and Scott (2019) to four maturities. We find that the optimal policy prescribes an active role for the newly added medium‐term maturities, enabling the planner to raise financial income without increasing its total borrowing in response to expenditure shocks. Through this mechanism, the government effectively subsidizes the private sector during recessions. [ABSTRACT FROM AUTHOR]

Different notions on regularity of sets and of collection of sets play an important role in the analysis of the convergence of projection algorithms in nonconvex scenarios. While some projection algorithms can be applied to feasibility problems defined by finitely many sets, some other require the use of a product space reformulation to construct equivalent problems with two sets. In this work we analyze how some regularity properties are preserved under a reformulation in a product space of reduced dimension. This allows us to establish local linear convergence of parallel projection methods which are constructed through this reformulation. [ABSTRACT FROM AUTHOR]

Dimensionality reduction methods can be used to project high-dimensional data into low-dimensional space. If the output space is restricted to two dimensions, the result is a scatter plot whose goal is to present insightful visualizations of distance- and density-based structures. The topological invariance of dimension indicates that the two-dimensional similarities in the scatter plot cannot coercively represent high-dimensional distances. In praxis, projections of several datasets with distance- and density-based structures show a misleading interpretation of the underlying structures. The examples outline that the evaluation of projections remains essential. Here, 19 unsupervised quality measurements (QM) are grouped into semantic classes with the aid of graph theory. We use three representative benchmark datasets to show that QMs fail to evaluate the projections of straightforward structures when common methods such as Principal Component Analysis (PCA), Uniform Manifold Approximation projection, or t-distributed stochastic neighbor embedding (t-SNE) are applied. This work shows that unsupervised QMs are biased towards assumed underlying structures. Based on insights gained from graph theory, we propose a new quality measurement called the Gabriel Classification Error (GCE). This work demonstrates that GCE can make an unbiased evaluation of projections. The GCE is accessible within the R package DR quality available on CRAN. [ABSTRACT FROM AUTHOR]

To enforce the conservation of mass principle, a pressure Poisson equation arises in the numerical solution of incompressible fluid flow using the pressure-based segregated algorithms such as projection methods. For unsteady flows, the pressure Poisson equation is solved at each time step usually in physical space using iterative solvers, and the resulting pressure gradient is then applied to make the velocity field divergence-free. It is generally accepted that this pressure-correction stage is the most time-consuming part of the flow solver and any meaningful acceleration would contribute significantly to the overall computational efficiency. The objective of the present work was to develop a fast hybrid pressure-correction algorithm for numerical simulation of incompressible flows around obstacles in the context of projection methods. The key idea is to adopt different numerical methods/discretisations in the sub-steps of projection methods. Here, a classical second-order time-marching projection method, which consists of two sub-steps, was chosen for the purposes of demonstration. In the first sub-step, the momentum equations were discretised on unstructured grids and solved by conventional numerical methods, here a meshless method. In the second sub-step (pressure-correction), the proposed algorithm adopts a double-discretisation system and combines the weighted least-squares approximation with the essence of immersed boundary methods. Such a design allowed us to develop an FFT-based solver to speed up the solution of the pressure Poisson equation for flow cases with obstacles, while keeping the implementation of the boundary conditions for the momentum equations as easy as conventional numerical methods do with unstructured grids. The numerical experiments of five test cases were performed to verify and validate the proposed hybrid algorithm and evaluate its computational performance. The results showed that the new FFT-based hybrid algorithm works and is robust, and it was significantly faster than the multigrid-based reference method. The hybrid algorithm opens an avenue for the development of next-generation high-performance parallel computational fluid dynamics solvers for incompressible flows. [ABSTRACT FROM AUTHOR]

In this paper, we mainly propose a new Bregman projection method with a different self-adaptive process for solving variational inequalities in a real reflexive Banach space. Exactly, we obtain that the iterative sequence generated by our new algorithm converges strongly to an element of solution set for the variational inequality problem. Our algorithm is interesting and easy to implement in numerical experiments because it has only one projection and does not need to know the Lipschitz constant of the considered operator in advance. The results obtained in this paper can be considered as an improvement and supplement of many recent ones in the field. [ABSTRACT FROM AUTHOR]

This paper proposes a modified D-iteration to approximate the solutions of three quasinonexpansive multivalued mappings in a real Hilbert space. Due to the incorporation of an inertial step in the iteration, the sequence generated by the modified method converges faster to the common fixed point of the mappings. Furthermore, the generated sequence strongly converges to the required solution using a shrinking technique. Numerical results obtained indicate that the proposed iteration is computationally effcient and outperforms the standard forward-backward with inertial step. [ABSTRACT FROM AUTHOR]

In this paper, we develop and analyze a finite element projection method for magnetohydrodynamics equations in Lipschitz domain. A fully discrete scheme based on Euler semi‐implicit method is proposed, in which continuous elements are used to approximate the Navier–Stokes equations and H(curl) conforming Nédélec edge elements are used to approximate the magnetic equation. One key point of the projection method is to be compatible with two different spaces for calculating velocity, which leads one to obtain the pressure by solving a Poisson equation. The results show that the proposed projection scheme meets a discrete energy stability. In addition, with the help of a proper regularity hypothesis for the exact solution, this paper provides a rigorous optimal error analysis of velocity, pressure and magnetic induction. Finally, several numerical examples are performed to demonstrate both accuracy and efficiency of our proposed scheme. [ABSTRACT FROM AUTHOR]

Neural field models are nonlinear integro-differential equations for the evolution of neuronal activity, and they are a prototypical large-scale, coarse-grained neuronal model in continuum cortices. Neural fields are often simulated heuristically and, in spite of their popularity in mathematical neuroscience, their numerical analysis is not yet fully established. We introduce generic projection methods for neural fields and derive a priori error bounds for these schemes. We extend an existing framework for stationary integral equations to the time-dependent case, which is relevant for neuroscience applications. We find that the convergence rate of a projection scheme for a neural field is determined to a great extent by the convergence rate of the projection operator. This abstract analysis, which unifies the treatment of collocation and Galerkin schemes, is carried out in operator form, without resorting to quadrature rules for the integral term, which are introduced only at a later stage, and whose choice depends on the choice of the projector. Using an elementary time stepper as an example, we demonstrate that the error in a time stepper has two separate contributions: one from the projector and one from the time discretization. We give examples of concrete projection methods: two collocation schemes (piecewise linear and spectral collocation) and two Galerkin schemes (finite element and spectral Galerkin); for each of them we derive error bounds from the general theory, introduce several discrete variants, provide implementation details, and present reproducible convergence tests. [ABSTRACT FROM AUTHOR]

The inertial projection and contraction method and the inertial forward-backward-forward method have been studied for solving variational inequality problems due to the ability of these methods to work well with only one projection per iteration during computations. The inertial factor in these methods is chosen to be less than 1 in most of the papers that studied these methods. It is known from a computational point of view that the efficiency of these methods improves as the inertial factor approaches 1. In this paper, we modify both the inertial projection and contraction method and the inertial forward-backward-forward method with the possibility of inertial factor taken as 1. Our proposed methods also involve a step-size rule which does not depend on the Lipschitz constant of the cost function and without any line search rule. We analyse the weak convergence of the methods under appropriate conditions. We also modify the proposed methods so that strong convergence is obtained. Preliminary computational results show that our proposed methods are promising and show better performance than some variants of inertial projection and contraction methods and inertial forward-backward-forward methods where the inertial factor is assumed to be less than 1. [ABSTRACT FROM AUTHOR]

We propose a new projection-type method with inertial extrapolation for solving pseudo-monotone and Lipschitz continuous variational inequalities in Hilbert spaces. The proposed method does not require the knowledge of the Lipschitz constant as well as the sequential weak continuity of the corresponding operator. We introduce a self-adaptive procedure, which generates dynamic step-sizes converging to a positive constant. It is proved that the sequence generated by the proposed method converges weakly to a solution of the considered variational inequality with the nonasymptotic O(1/n) convergence rate. Moreover, the linear convergence is established under strong pseudo-monotonicity and Lipschitz continuity assumptions. Numerical a exmples for solving a class of Nash–Cournot oligopolistic market equilibrium model and a network equilibrium flow problem are given illustrating the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

For many problems, some of which are reviewed in the paper, popular algorithms like Douglas–Rachford (DR), ADMM, and FISTA produce approximating sequences that show signs of spiraling toward the solution. We present a meta-algorithm that exploits such dynamics to potentially enhance performance. The strategy of this meta-algorithm is to iteratively build and minimize surrogates for the Lyapunov function that captures those dynamics. As a first motivating application, we show that for prototypical feasibility problems the circumcentered-reflection method, subgradient projections, and Newton–Raphson are all describable as gradient-based methods for minimizing Lyapunov functions constructed for DR operators, with the former returning the minimizers of spherical surrogates for the Lyapunov function. As a second motivating application, we introduce a new method that shares these properties but with the added advantages that it: (1) does not rely on subproblems (e.g. reflections) and so may be applied for any operator whose iterates have the spiraling property; (2) provably has the aforementioned Lyapunov properties with few structural assumptions and so is generically suitable for primal/dual implementation; and (3) maps spaces of reduced dimension into themselves whenever the original operator does. This makes possible the first primal/dual implementation of a method that seeks the center of spiraling iterates. We describe this method, and provide a computed example (basis pursuit). [ABSTRACT FROM AUTHOR]

In this work, we construct two numerical approximations methods to deal with approximations of a linear Fredholm integro-differential equation. In order to show the existence and uniqueness of the solution we start by the reformulation of the integro-differential equation to a system of integral equations. We explain the general framework of the projection method which helps to clarify the basic ideas of the Collocation and Kantorovich methods. We introduce a new iterative method to avoid inversing the block operator matrix. Next, we apply the iterative projection methods and we present theorems to show the convergence of the constructed solutions to the exact solution. Finally, to observe the error behavior of the two methods, we give two numerical examples. [ABSTRACT FROM AUTHOR]

In this paper we propose a product space reformulation to transform monotone inclusions described by finitely many operators on a Hilbert space into equivalent two-operator problems. Our approach relies on Pierra's classical reformulation with a different decomposition, which results in a reduction of the dimension of the outcoming product Hilbert space. We discuss the case of not necessarily convex feasibility and best approximation problems. By applying existing splitting methods to the proposed reformulation we obtain new parallel variants of them with a reduction in the number of variables. The convergence of the new algorithms is straightforwardly derived with no further assumptions. The computational advantage is illustrated through some numerical experiments. [ABSTRACT FROM AUTHOR]

Recently, circumcentering reflection method (CRM) has been introduced for solving the feasibility problem of finding a point in the intersection of closed constraint sets. It is closely related with Douglas–Rachford method (DR). We prove local convergence of CRM in the same prototypical settings of most theoretical analysis of regular nonconvex DR, whose consideration is made natural by the geometry of the phase retrieval problem. For the purpose, we show that CRM is related to the method of subgradient projections. For many cases when DR is known to converge to a feasible point, we establish that CRM locally provides a better convergence rate. As a root finder, we show that CRM has local convergence whenever Newton–Raphson method does, has quadratic rate whenever Newton–Raphson method does, and exhibits superlinear convergence in many cases when Newton–Raphson method fails to converge at all. We also obtain explicit regions of convergence. As an interesting aside, we demonstrate local convergence of CRM to feasible points in cases when DR converges to fixed points that are not feasible. We demonstrate an extension in higher dimensions, and use it to obtain convergence rate guarantees for sphere and subspace feasibility problems. Armed with these guarantees, we experimentally discover that CRM is highly sensitive to compounding numerical error that may cause it to achieve worse rates than those guaranteed by theory. We then introduce a numerical modification that enables CRM to achieve the theoretically guaranteed rates. Any future works that study CRM for product space formulations of feasibility problems should take note of this sensitivity and account for it in numerical implementations. [ABSTRACT FROM AUTHOR]

This work proposes a valuable and successful strategy for approximating the solutions to the Bagley–Torvik system, which plays an essential role in fractional calculus. The Caputo sense is used to derive the basic conformable fractional. The Bagley–Torvik problem is numerically solved in this study using an effective symmetric projection method. From this symmetry, there are some interesting original results. The proposed approach has two key benefits. We began by converting the connected fractional Bagley–Torvik equations into two fractional-order Bagley–Torvik equations, which we then solved using the current method. Second, two linear equation systems are solved to obtain approximate solutions. [ABSTRACT FROM AUTHOR]

Training a neural network for semantic segmentation with many images and pixel-level segmentations is a well-established computer-vision technique. When pixel-level segmentations are unavailable, weak supervision or prior information like bounding boxes and the size/shape of objects still enables training a network. Directly including prior knowledge on the segmentations means constraining the network output. This complicates the possible optimization strategies because the regularization then acts on the non-linear neural-network function output and not on the optimization variables. We present a new algorithm to include prior information via constraints on the network output, implemented via projection-based point-to-set distance functions, that are differentiable and always have the same functional form for the derivative. Thus, there is no need to adapt penalty functions or algorithms to various constraints. The distance function's differentiability also avoids issues related to constraining properties typically associated with non-differentiable penalties. We show that by explicitly taking a general neural network structure into account, the Lagrangian for the problem 'naturally' decouples the constraints and neural network, which allows for a gradient computation via backpropagation/adjoint-state as is common in deep learning. We present a suite of constraint sets suitable for segmentation problems. The numerical experiments show that learning from constraint sets applies to the broader imaging sciences, including visual and non-visual imagery, even when training a network for a single example. [ABSTRACT FROM AUTHOR]

Iterative projection methods may become trapped at non-solutions when the constraint sets are nonconvex. Two kinds of parameters are available to help avoid this behavior and this study gives examples of both. The first kind of parameter, called a hyperparameter, includes any kind of parameter that appears in the definition of the iteration rule itself. The second kind comprises metric parameters in the definition of the constraint sets, a feature that arises when the problem to be solved has two or more kinds of variables. Through examples we show the importance of properly tuning both kinds of parameters and offer heuristic interpretations of the observed behavior. [ABSTRACT FROM AUTHOR]

We develop a continuum model for the dynamics of grain boundaries in three dimensions that incorporates the motion and reaction of the constituent dislocations. The continuum model is based on a simple representation of densities of curved dislocations on the grain boundary. Illposedness due to nonconvexity of the total energy is fixed by a numerical treatment based on a projection method that maintains the connectivity of the constituent dislocations. An efficient simulation method is developed, in which the critical but computationally expensive long-range interaction of dislocations is replaced by another projection formulation that maintains the constraint of equilibrium of the dislocation structure described by the Frank's formula. This continuum model is able to describe the grain boundary motion and grain rotation due to both coupling and sliding effects, to which the classical motion by mean curvature model does not apply. Comparisons with atomistic simulation results show that our continuum model is able to give excellent predictions of evolutions of low angle grain boundaries and their dislocation structures. [ABSTRACT FROM AUTHOR]

Classical Krylov subspace projection methods for the solution of linear problem A x = b output an approximate solution x ~ ≃ x . Recently, it has been recognized that projection methods can be understood from a statistical perspective. These probabilistic projection methods return a distribution p (x ~) in place of a point estimate x ~ . The resulting uncertainty, codified as a distribution, can, in theory, be meaningfully combined with other uncertainties, can be propagated through computational pipelines, and can be used in the framework of probabilistic decision theory. The problem we address is that the current probabilistic projection methods lead to the poorly calibrated posterior distribution. We improve the covariance matrix from previous works in a way that it does not contain such undesirable objects as A - 1 or A - 1 A - T , results in nontrivial uncertainty, and reproduces an arbitrary projection method as a mean of the posterior distribution. We also propose a variant that is numerically inexpensive in the case the uncertainty is calibrated a priori. Since it usually is not, we put forward a practical way to calibrate uncertainty that performs reasonably well, albeit at the expense of roughly doubling the numerical cost of the underlying projection method. [ABSTRACT FROM AUTHOR]

A common representation of a three dimensional object in computer applications, such as graphics and design, is in the form of a triangular mesh. In many instances, individual or groups of triangles in such representation need to satisfy spatial constraints that are imposed either by observation from the real world, or by concrete design specifications of the object. As these problems tend to be of large scale, choosing a mathematical optimization approach can be particularly challenging. In this paper, we model various geometric constraints as convex sets in Euclidean spaces, and find the corresponding projections in closed forms. We also present an interesting idea to successfully manoeuvre around some important non-convex constraints while still preserving the intrinsic nature of the original design problem. We then use these constructions in modern first-order splitting methods to find optimal solutions. [ABSTRACT FROM AUTHOR]

The proof of Lemma 5 in de Frutos et al. (J Sci Comput 80: 1330–1368, 2019) is not correct. An alternative statement of Lemma 5 and its proof is provided. With this new statement the order of convergence of the pressure is reduced by one half order in the spatial mesh size. Changes in the results relying Lemma 5 are also provided. [ABSTRACT FROM AUTHOR]

In this paper, we propose and analyze a variant of the proximal point method for obtaining weakly efficient solutions of convex vector optimization problems in real Hilbert spaces, with respect to a partial order induced by a closed, convex and pointed cone with nonempty interior. The proposed method is a hybrid scheme that combines proximal point type iterations and projections onto some special halfspaces in order to achieve the strong convergence to a weakly efficient solution. To the best of our knowledge, this is the first time that proximal point type method with strong convergence has been considered in the literature for solving vector/multiobjective optimization problems in infinite dimensional Hilbert spaces. [ABSTRACT FROM AUTHOR]

We show that the discrete operator stemming from time-space discretization of evolutionary partial differential equations can be represented in terms of a single Sylvester matrix equation. A novel solution strategy that combines projection techniques with the full exploitation of the entry-wise structure of the involved coefficient matrices is proposed. The resulting scheme is able to efficiently solve problems with a tremendous number of degrees of freedom while maintaining a low storage demand as illustrated in several numerical examples. [ABSTRACT FROM AUTHOR]

This paper compares the conventional Calvo and Rotemberg price adjustments at the zero lower bound (ZLB) on nominal interest rates. Although the two pricing mechanisms are equivalent to a first-order approximation around the zero-inflation steady state, they produce very different results, based on a fully-nonlinear method. Specifically, the nominal interest rate hits the ZLB more frequently in the Calvo model than in the Rotemberg model. At the ZLB, deflation is larger and recessions are more severe in the Calvo model. The main reason for the difference in results is that price adjustment costs show up in the resource constraints in the Rotemberg. When they are rebated to the household, the two models behave similarly. [ABSTRACT FROM AUTHOR]

We study the split common fixed point problem with multiple output sets in Hilbert spaces. In order to solve this problem, we propose two new algorithms and establish two strong convergence theorems for both of them. Moreover, using these methods, we also remove the assumptions imposed on the norm of the transfer operators. [ABSTRACT FROM AUTHOR]

String-averaging is an algorithmic structure used when handling a family of operators in situations where the algorithm in hand requires to employ the operators in a specific order. Sequential orderings are well known, and a simultaneous order means that all operators are used simultaneously (in parallel). String-averaging allows to use strings of indices, constructed by subsets of the index set of all operators, to apply the operators along these strings, and then to combine their end-points in some agreed manner to yield the next iterate of the algorithm. String-averaging methods were discussed and used for solving the common fixed point problem or its important special case of the convex feasibility problem. In this paper we propose and investigate string-averaging methods for the problem of best approximation to the common fixed point set of a family of operators. This problem involves finding a point in the common fixed point set of a family of operators that is closest to a given point, called an anchor point, in contrast with the common fixed point problem that seeks any point in the common fixed point set. We construct string-averaging methods for solving the best approximation problem to the common fixed points set of either finite or infinite families of firmly nonexpansive operators in a real Hilbert space. We show that the simultaneous Halpern–Lions–Wittman–Bauschke algorithm, the Halpern–Wittman algorithm, and the Combettes algorithm, which were not labeled as string-averaging methods, are actually special cases of these methods. Some of our string-averaging methods are labeled as "static" because they use a fixed pre-determined set of strings. Others are labeled as "quasi-dynamic" because they allow the choices of strings to vary, between iterations, in a specific manner and belong to a finite fixed pre-determined set of applicable strings. For the problem of best approximation to the common fixed point set of a family of operators, the full dynamic case that would allow strings to unconditionally vary between iterations remains unsolved, although it exists and is validated in the literature for the convex feasibility problem where it is called "dynamic string-averaging". [ABSTRACT FROM AUTHOR]

Constrained convex optimization problems arise naturally in many real-world applications. One strategy to solve them in an approximate way is to translate them into a sequence of convex feasibility problems via the recently developed level set scheme and then solve each feasibility problem using projection methods. However, if the problem is ill-conditioned, projection methods often show zigzagging behavior and therefore converge slowly. To address this issue, we exploit the bounded perturbation resilience of the projection methods and introduce two new perturbations which avoid zigzagging behavior. The first perturbation is in the spirit of k-step methods and uses gradient information from previous iterates. The second uses the approach of surrogate constraint methods combined with relaxed, averaged projections. We apply two different projection methods in the unperturbed version, as well as the two perturbed versions, to linear feasibility problems along with nonlinear optimization problems arising from intensity-modulated radiation therapy (IMRT) treatment planning. We demonstrate that for all the considered problems the perturbations can significantly accelerate the convergence of the projection methods and hence the overall procedure of the level set scheme. For the IMRT optimization problems the perturbed projection methods found an approximate solution up to 4 times faster than the unperturbed methods while at the same time achieving objective function values which were 0.5 to 5.1% lower. [ABSTRACT FROM AUTHOR]

In the present research, we take another look at the relatively new method for computing eigenvalues and eigenvectors of the Frobenius companion matrix. The purpose of the paper is to interpret the method considered in terms of oblique projection methods, i.e. as a Galerkin type method. Based on this dependence, we derive some new theoretical results. We establish certain error estimates, which will contribute to further studies of the convergence analysis of the method under consideration. [ABSTRACT FROM AUTHOR]

In this paper we propose a new class of Hermite series-based direct plug-in bandwidth selectors for kernel density estimation and we describe their asymptotic and finite sample behaviours. Unlike the direct plug-in bandwidth selectors considered in the literature, the proposed methodology does not involve multistage strategies and reference distributions are no longer needed. The new bandwidth selectors show a good finite sample performance when the underlying probability density function presents not only 'easy-to-estimate' but also 'hard-to-estimate' distribution features. This quality, that is not shared by other widely used bandwidth selectors as the classical plug-in or the least-square cross-validation methods, is the most significant aspect of the Hermite series-based direct plug-in approach to bandwidth selection. [ABSTRACT FROM AUTHOR]

Iterative learning control (ILC) is a high-performance control design method for systems working in a repetitive manner. ILC has traditionally focused on tracking a reference defined at all points over a finite-time interval; recent developments have begun to exploit the design freedom unlocked by tracking only a finite number of distinct time instants driven by the needs of, e.g., robotic pick-and-place tasks. This paper proposes a generalized ILC paradigm, which extends and unifies the scope of existing design frameworks by amalgamating previous task descriptions and embedding system constraints on the input and output. A novel solution is then derived using a successive projection method, which provides well-defined convergence properties. The proposed design framework is illustrated by applying it to a spatial reference tracking problem with experimental results on a gantry robot testing platform demonstrating its effectiveness. [ABSTRACT FROM AUTHOR]

We study the problem of estimating the parameters of a linear median regression without any assumption on the shape of the error distribution – including no condition on the existence of moments – allowing for heterogeneity (or heteroskedasticity) of unknown form, noncontinuous distributions, and very general serial dependence (linear and nonlinear). This is done through a reverse inference approach, based on a distribution-free sign-based testing theory, from which confidence sets and point estimators are subsequently generated. We propose point estimators, which have a natural association with confidence distributions. These estimators are based on maximizing test p-values and inherit robustness properties from the generating distribution-free tests. Both finite-sample and large-sample properties of the proposed estimators are established under weak regularity conditions. We show that they are median-unbiased (under symmetry and estimator unicity) and possess equivariance properties. Consistency and asymptotic normality are established without any moment existence assumption on the errors. A Monte Carlo study of bias and RMSE shows sign-based estimators perform better than LAD-type estimators in various heteroskedastic settings. We illustrate the use of sign-based estimators on an example of β-convergence of output levels across U.S. states. [ABSTRACT FROM AUTHOR]

The Douglas–Rachford algorithm is an optimization method that can be used for solving feasibility problems. To apply the method, it is necessary that the problem at hand is prescribed in terms of constraint sets having efficiently computable nearest points. Although the convergence of the algorithm is guaranteed in the convex setting, the scheme has demonstrated to be a successful heuristic for solving combinatorial problems of different type. In this self-contained tutorial, we develop the convergence theory of projection algorithms within the framework of fixed point iterations, explain how to devise useful feasibility problem formulations, and demonstrate the application of the Douglas–Rachford method to said formulations. The paradigm is then illustrated on two concrete problems: a generalization of the "eight queens puzzle" known as the "(m, n)-queens problem", and the problem of constructing a probability distribution with prescribed moments. [ABSTRACT FROM AUTHOR]

This paper applies the explicit aggregation (Xpa) algorithm developed by den Haan and Rendahl (J Econ Dyn Control 34:69–78, 2010) to the heterogeneous-firm models of Khan and Thomas (J Monet Econ 50:331–360, 2003; Econometrica 76:395–436, 2008) and the heterogeneous-household models of Krueger et al. (in: Taylor, Uhlig (eds) Handbook of macroeconomics, vol 2. Elsevier, Amsterdam, pp 843–921, 2016). We find the Xpa algorithm is an order of magnitude faster for solving these models than the standard Krusell–Smith (KS) algorithm because it does not need to simulate the distribution of individual capital and productivity when updating the aggregate forecasting rules. However, the simulation results in the Xpa and KS algorithms in terms of both the micro- and macro-level moments are almost identical, even in levels, because of bias correction. The Xpa algorithm also exhibits accuracy comparable with the KS algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we present a mixed projection- and density-based topology optimization approach. The aim is to combine the benefits of both parametrizations: the explicit geometric representation provides specific controls on certain design regions while the implicit density representation provides the ultimate design freedom elsewhere. While mixing projection and density representations in one formulation can serve many applications, our focus herein is only on the particular case of structural assemblies: the optimization of the structural topology is coupled with the optimization of the shape of the interface between the sub-components in a unified formulation. The interface between the assemblies is defined by a segmented profile made of linear geometric entities. The geometric coordinates of the nodes connecting the profile segments are used as shape variables in the problem, together with density variables as in conventional topology optimization. The variable profile is used to locally impose specific geometric constraints or to project particular material properties. Examples of the properties considered herein are a local volume constraint, a local maximum length scale control, a variable Young's modulus for the distributed solid material, and spatially variable minimum and maximum length scale. The resulting optimization approach is general and various geometric entities can be used. The potential for complex design manipulations is demonstrated through several numerical examples. [ABSTRACT FROM AUTHOR]

A machine-process combination may be seen as two dynamic systems cooperating for the realization of some process. A well-known example is the projection of arbitrary many robot degrees of freedom on a given path resulting in a set of nonlinear equations with one degree of freedom only, for example the path coordinate s. The set with the complete number of robot degrees of freedom is used to construct a mobility space (s ¨ , s ˙ , s) for any combination of coordinates and constraints. The paper presents a corresponding approach for more general machine and process configurations applying multibody system theory put in Lagrange I that form together with transformations from the machine side to the process side and vice versa. Practical aspects are discussed and examples given. [ABSTRACT FROM AUTHOR]

This paper treats iterative solution methods for the generalized Lyapunov equation. Specifically, a residual-based generalized rational-Krylov-type subspace is proposed. Furthermore, the existing theoretical justification for the alternating linear scheme (ALS) is extended from the stable Lyapunov equation to the stable generalized Lyapunov equation. Further insights are gained by connecting the energy-norm minimization in ALS to the theory of H2-optimality of an associated bilinear control system. Moreover it is shown that the ALS-based iteration can be understood as iteratively constructing rank-1 model reduction subspaces for bilinear control systems associated with the residual. Similar to the ALS-based iteration, the fixed-point iteration can also be seen as a residual-based method minimizing an upper bound of the associated energy norm. [ABSTRACT FROM AUTHOR]

Sub-national population projections help allocate national funding to local areas for planning local services. For example, water utilities prepare plans to meet future water demand over long-term horizons. Future demand depends on projected populations and households and forecasts of per household and per capita domestic water consumption in supply zones. This paper reports on population projections prepared for a water utility, Thames Water, which supplies water to over nine million people in London and the Thames Valley. Thames Water required an evaluation of the accuracy of the delivered projections against alternatives and estimates of uncertainty. The paper reviews how such evaluations have been made by researchers. The factors leading to variation in sub-national projections are identified. The methods, assumptions and results for English sub-national areas, used in five sets of projections, are compared. There is a consensus across projections about the future fertility and mortality but varying views about the future impact of internal and international migration flows. However, the greatest differences were between projections using ethnic populations and those using homogeneous populations. Areas with high populations of ethnic minorities were projected to grow faster when an ethnic-specific model was used. This result is important for assessing projections for countries housing diverse populations with different demographic profiles. Historic empirical prediction intervals are used to assess the uncertainty of the London and the Thames Valley projections. By 2101 the preferred projection suggests that the population of the Thames Water region will have grown by 85% within an 80% empirical prediction interval between 45 and 125%. [ABSTRACT FROM AUTHOR]

Several recent works have developed a new, probabilistic interpretation for numerical algorithms solving linear systems in which the solution is inferred in a Bayesian framework, either directly or by inferring the unknown action of the matrix inverse. These approaches have typically focused on replicating the behaviour of the conjugate gradient method as a prototypical iterative method. In this work, surprisingly general conditions for equivalence of these disparate methods are presented. We also describe connections between probabilistic linear solvers and projection methods for linear systems, providing a probabilistic interpretation of a far more general class of iterative methods. In particular, this provides such an interpretation of the generalised minimum residual method. A probabilistic view of preconditioning is also introduced. These developments unify the literature on probabilistic linear solvers and provide foundational connections to the literature on iterative solvers for linear systems. [ABSTRACT FROM AUTHOR]

We advocate the use of an Indirect Inference method to estimate the parameter of a COGARCH(1,1) process for equally spaced observations. This requires that the true model can be simulated and a reasonable estimation method for an approximate auxiliary model. We follow previous approaches and use linear projections leading to an auxiliary autoregressive model for the squared COGARCH returns. The asymptotic theory of the Indirect Inference estimator relies on a uniform strong law of large numbers and asymptotic normality of the parameter estimates of the auxiliary model, which require continuity and differentiability of the COGARCH process with respect to its parameter and which we prove via Kolmogorov's continuity criterion. This leads to consistent and asymptotically normal Indirect Inference estimates under moment conditions on the driving Lévy process. A simulation study shows that the method yields a substantial finite sample bias reduction compared with previous estimators. [ABSTRACT FROM AUTHOR]

This paper studies fully discrete approximations to the evolutionary Navier–Stokes equations by means of inf-sup stable H 1 -conforming mixed finite elements with a grad-div type stabilization and the Euler incremental projection method in time. We get error bounds where the constants do not depend on negative powers of the viscosity. We get the optimal rate of convergence in time of the projection method. For the spatial error we get a bound O (h k) for the L 2 error of the velocity, k being the degree of the polynomials in the velocity approximation. We prove numerically that this bound is sharp for this method. [ABSTRACT FROM AUTHOR]

We consider dynamical low-rank approximation on the manifold of fixed-rank matrices and tensor trains (also called matrix product states), and analyse projection methods for the time integration of such problems. First, under suitable approximability assumptions, we prove error estimates for the explicit Euler method equipped with quasi-optimal projections to the manifold. Then we discuss the possibilities and difficulties with higher-order explicit methods. In particular, we discuss ways for limiting rank growth in the increments, and robustness with respect to small singular values. [ABSTRACT FROM AUTHOR]

Many reduced-order models are neither robust with respect to parameter changes nor cost-effective enough for handling the nonlinear dependence of complex dynamical systems. In this study, we put forth a robust machine learning framework for projection-based reduced-order modeling of such nonlinear and nonstationary systems. As a demonstration, we focus on a nonlinear advection-diffusion system given by the viscous Burgers equation, which is a prototypical setting of more realistic fluid dynamics applications due to its quadratic nonlinearity. In our proposed methodology the effects of truncated modes are modeled using a single layer feed-forward neural network architecture. The neural network architecture is trained by utilizing both the Bayesian regularization and extreme learning machine approaches, where the latter one is found to be more computationally efficient. A significant emphasis is laid on the selection of basis functions through the use of both Fourier bases and proper orthogonal decomposition. It is shown that the proposed model yields significant improvements in accuracy over the standard Galerkin projection methodology with a negligibly small computational overhead and provide reliable predictions with respect to parameter changes. [ABSTRACT FROM AUTHOR]