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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]

We study the stability and convergence properties of a semi-implicit time stepping scheme for the incompressible Navier–Stokes equations with variable density and viscosity. The density is assumed to be approximated in a way that conserves the minimum-maximum principle. The scheme uses a fractional time-stepping method and the momentum, which is equal to the product of the density and velocity, as a primary unknown. The semi-implicit algorithm for the coupled momentum-pressure is shown to be conditionally stable and the velocity is shown to converge in L 2 norm with order one in time. Numerical illustrations confirm that the algorithm is stable and convergent under classic CFL condition even for sharp density profiles. [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]

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.

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]

This paper proposes a modified D-iteration to approximate the solutions of three quasi-nonexpansive 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 efficient and outperforms the standard forward-backward with inertial step.

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]

We consider a pressure correction temporal discretization for the incompressible Navier-Stokes equations in EMAC form. We prove stability and error estimates for the case of mixed finite element spatial discretization, and in particular that the Gronwall constant's exponential dependence on the Reynolds number is removed (for sufficiently smooth true solutions) or at least significantly reduced compared to the commonly used skew-symmetric formulation. We also show the method preserves momentum and angular momentum, and while it does not preserve energy it does admit an energy inequality. Several numerical tests show the advantages EMAC can have over other commonly used formulations of the nonlinearity. Additionally, we discuss extensions of the results to the usual Crank-Nicolson temporal discretization. [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]

This paper compares and contrasts higher-order, semi-implicit temporal integration strategies for the incompressible Navier–Stokes methods based on spectral deferred corrections applied to certain gauge or auxiliary variable formulations of the equations. Particular focus is placed on the imposition of boundary conditions in the semi-implicit formulation, the accurate treatment of the pressure term, and the smoothness of the numerical solution for different formulations. The main result presented here is the formulation and numerical validation of a single-step, semi-implicit method called spectral deferred pressure corrections, which can in theory obtain arbitrary formal order of accuracy in time. Numerical results demonstrate up to eighth-order accuracy, scenarios where optimal temporal accuracy is attained, and scenarios where order reduction is observed due to time-dependent boundary conditions.

Flows in which the primary features of interest do not rely on high-frequency acoustic effects, but in which long-wavelength acoustics play a nontrivial role, present a computational challenge. Integrating the entire domain with low-Mach-number methods would remove all acoustic wave propagation, while integrating the entire domain with the fully compressible equations can in some cases be prohibitively expensive due to the CFL time step constraint. For example, simulation of thermoacoustic instabilities might require fine resolution of the fluid/chemistry interaction but not require fine resolution of acoustic effects, yet one does not want to neglect the long-wavelength wave propagation and its interaction with the larger domain. The present paper introduces a new multi-level hybrid algorithm to address these types of phenomena. In this new approach, the fully compressible Euler equations are solved on the entire domain, potentially with local refinement, while their low-Mach-number counterparts are solved on subregions of the domain with higher spatial resolution. The finest of the compressible levels communicates inhomogeneous divergence constraints to the coarsest of the low-Mach-number levels, allowing the low-Mach-number levels to retain the long-wavelength acoustics. The performance of the hybrid method is shown for a series of test cases, including results from a simulation of the aeroacoustic propagation generated from a Kelvin–Helmholtz instability in low-Mach-number mixing layers. It is demonstrated that compared to a purely compressible approach, the hybrid method allows time-steps two orders of magnitude larger at the finest level, leading to an overall reduction of the computational time by a factor of 8.

Flows in which the primary features of interest do not rely on high-frequency acoustic effects, but in which long-wavelength acoustics play a nontrivial role, present a computational challenge. Integrating the entire domain with low-Mach-number methods would remove all acoustic wave propagation, while integrating the entire domain with the fully compressible equations can in some cases be prohibitively expensive due to the CFL time step constraint. For example, simulation of thermoacoustic instabilities might require fine resolution of the fluid/chemistry interaction but not require fine resolution of acoustic effects, yet one does not want to neglect the long-wavelength wave propagation and its interaction with the larger domain. The present paper introduces a new multi-level hybrid algorithm to address these types of phenomena. In this new approach, the fully compressible Euler equations are solved on the entire domain, potentially with local refinement, while their low-Mach-number counterparts are solved on subregions of the domain with higher spatial resolution. The finest of the compressible levels communicates inhomogeneous divergence constraints to the coarsest of the low-Mach-number levels, allowing the low-Mach-number levels to retain the long-wavelength acoustics. The performance of the hybrid method is shown for a series of test cases, including results from a simulation of the aeroacoustic propagation generated from a Kelvin–Helmholtz instability in low-Mach-number mixing layers. It is demonstrated that compared to a purely compressible approach, the hybrid method allows time-steps two orders of magnitude larger at the finest level, leading to an overall reduction of the computational time by a factor of 8.

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]

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.

Given two nonempty and disjoint intersections of closed and convex subsets, we look for a best approximation pair relative to them, i.e., a pair of points, one in each intersection, attaining the minimum distance between the disjoint intersections. We propose an iterative process based on projections onto the subsets which generate the intersections. The process is inspired by the Halpern–Lions–Wittmann–Bauschke algorithm and the classical alternating process of Cheney and Goldstein, and its advantage is that there is no need to project onto the intersections themselves, a task which can be rather demanding. We prove that under certain conditions the two interlaced subsequences converge to a best approximation pair. These conditions hold, in particular, when the space is Euclidean and the subsets which generate the intersections are compact and strictly convex. Our result extends the one of Aharoni, Censor and Jiang ["Finding a best approximation pair of points for two polyhedra", Computational Optimization and Applications 71 (2018), 509–23] who considered the case of finite-dimensional polyhedra. [ABSTRACT FROM AUTHOR]

• An innovative quaternary hybrid light modulation (QHLM) based projection display is proposed. • A quaternary spatial light modulator is constructed by illuminating two parallel digital micro-mirror devices (DMDs) with different light intensities. • The quaternary digit-plane decomposition and the pixel-level alignment of two DMDs are realized. • The prototype is designed and implemented to verify that the QHLM can achieve 16-bit projection at 220 fps. Conventional spatial light modulators (SLM) can only be used for projecting 8-bit or 10-bit images at normal frame rate. Therefore, commercial high dynamic range (HDR) displays typically focus on boosting the contrast while neglecting to raise the bit-depth. Existing methods for high bit-depth display generally rely on stacking two SLMs to modulate the outgoing beam twice, namely multiplicative modulation, resulting in many troubles such as low optical efficiency, difficulty in pixel-level alignment, and complex image rendering algorithm. In this paper, an innovative quaternary hybrid light modulation (QHLM) based projection display is proposed and realized. By illuminating two parallel digital micro-mirror devices (DMDs) with different light intensities, the quaternary digit-planes (QDs) with four gray levels are able to be modulated rapidly. Aiming at this ability, the quaternary pulse width modulation (QPWM) is incorporated with the quaternary light-intensity modulation (QLM) to fundamentally improve the modulation efficiency compared to the binary light modulation mode. Furthermore, the quaternary digit-plane decomposition (QDD) based image splitting algorithm is adopted to split a high bit-depth image into two images that drive two DMDs respectively. The prototype is designed and built to verify the feasibility of the QHLM based projection display. The experimental results demonstrate that the prototype can project 16-bit images at 220 fps. Through additive modulation of two DMDs in parallel, the QHLM entirely avoids the drawbacks of multiplicative modulation. As a completely different technology, the QHLM has a great potential for HDR projection applications. [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]

Designing numerical methods for incompressible fluid flow involving moving interfaces, for example, in the computational modeling of bubble dynamics, swimming organisms, or surface waves, presents challenges due to the coupling of interfacial forces with incompressibility constraints. A class of methods, denoted interfacial gauge methods, is introduced for computing solutions to the corresponding incompressible Navier-Stokes equations. These methods use a type of "gauge freedom" to reduce the numerical coupling between fluid velocity, pressure, and interface position, allowing high-order accurate numerical methods to be developed more easily. Making use of an implicit mesh discontinuous Galerkin framework, developed in tandem with this work, high-order results are demonstrated, including surface tension dynamics in which fluid velocity, pressure, and interface geometry are computed with fourth-order spatial accuracy in the maximum norm. Applications are demonstrated with two-phase fluid flow displaying fine-scaled capillary wave dynamics, rigid body fluid-structure interaction, and a fluid-jet free surface flow problem exhibiting vortex shedding induced by a type of Plateau-Rayleigh instability. The developed methods can be generalized to other types of interfacial flow and facilitate precise computation of complex fluid interface phenomena.

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]

In this paper, we investigate the projected dynamical system associated with the generalized variational inequality with hesitant fuzzy relation. We establish the equivalence between the generalized variational inequality with hesitant fuzzy relation and the fuzzy fixed point problem. And we analyze the existence theorem and iterative algorithm of solutions to such problem. Furthermore, using the projection method, we propose a projection neural network for solving the generalized variational inequality with hesitant fuzzy relation and discuss the stability of the proposed projected dynamical system.

Objectives This work is the conclusion of considerations related to the subject of communication in an assistance situation. Patient communication - a volunteer in a psychiatric institution was analyzed. Material and methods The work will present one, selected, projection method from the battery of questionnaires carried out from 2014. The study was attended by people at the age of emerging adulthood (19-26 years), most of them come from the Lublin province. Finally, the answers of 120 volunteers operating in the Volunteering Center Association in Lublin (Voluntary Psychiatric Clinic) were taken into account. Results The results of the research also testify to the importance of other psychological resources in coping with stress, for example in the workplace. The results of the conducted analyzes may be included in the discussion on the definition and functioning of volunteering as a phenomenon that is still not in the interest of research in the social sciences. Volunteers are a characteristic group that deals free with a variety of assistance activities, with quality and commitment to the projects being implemented. Conclusions Therefore, the work has a theoretical and practical dimension in relation to the attached elements concerning projection methods, with the help of which more information can be obtained from the research group. Communication in the context of hospitalization and treatment is a very complex aspect. The model was created as a result of undertaken scientific research in psychiatric facilities. A person threatened with social exclusion, after a suicide attempt or when suffering from mental disorders is associated with many institutions. Very often, the treatment involves not only pharmacotherapy, psychotherapy, but also occupational therapy.

Our aim through this paper is to describe a Krylov based projection method in order to reduce high-order dynamical systems. We focus on differential algebraic equations (DAEs) of index-2 that arise from spatial discretization of Stokes equations. An efficient algorithm based on a projection technique onto an extended block Krylov subspace that appropriately allows us to construct a reduced order system is described. Numerical results are provided to confirm the performance of the derived method compared with other known ones. [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]

The backward substitution method is a newly developed meshless method that has been used for the simulation of many problems in science and engineering with high accuracy and efficiency. In this paper, we explore the feasibility of employing the backward substitution method for simulating two-dimensional incompressible flows. Two non-increment pressure correction projection methods are considered to decompose the original velocity and pressure coupling system into two boundary value problems of intermediate velocity and pressure. Then, two boundary value problems are solved by the backward substitution method in each time iteration step. Five numerical examples are provided to demonstrate the accuracy, computational efficiency and convergence of the method. Comparisons with some existing meshless methods verify the method's advantages and potential applications to engineering. • The backward substitution method is first raised for the simulation of incompressible flows. • Investigated effects of the time step and the number of collocation points on the accuracy. • Verification benchmark comprising several fluid flows problems. • Two projection methods are considered to verify the algorithm. • The algorithm is easily extended to three-dimensional incompressible flows. [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.