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Projection Method

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The Projection Method is a mathematical technique used to solve partial differential equations by transforming the problem into a simpler form, typically through the use of orthogonal projections. This method facilitates the analysis and numerical approximation of solutions by projecting functions onto a subspace defined by basis functions.
lightbulbAbout this topic
The Projection Method is a mathematical technique used to solve partial differential equations by transforming the problem into a simpler form, typically through the use of orthogonal projections. This method facilitates the analysis and numerical approximation of solutions by projecting functions onto a subspace defined by basis functions.

Key research themes

1. How can projection methods be adapted and visualized to better understand distortions and topological preservation in multidimensional data projections?

This research area focuses on enhancing continuous projection techniques, such as Principal Component Analysis and Curvilinear Component Analysis, by developing tools to visualize and quantify the distortions and topological changes that occur when projecting high-dimensional data onto lower-dimensional spaces. Understanding these distortions is crucial for accurate data interpretation, especially in exploratory data analysis and manifold learning.

Key finding: Introduced the use of coloring Voronoi cells in the projection space based on measures related to projected data to visually estimate the faithfulness of a projection to original data topology. This approach enables... Read more
Key finding: Developed an improved symmetric projection method which addresses the limitations of existing projection methods in group decision-making contexts (using AHP). By focusing on homogeneity of decision makers' opinions, the... Read more
Key finding: Established convergence criteria for an algorithm that combines alternating projection and Douglas–Rachford operators, improving phase retrieval solutions. By mathematically interpreting DRAP as a convex combination of two... Read more

2. What advances in projection methods facilitate efficient and convergent solutions for large-scale linear and nonlinear systems, including inverse problems and optimal control?

This theme addresses recent methodological developments that leverage projection algorithms, including alternating projection, relaxed and successive projection, and Krylov subspace methods, to solve large-scale linear systems, inverse tomography problems, and nonlinear equations arising in optimal control and imaging. Research encompasses theoretical convergence guarantees, relaxation schemes, and iterative solvers designed to enhance computational efficiency and accuracy in high-dimensional or constrained settings.

Key finding: Generalized the classic von Neumann alternating projection method by introducing relaxation and step-size functions, proving weak convergence and Fejér monotonicity properties for sequences generated by these relaxed... Read more
Key finding: Proposed iterative projection methods that append one column to the sketching matrix each iteration, guaranteeing finite convergence for consistent square or rectangular linear systems. The method generalizes Kaczmarz and... Read more
Key finding: Extended m-dimensional successive projection methods (mD-SPM) from symmetric positive definite matrices to general non-singular systems. Demonstrated theoretically and experimentally that increasing block sizes (m) in... Read more
Key finding: Presented a novel theoretical algebraic reconstruction algorithm that recovers a discretized image exactly and uniquely from the Radon transform at a single selected projection angle combined with a specifically chosen family... Read more
Key finding: Applied a projection function within an iterative method to approximate control laws in bounded optimal control problems. By constructing approximations of state and co-state variables through the Hamiltonian, the paper... Read more

3. How do projection-based numerical methods, including meshless and immersed boundary projection algorithms, enhance simulations of complex physical and fluid-structure interaction problems?

This theme explores computational techniques employing projection methods to handle complex boundary conditions and coupling in fluid dynamics and fluid-structure interaction (FSI) simulations. These methods improve robustness and efficiency in 2D and 3D simulations involving incompressible flows, multiphase media, and deformable structures by mathematically decoupling constraints, enforcing boundary conditions weakly, or developing monolithic solvers that preserve accuracy and stability.

Key finding: Addressed the challenge of imposing complicated boundary conditions on auxiliary vector fields in the gauge reformulation of Navier-Stokes via a weak enforcement strategy using symmetric Nitsche method within finite elements.... Read more
Key finding: Developed a projection algorithm combined with a co-located diffuse approximation meshless method to numerically solve incompressible Navier-Stokes equations in 2D and 3D domains. Testing on classical flow problems showed the... Read more
Key finding: Presented a monolithic immersed boundary projection method (IBPM) using continuous forcing and nested approximate LU decomposition to simulate highly dynamic fluid-structure interaction problems involving complex geometries... Read more

All papers in Projection Method

The physical origin of spiral vortex breakdown is investigated using the direct and adjoint Navier-Stokes equations linearized around axisymmetric vortex breakdown. The wave modes region, defined as the overlap region between adjoint and... more
The American Physical Society Quantum dynamics of H2+ in an atto-second laser pulse NICHOLAS VENCE, ROBERT HARRISON, University of Tennessee, PREDRAG KRSTIC, Oak Ridge National Laboratory -We demonstrate a highly-accurate numerical... more
Meshfree solution schemes for the incompressible Navier-Stokes equations are usually based on algorithms commonly used in finite volume methods, such as projection methods, SIMPLE and PISO algorithms. However, drawbacks of these... more
A meshfree projection method for compressible as well as incompressible flows and the coupling of two phase flows with high density and viscosity ratios is presented. The Navier-Stokes equations are considered as the basic mathematical... more
Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This paper has been... more
Projection operators derived from Gabor multipliers are suggested as a time-frequency localization tool. We give a description of the numerical realization of such projection operators and investigate the dependence of their localization... more
We establish a bijective correspondence between affine connections and a class of semi-holonomic jets of local diffeomorphisms of the underlying manifold called symmetry jets in the text. The symmetry jet corresponding to a torsion free... more
We establish a bijective correspondence between affine connections and a class of semi-holonomic jets of local diffeomorphisms of the underlying manifold called symmetry jets in the text. The symmetry jet corresponding to a torsion free... more
Abstract. This paper discusses a method based on Laguerre polynomials co mbined with a Filtered Conjugate Residual (FCR) framework to compute the product of the exponen tial of a matrix by a vector. The method implicitly uses an expansion... more
How to properly specify boundary conditions for pressure is a longstanding problem for the incompressible Navier-Stokes equations with no-slip boundary conditions. An analytical resolution of this issue stems from a recently developed... more
To present individual body identification efforts, as part of the World Trade Center (WTC) mass disaster identification project. More than 500 samples were tested by using polymerase chain reaction (PCR) amplification and short tandem... more
Data projection is a commonly used technique applied to analyse high dimensional data. In the present work, we propose a new data projection method that uses genetic algorithms to find linear projections, providing meaningful... more
In the present paper we have used the Differential forms also known as exterior calculus of E. in Pullback calculations and proving the main theorems of advanced calculus i.e. Green, Gauss and Stoke's theorems. In particular a convenient... more
An application of the score test of Rao is developed and presented for the overlap coefficient in the Draper and Guttman´s model using the response surface g...
RESUMEN: En este artículo ha sido propuesto un test para el coeficiente de solapamiento para el modelo de Draper y Guttman utilizando modelos de superficies de respuesta de primer y segundo orden. El test está basado en el test de score... more
Two different dual-energy projection radiography techniques were used to quantitate bone mineral density in the femoral neck. A heterogeneous population of normal aging individuals of both genders was studied. Using a dual-energy scanned... more
This paper describes ongoing multidisciplinary research which aims to analyse an to apply neural networks architectures to the emerging field of Knowledge Management. In this case, we illustrate the particular use of a novel connectionist... more
The corrective smoothed particle method (CSPM) is an improvement to the conventional smoothed particle hydrodynamics (SPH) method. It overcomes the problems of tensile instability and boundary deficiency in SPH. As a specially designed... more
This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will... more
We study the general dynamics of the spherically symmetric gravitational collapse of a massless scalar field. We apply the Galerkin projection method to transform a system of partial differential equations into a set of ordinary... more
the near future, KEPLER and GAIA, are missions that are producing, or will produce, light curves as we have never seen before, because of their quantity or quality. The exploitation of the scientific potential hidden in these datasets is... more
The two-step regression process for whole stand survival modeling has been widely used for multiple species. However, estimating the parameters of a probability model for the discrete event of mortality and a whole stand survival model... more
LSMR (Least Squares Minimal Residual) is an iterative method for the solution of the linear system of equations and leastsquares problems. This paper presents a block version of the LSMR algorithm for solving linear systems with multiple... more
This paper presents some further results on proximal and asymptotic proximal contractions and on a class of generalized weak proximal contractions in metric spaces. The generalizations are stated for non-self-mappings of the forms T n : ,... more
The purpose of this paper is to introduce a new algorithm to approximate a common solution for a system of generalized mixed equilibrium problems, split variational inclusion problems of a countable family of multivalued maximal monotone... more
This paper presents some further results on proximal and asymptotic proximal contractions and on a class of generalized weak proximal contractions in metric spaces. The generalizations are stated for non-self-mappings of the forms T n : ,... more
We present and validate a numerical technique for computing dendritic growth of crystals from pure melts in the presence of forced convection. The Navier-Stokes equations are solved on a fixed Cartesian mesh and a mixed... more
This paper explores risk aversion among Australian households using panel data from the Household Income and Labour Dynamics in Australia (HILDA) survey. Using households’ share of risky assets, we test whether relative risk aversion is... more
In this paper, design and analysis of Iterative Learning Control (ILC) based on partial information from previous cycles is developed. Typically, in a discrete-time repetitive process, ILC schemes use error from the entire previous cycle... more
Combining the projection method of Solodov and Svaiter with the Liu-Storey and Fletcher Reeves conjugate gradient algorithm of Djordjević for unconstrained minimization problems, a hybrid conjugate gradient algorithm is proposed and... more
The 2s2p 'P and 'P autoionizing states of heliumlike ions are studied via an implementation of the Feshbach projection method within the framework of Z-dependent perturbation theory. These same states are also studied by a more... more
This paper reviews the empire problem for quasiperiodic tilings and the existing methods for generating the empires of the vertex configurations in quasicrystals, while introducing a new and more efficient method based on the... more
We use q-Pascal's triangle to define a family of representations of dimension 6 of the braid group B 3 on three strings. Then we give a necessary and sufficient condition for these representations to be irreducible.
ALEGRA is an arbitrary Lagrangian-Eulerian multi-material finite element code used for modeling solid dynamics problems involving large distortion and shock propagation. This document describes the basic user input language and... more
ALEGRA is an arbitrary Lagrangian-Eulerian multi-material finite element code used for modeling solid dynamics problems involving large distortion and shock propagation. This document describes the basic user input language and... more
In this paper, two-step inertial extrapolation and self-adaptive step sizes is proposed to solve the split common null point problem with multiple output sets in Hilbert spaces. Weak convergence analysis are obtained under some easy to... more
Nonmonotone spectral gradient (NSG) techniques are considered for unconstrained optimization of differentiable functions. They combine a nonmonotone step length strategy, that is based on the Grippo-Lampariello-Lucidi nonmonotone line... more
Fringe projection profilometry (FPP) has evolved dramatically, with many highly demanded features for threedimensional (3D) imaging, such as high accuracy, easy implementation, and capability of measuring multiple objects with complex... more
We explore the rich nature of correlations in the ground state of ultracold atoms trapped in state-dependent optical lattices. In particular, we consider interacting fermionic ytterbium or strontium atoms, realizing a twoorbital Hubbard... more
A new projection method based on radial basis functions (RBFs) is presented for discretizing the incompressible unsteady Stokes equations in irregular geometries. The novelty of the method comes from the application of a new technique for... more
A new projection method based on radial basis functions (RBFs) is presented for discretizing the incompressible unsteady Stokes equations in irregular geometries. The novelty of the method comes from the application of a new technique for... more
A new concept of the asymptotically weakG-pseudo-Ψ-contractive non-self-mappingT:G↦Bis introduced and some strong convergence theorems for the mapping are proved by using the generalized projection method combined with the modified... more
This paper focuses on the treatment of volume constraints which in the context of elasto-plasticity typically arise as a result of assuming volume-preserving plastic flow. Projection methods based on the modification of the discrete... more
The imbibition of the injection water into shale matrix may result in shale swelling affecting oil recovery from the rock matrix and the transport properties of the fractures. The expansion of rock due to the swelling of shale matrix... more
In this paper, a continuous projection method is designed and analyzed. The continuous projection method consists of a set of partial differential equations which can be regarded as an approximation of the Navier-Stokes (N-S) equations in... more
This present study focuses on three topics: (i) formulation of the firstorder perturbation problem of the Earth's normal modes; (ii) formulation of a variational method based on the solutions of first-order quasidegenerate perturbation... more
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