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Projection method

About: Projection method is a research topic. Over the lifetime, 3685 publications have been published within this topic receiving 85244 citations.


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TL;DR: This paper proposes an alternating projection method that is versatile enough to solve a huge class of inverse eigenvalue problems (IEPs), which includes the frame design problem, and addresses the most basic design problem: constructing tight frames with prescribed vector norms.
Abstract: Tight frames, also known as general Welch-bound- equality sequences, generalize orthonormal systems. Numerous applications - including communications, coding, and sparse approximation- require finite-dimensional tight frames that possess additional structural properties. This paper proposes an alternating projection method that is versatile enough to solve a huge class of inverse eigenvalue problems (IEPs), which includes the frame design problem. To apply this method, one needs only to solve a matrix nearness problem that arises naturally from the design specifications. Therefore, it is the fast and easy to develop versions of the algorithm that target new design problems. Alternating projection will often succeed even if algebraic constructions are unavailable. To demonstrate that alternating projection is an effective tool for frame design, the paper studies some important structural properties in detail. First, it addresses the most basic design problem: constructing tight frames with prescribed vector norms. Then, it discusses equiangular tight frames, which are natural dictionaries for sparse approximation. Finally, it examines tight frames whose individual vectors have low peak-to-average-power ratio (PAR), which is a valuable property for code-division multiple-access (CDMA) applications. Numerical experiments show that the proposed algorithm succeeds in each of these three cases. The appendices investigate the convergence properties of the algorithm.

496 citations

Journal ArticleDOI
TL;DR: In this paper, the authors proposed a projection algorithm for solving the variational inequality problem, where the underlying function is continuous and satisfies a certain generalized monotonicity assumption (e.g., it can be pseudomonotone).
Abstract: We propose a new projection algorithm for solving the variational inequality problem, where the underlying function is continuous and satisfies a certain generalized monotonicity assumption (e.g., it can be pseudomonotone). The method is simple and admits a nice geometric interpretation. It consists of two steps. First, we construct an appropriate hyperplane which strictly separates the current iterate from the solutions of the problem. This procedure requires a single projection onto the feasible set and employs an Armijo-type linesearch along a feasible direction. Then the next iterate is obtained as the projection of the current iterate onto the intersection of the feasible set with the halfspace containing the solution set. Thus, in contrast with most other projection-type methods, only two projection operations per iteration are needed. The method is shown to be globally convergent to a solution of the variational inequality problem under minimal assumptions. Preliminary computational experience is also reported.

490 citations

01 Jan 1998
TL;DR: A method for solving the equations governing time-dependent, variable density incompressible flow in two or three dimensions on an adaptive hierarchy of grids based on a projection formulation in which the first step is to solve advection?diffusion equations to predict intermediate velocities, and then project these Velocities onto a space of approximately divergence-free vector fields.
Abstract: In this paper we present a method for solving the equations governing timedependent, variable density incompressible flow in two or three dimensions on an adaptive hierarchy of grids. The method is based on a projection formulation in which we first solve advection‐diffusion equations to predict intermediate velocities, and then project these velocities onto a space of approximately divergence-free vector fields. Our treatment of the first step uses a specialized second-order upwind method for differencing the nonlinear convection terms that provides a robust treatment of these terms suitable for inviscid and high Reynolds number flow. Density and other scalars are advected in such a way as to maintain conservation, if appropriate, and free-stream preservation. Our approach to adaptive refinement uses a nested hierarchy of logically-rectangular girds with simultaneous refinement of the girds in both space and time. The integration algorithm on the grid hierarchy is a recursive procedure in which coarse grids are advanced in time, fine grids are advanced multiple steps to reach the same time as the coarse grids and the data at different levels are then synchronized. The single grid algorithm is described briefly, but the emphasis here is on the time-stepping procedure for the adaptive hierarchy. Numerical examples are presented to demonstrate the algorithms’s accuracy and convergence properties, and illustrate the behavior of the method. An additional example demonstrates the performance of the method on a more realistic problem, namely, a three-dimensional variable density shear layer. c ∞ 1998 Academic Press

483 citations

Journal ArticleDOI
TL;DR: In this article, the authors present a method for solving the equations governing time-dependent, variable density incompressible flow in two or three dimensions on an adaptive hierarchy of grids.

477 citations

Journal ArticleDOI
TL;DR: In this paper, the iterative method for solving system of linear equations, due to Kaczmarz [2], is investigated and it is shown that the method works well for both singular and non-singular systems and it determines the affine space formed by the solutions if they exist.
Abstract: The iterative method for solving system of linear equations, due to Kaczmarz [2], is investigated. It is shown that the method works well for both singular and non-singular systems and it determines the affine space formed by the solutions if they exist. The method also provides an iterative procedure for computing a generalized inverse of a matrix.

468 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202331
202279
2021153
2020209
2019182
2018207