Eigenvalue Estimates via Pseudospectra

In this note, given a matrix A∈Cn×n (or a general matrix polynomial P(z), z∈C) and an arbitrary scalar λ0∈C, we show how to define a sequence μkk∈N which converges to some element of its spectrum. The scalar λ0 serves as initial term (μ0=λ0), while additional terms are constructed through a recursive procedure, exploiting the fact that each term μk of this sequence is in fact a point lying on the boundary curve of some pseudospectral set of A (or P(z)). Then, the next term in the sequence is detected in the direction which is normal to this curve at the point μk. Repeating the construction for additional initial points, it is possible to approximate peripheral eigenvalues, localize the spectrum and even obtain spectral enclosures. Hence, as a by-product of our method, a computationally cheap procedure for approximate pseudospectra computations emerges. An advantage of the proposed approach is that it does not make any assumptions on the location of the spectrum. The fact that all computations are performed on some dynamically chosen locations on the complex plane which converge to the eigenvalues, rather than on a large number of predefined points on a rigid grid, can be used to accelerate conventional grid algorithms. Parallel implementation of the method or use in conjunction with randomization techniques can lead to further computational savings when applied to large-scale matrices.

Comparative Analysis of Optimal Path Search Algorithms

This article discusses widely used algorithms for finding optimal paths. Currently, there is a fairly wide list of algorithms for the problem of finding the shortest path, and is actively used in mobile robotics to find the optimal route. The article offers a two-level system that performs traffic planning. Comparative analysis of various search methods was carried out: their length, complexity, and a number of turning points. The purpose of the article is to study and compare algorithms from the field of artificial intelligence for finding the shortest path in a maze and a hexagonal grid. Algorithms under study: A* (star), Dijkstra algorithm, BFS, DFS, and Greedy algorithm. Algorithms are compared based on two criteria: the length of the found path and the time it takes to find the path. The results, presented analytically and graphically, show the application of five algorithms for mazes with different size and number of obstacles.

Grid Reconfiguration Method for Off-Grid DOA Estimation

Off-grid algorithms for direction of arrival (DOA) estimation have become attractive because of their advantages in resolution and efficiency over conventional ones. In this paper, we propose a grid reconfiguration direction of arrival (GRDOA) estimation method based on sparse Bayesian learning. Unlike other off-grid methods, the grid points of GRDOA are treated as dynamic parameters. The number and position of the grid points are varied iteratively via a root method and a fission process. Then, the grid gets reconfigured through some criteria. By iteratively updating the reconfigured grid, DOAs are estimated completely. Since GRDOA has fewer grid points, it has better computational efficiency than the previous methods. Moreover, GRDOA can achieve better resolution and relatively higher accuracy. Numerical simulation results validate the effectiveness of GRDOA.

Multigrid Backprojection Super–Resolution and Deep Filter Visualization

We introduce a novel deep–learning architecture for image upscaling by large factors (e.g. 4×, 8×) based on examples of pristine high–resolution images. Our target is to reconstruct high–resolution images from their downscale versions. The proposed system performs a multi–level progressive upscaling, starting from small factors (2×) and updating for higher factors (4× and 8×). The system is recursive as it repeats the same procedure at each level. It is also residual since we use the network to update the outputs of a classic upscaler. The network residuals are improved by Iterative Back–Projections (IBP) computed in the features of a convolutional network. To work in multiple levels we extend the standard back– projection algorithm using a recursion analogous to Multi– Grid algorithms commonly used as solvers of large systems of linear equations. We finally show how the network can be interpreted as a standard upsampling–and–filter upscaler with a space–variant filter that adapts to the geometry. This approach allows us to visualize how the network learns to upscale. Finally, our system reaches state of the art quality for models with relatively few number of parameters.

Application of a new single staggered grid method to the heat transfer problems

In this paper, a new single staggered grid method is proposed to solve the fluid dynamic problems numerically. The advantages of the new grid method are analyzed in comparison with the classical grid algorithms such as the staggered grids, collocated grids and semi-staggered grids. The discretization of the basic equations for the fluid dynamics on the new single staggered grids is derived and the corresponding SIMPLE algorithm is introduced. As an example, the heat transfer problem of fluid-flow at a right angle is solved to prove the validity of the new single staggered grid method.

Processing conversion and parallel control platform: a parallel approach to serial hydrodynamic simulators for complex hydrodynamic simulations

In this paper, a processing conversion and parallel control platform (PCsP) is proposed for transitioning serial hydrodynamic simulators to a cluster-computing system. We previously undertook efforts to promote the research and development of this type of platform and to demonstrate and commercialize it. Our PCsP provide distributed and parallel patterns, a centralized architecture, and user support. To validate our employed methodology and highlight its simplicity, we adopted the technology in various applications based on multi-grid algorithms. The methodology was shown to be reliable and feasible across computational domains, partitioning strategies, and multi-grid codes. Furthermore, its effectiveness was demonstrated using a complex engineering case in addition to code based on slightly less complex mathematical models. Eventual transition to a cluster-computing system will require further investigation of the impact of different model combinations on calculation accuracy, efficiency of operating models, and PCsP functional development.