Numerical Algorithms for Computing an Arbitrary Singular Value of a Tensor Sum

We consider computing an arbitrary singular value of a tensor sum: T:=In⊗Im⊗A+In⊗B⊗Iℓ+C⊗Im⊗Iℓ∈Rℓmn×ℓmn, where A∈Rℓ×ℓ, B∈Rm×m, C∈Rn×n. We focus on the shift-and-invert Lanczos method, which solves a shift-and-invert eigenvalue problem of (TTT−σ˜2Iℓmn)−1, where σ˜ is set to a scalar value close to the desired singular value. The desired singular value is computed by the maximum eigenvalue of the eigenvalue problem. This shift-and-invert Lanczos method needs to solve large-scale linear systems with the coefficient matrix TTT−σ˜2Iℓmn. The preconditioned conjugate gradient (PCG) method is applied since the direct methods cannot be applied due to the nonzero structure of the coefficient matrix. However, it is difficult in terms of memory requirements to simply implement the shift-and-invert Lanczos and the PCG methods since the size of T grows rapidly by the sizes of A, B, and C. In this paper, we present the following two techniques: (1) efficient implementations of the shift-and-invert Lanczos method for the eigenvalue problem of TTT and the PCG method for TTT−σ˜2Iℓmn using three-dimensional arrays (third-order tensors) and the n-mode products, and (2) preconditioning matrices of the PCG method based on the eigenvalue and the Schur decomposition of T. Finally, we show the effectiveness of the proposed methods through numerical experiments.

Accelerating the solution of linear systems appearing in two-phase reservoir simulation by the use of POD-based deflation methods

We explore and develop a Proper Orthogonal Decomposition (POD)-based deflation method for the solution of ill-conditioned linear systems, appearing in simulations of two-phase flow through highly heterogeneous porous media. We accelerate the convergence of a Preconditioned Conjugate Gradient (PCG) method achieving speed-ups of factors up to five. The up-front extra computational cost of the proposed method depends on the number of deflation vectors. The POD-based deflation method is tested for a particular problem and linear solver; nevertheless, it can be applied to various transient problems, and combined with multiple solvers, e.g., Krylov subspace and multigrid methods.

A Barotropic Solver for High-Resolution Ocean General Circulation Models

High-resolution global ocean general circulation models (OGCMs) play a key role in accurate ocean forecasting. However, the models of the operational forecasting systems are still not in high resolution due to the subsequent high demand for large computation, as well as the low parallel efficiency barrier. Good scalability is an important index of parallel efficiency and is still a challenge for OGCMs. We found that the communication cost in a barotropic solver, namely, the preconditioned conjugate gradient (PCG) method, is the key bottleneck for scalability due to the high frequency of the global reductions. In this work, we developed a new algorithm—a communication-avoiding Krylov subspace method with a PCG (CA-PCG)—to improve scalability and then applied it to the Nucleus for European Modelling of the Ocean (NEMO) as an example. For PCG, inner product operations with global communication were needed in every iteration, while for CA-PCG, inner product operations were only needed every eight iterations. Therefore, the global communication cost decreased from more than 94.5% of the total execution time with PCG to less than 63.4% with CA-PCG. As a result, the execution time of the barotropic modes decreased from more than 17,000 s with PCG to less than 6000 s with CA-PCG, and the total execution time decreased from more than 18,000 s with PCG to less than 6200 s with CA-PCG. Besides, the ratio of the speedup can also be increased from 3.7 to 4.6. In summary, the high process count scalability when using CA-PCG was effectively improved from that using the PCG method, providing a highly effective solution for accurate ocean simulation.

Convergence behavior of single-step GBLUP and SNPBLUP for different termination criteria

Background The preconditioned conjugate gradient (PCG) method is the current method of choice for iterative solving of genetic evaluations. The relative difference between two successive iterates and the relative residual of the system of equations are usually chosen as a termination criterion for the PCG method in animal breeding. However, our initial analyses showed that these two commonly used termination criteria may report that a PCG method applied to a single-step single nucleotide polymorphism best linear unbiased prediction (ssSNPBLUP) is not converged yet, whereas the solutions are accurate enough for practical use. Therefore, the aim of this study was to propose two termination criteria that have been (partly) developed in other fields, but are new in animal breeding, and to compare their behavior to that of the two termination criteria widely used in animal breeding for the PCG method applied to ssSNPBLUP. The convergence patterns of ssSNPBLUP were also compared to the convergence patterns of single-step genomic BLUP (ssGBLUP). Results Building upon previous work, we propose two termination criteria that take the properties of the system of equations into account. These two termination criteria are directly related to the relative error of the iterates with respect to the true solutions. Based on pig and dairy cattle datasets, we show that the preconditioned coefficient matrices of ssSNPBLUP and ssGBLUP have similar properties when using a second-level preconditioner for ssSNPBLUP. Therefore, the PCG method applied to ssSNPBLUP and ssGBLUP converged similarly based on the relative error of the iterates with respect to the true solutions. This similar convergence behavior between ssSNPBLUP and ssGBLUP was observed for both proposed termination criteria. This was, however, not the case for the termination criterion defined as the relative residual when applied to the dairy cattle evaluations. Conclusion Our results showed that the PCG method can converge similarly when applied to ssSNPBLUP and to ssGBLUP. The two proposed termination criteria always depicted these similar convergence behaviors, and we recommend them for comparing convergence properties of different models and for routine evaluations.

Description and implementation of an algebraic multigrid preconditioner for H1-conforming finite element schemes

This paper presents detailed aspects regarding the implementation of the Finite Element Method (FEM) to solve a Poisson’s equation with homogeneous boundary conditions. The aim of this paper is to clarify details of this implementation, such as the construction of algorithms, implementation of numerical experiments, and their results. For such purpose, the continuous problem is described, and a classical FEM approach is used to solve it. In addition, a multilevel technique is implemented for an efficient resolution of the corresponding linear system, describing and including some diagrams to explain the process and presenting the implementation codes in MATLAB®. Finally, codes are validated using several numerical experiments. Results show an adequate behavior of the preconditioner since the number of iterations of the PCG method does not increase, even when the mesh size is reduced.

Island Generator pada Game Open world Menggunakan Algoritma Perlin noise

AbstrakPerkembangan game pada saat ini berkembang dengan sangat cepat, dalam perkermbangan game topik AI adalah topik yang paling banyak diteliti oleh beberapa peneliti khususnya pada pembuatan suatu konten game menggunakan metode PCG (procedural content generation). Pada pembuatan sebuah game world menggunakan metode PCG sudah banyak developer game yang sukses dengan mengimplementasikan metode ini, metode ini banyak digunkan pada geme dengan genre RPG, Rouglikes, Platformer, SandBox, Simulation dan lain sebagainya, Pada penelitian ini berfokus pada pengembangan sebuah game world generator untuk game berjenis open world yang berupa sebuah kepulauan dengan metode PCG dengan menggunakan algoritma perlin noise sebagai algoritma pembentuk textur utama pulau yang dimana pada penelitian ini memanfaatkan beberapa variable noise seperti octave, presistance dan lacunarity guna untuk menambah kontrol dari hasil textur yang dihasilkan serta algoritma penempatan pulau untuk membuat sebuah game world yang menyerupai sebuah kepulauan. Dari hasil uji generator terkait degan pengujian playability dan performa dapat disimpulkan bahwa generator yang dikembangkan playable serta performa yang dianaliasa menggunakan notasi Big O menunjukkan (linear). Abstract Game development is currently growing very fast, game development AI is the most discussed topic by most researchers especially in the developing of game content using the PCG (procedural content generation) method. In making a game world using the PCG method, many game developers have succeeded by implementing this method, this method is widely used on RPGs, Rouglikes, Platformers, SandBox, Simulations and ect,. This study focuses on developing a game world generator game for open world type games in the form of an archipelago using the PCG method using the noise perlin algorithm as the island's main texturizing algorithm which in this study utilizes several noise variables such as octave, presistance and use for add control of the texture results as well as the island placement algorithm’s to create a game world that resembles an archipelago form. From the generator test results related to the playability and performance testing, it shows that map are being generated by the generators are playable and performance that are analyzed using Big O notation show O (n) (linear).

A literature review of Empirical Evidence on Procedural Content Generation in Game-Related Implementation

Procedural Content Generation (PCG) is an emerging field of study in computer science that focuses on automating the process of generating content by using algorithm, making the content generation process with less human effort. However, a more specific empirical evidence on how it is being used in a game-related implementation are still lacking. This paper presents the findings of review performed in the past 5 years looking on how PCG are being applied in game-related content, whether it is from the basic paper characteristic to analyze the trends, the field of PCG itself, and the game domain of game model and game genre. The studies had shown that PCG are being used extensively in game-related content but has seen more uses on specific type of contents rather than being used ubiquitously in all sorts of contents. The result shown that there are not specific best type of PCG method or algorithm being used instead an array of approach can be used based on what content being created. Result also shown that PCG are being used in multiple type of games, but similarly, based on the paper found, only certain types of game benefits PCG extensively such as action and platforming games while other model and genre of games have not seen much PCG application yet. Further studies are also required to analyze how experimentation and evaluation of PCG are being done as well as PCG domain in educational games as well as game-based learning, the quality catachrestic being analyzed on the papers are also worth mentioning to understand the underlying result of PCG usage in game-related contents.

On single-step HSS iterative method with circulant preconditioner for fractional diffusion equations

By exploiting Toeplitz-like structure and non-Hermitian dense property of the discrete coefficient matrix, a new double-layer iterative method called SHSS-PCG method is employed to solve the linear systems originating from the implicit finite difference discretization of fractional diffusion equations (FDEs). The method is a combination of the single-step Hermitian and skew-Hermitian splitting (SHSS) method with the preconditioned conjugate gradient (PCG) method. Further, the new circulant preconditioners are proposed to improve the efficiency of SHSS-PCG method, and the computation cost is further reduced via using the fast Fourier transform (FFT). Theoretical analysis shows that the SHSS-PCG iterative method with circulant preconditioners is convergent. Numerical experiments are given to show that our SHSS-PCG method with circulant preconditioners preforms very well, and the proposed circulant preconditioners are very efficient in accelerating the convergence rate.

Reinforcement Learning Content Generation for Virtual Reality Applications

This work presents a Procedural Content Generation (PCG) method based on a Neural Network Reinforcement Learning (RL) approach that generates new environments for Virtual Reality (VR) learning applications. The primary objective of PCG methods is to algorithmically generate new content (e.g., environments, levels) in order to improve user experience. Researchers have started exploring the integration of Machine Learning (ML) algorithms into their PCG methods. These ML approaches help explore the design space and generate new content more efficiently. The capability to provide users with new content has great potential for learning applications. However, these ML algorithms require large datasets to train their generative models. In contrast, RL based methods take advantage of simulation to train their models. Moreover, even though VR has become an emerging technology to engage users, there have been few studies that explore PCG for learning purposes and fewer in the context of VR. Considering these limitations, this work presents a method that generates new VR environments by training an RL agent using a simulation platform. This PCG method has the potential to maintain users’ engagement over time by presenting them with new environments in VR learning applications.