Browsing by Subject "CPD"
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Item Open Access Parafac-spark: parallel tensor decompositions on spark(Bilkent University, 2019-08) Bekçe, Selim ErenTensors are higher order matrices, widely used in many data science applications and scienti c disciplines. The Canonical Polyadic Decomposition (also known as CPD/PARAFAC) is a widely adopted tensor factorization to discover and extract latent features of tensors usually applied via alternating squares (ALS) method. Developing e cient parallelization methods of PARAFAC on commodity clusters is important because as common tensor sizes reach billions of nonzeros, a naive implementation would require infeasibly huge intermediate memory sizes. Implementations of PARAFAC-ALS on shared and distributedmemory systems are available, but these systems require expensive cluster setups, are too low level, not compatible with modern tooling and not fault tolerant by design. Many companies and data science communities widely prefer Apache Spark, a modern distributed computing framework with in-memory caching, and Hadoop ecosystem of tools for their ease of use, compatibility, ability to run on commodity hardware and fault tolerance. We developed PARAFAC-SPARK, an e cient, parallel, open-source implementation of PARAFAC on Spark, written in Scala. It can decompose 3D tensors stored in common coordinate format in parallel with low memory footprint by partitioning them as grids and utilizing compressed sparse rows (CSR) format for e cient traversals. We followed and combined many of the algorithmic and methodological improvements of its predecessor implementations on Hadoop and distributed memory, and adapted them for Spark. During the kernel MTTKRP operation, by applying a multi-way dynamic partitioning scheme, we were also able to increase the number of reducers to be on par with the number of cores to achieve better utilization and reduced memory footprint. We ran PARAFAC-SPARK with some real world tensors and evaluated the e ectiveness of each improvement as a series of variants compared with each other, as well as with some synthetically generated tensors up to billions of rows to measure its scalability. Our fastest variant (PS-CSRSX ) is up to 67% faster than our baseline Spark implementation (PS-COO) and up to 10 times faster than the state of art Hadoop implementations.Item Open Access Relationships between the material parameters of continuum-kinematics-inspired peridynamics and isotropic linear elasticity for two-dimensional problems(Elsevier Ltd, 2021-12-06) Ekiz, Ekim; Steinmann, P.; Javili, AliContinuum-kinematics-inspired Peridynamics (CPD) has been recently proposed as a geometrically exact formulation of peridynamics that is also thermodynamically and variationally consistent. CPD can capture the Poisson effect exactly, unlike the original formulation of peridynamics (PD). Due to its geometrically exact nature, CPD does not suffer from zero-energy modes and displacement oscillations that may be observed in state-based PD formulations. For a two-dimensional analysis, CPD builds upon one-neighbor and two-neighbor interactions. The one-neighbor interactions of CPD are equivalent to the bond-based interactions of the original PD formalism. Two-neighbor interactions, however, are key in CPD since the basic notions of classical continuum kinematics, namely length and area, are preserved exactly. The isotropic two-dimensional CPD formulation of non-local elasticity therefore involves two material constants, namely C1 and C2, associated with length and area, respectively. This manuscript aims to establish relationships between the material parameters of CPD and isotropic linear elasticity for an affine deformation in a two-dimensional setting. It is shown that each of the CPD material parameters can be expressed in terms of any pairs of isotropic linear elasticity constants, such as Lamé parameters. Finally, we establish the admissible ranges for CPD material parameters.