Keerthi Gaddameedi | Searcharxiv

arXiv2026

Adaptive Parallel-in-Time Integration with Dynamic Resource Management

As computational resources continue to grow, the strong-scaling limitations of spatial parallelism motivate the pursuit of additional concurrency in the temporal dimension, particularly for applications with hard time constraints, such as weather and climate simulations. The Parallel Full Approximation Scheme in Space and Time (PFASST) is a parallel-in-time method based on Spectral Deferred Corrections (SDC). It computes multiple timesteps concurrently by coupling fine- and coarse-grid SDC sweeps using multigrid Full Approximation Scheme (FAS) corrections. However, PFASST's convergence is often problem-dependent, demanding a variable number of parallel timesteps and, hence, computing resources at different times throughout the simulation. Dynamic Resource Management (DRM) provides a remedy for this challenge by enabling the adaptive adjustment of computational resources and algorithmic parameters at runtime. In this work, we present our novel approach to extending PFASST with DRM, which enables (a) dynamic adaptation of computing resources, (b) adaptive selection of the number of PFASST iterations based on local convergence behavior, and (c) coupling of these two adaptations into a single resizing strategy. With this approach, we demonstrate for the first time that optimal configurations can be identified in real time for each application, rather than relying on static allocation. Furthermore, we show that convergence-informed tuning of PFASST improves resource utilization and convergence efficiency.

cs.CE↗

arXiv2023

Efficient and Scalable Kernel Matrix Approximations using Hierarchical Decomposition

With the emergence of Artificial Intelligence, numerical algorithms are moving towards more approximate approaches. For methods such as PCA or diffusion maps, it is necessary to compute eigenvalues of a large matrix, which may also be dense depending on the kernel. A global method, i.e. a method that requires all data points simultaneously, scales with the data dimension N and not with the intrinsic dimension d; the complexity for an exact dense eigendecomposition leads to $\mathcal{O}(N^{3})$. We have combined the two frameworks, $\mathsf{datafold}$ and $\mathsf{GOFMM}$. The first framework computes diffusion maps, where the computational bottleneck is the eigendecomposition while with the second framework we compute the eigendecomposition approximately within the iterative Lanczos method. A hierarchical approximation approach scales roughly with a runtime complexity of $\mathcal{O}(Nlog(N))$ vs. $\mathcal{O}(N^{3})$ for a classic approach. We evaluate the approach on two benchmark datasets -- scurve and MNIST -- with strong and weak scaling using OpenMP and MPI on dense matrices with maximum size of $100k\times100k$.

math.NA↗