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UID:216@cds.iisc.ac.in
DTSTART;TZID=Asia/Kolkata:20260730T153000
DTEND;TZID=Asia/Kolkata:20260730T163000
DTSTAMP:20260723T065407Z
URL:https://cds.iisc.ac.in/events/ph-d-thesis-colloquium-102-cds-30-july-2
 026-hardware-aware-matrix-free-algorithms-for-accelerating-large-scale-fin
 ite-element-simulations-on-distributed-cpu-and-gpu-architectures/
SUMMARY:Ph.D: Thesis Colloquium: 102: CDS: 30\, July 2026 “Hardware-Aware
  Matrix-Free Algorithms for Accelerating Large-Scale Finite-Element Simula
 tions on Distributed CPU and GPU Architectures”
DESCRIPTION:DEPARTMENT OF COMPUTATIONAL AND DATA SCIENCES\nPh.D. Thesis Col
 loquium\n\n\n\nSpeaker : Mr. Gourab Panigrahi\nS.R. Number : 06-18-01-10-1
 2-21-1-19282\nTitle : “Hardware-Aware Matrix-Free Algorithms for Acceler
 ating Large-Scale\nFinite-Element Simulations on Distributed CPU and GPU A
 rchitectures”\nResearch Supervisor : Dr. Phani Motamarri\nDate &amp\; Ti
 me : July 30\, 2026 (Thursday)\, 15:30 PM\nVenue : #102\, CDS Seminar Hall
 \n\n\n\nABSTRACT\nMost large-scale simulations governed by partial differe
 ntial equations (PDEs) reduce\, after numerical discretization\, to two re
 curring computational problems: eigenproblems solved by subspace iteration
  approaches\, and linear systems by preconditioned Krylov methods. On pres
 ent-day supercomputers\, the performance of the underlying solvers is limi
 ted by data movement through deep memory hierarchies and across interconne
 cts rather than by floating-point throughput. This thesis addresses these 
 issues within the framework of higher-order finite-element (FE) discretiza
 tions\, where their dominant kernels are the action of the FE-discretized 
 operator on large blocks of vectors within eigensolvers and the solution o
 f FE-discretized generalized Poisson or Helmholtz systems within linear so
 lvers. These kernels have traditionally relied on assembled global sparse-
 matrix or cell-matrix strategies\, both of which scale poorly at high FE p
 olynomial orders: the former limited by data movement\, the latter by an o
 peration count that grows as the sixth power of the polynomial order. Addr
 essing these limitations\, we develop hardware-aware\, matrix-free algorit
 hms that evaluate operator actions on the fly through sum factorization\, 
 exploiting the tensor-product structure of the FE basis to reduce both ope
 ration count and memory footprint. Combined with communication-efficient a
 pproximation-tolerant eigensolvers\, tensor-structured direct linear solve
 rs\, and device-native multigrid preconditioners\, the performance of thes
 e methods has been demonstrated on the world's foremost supercomputing pla
 tforms. As the driving application\, we adopt the FE discretization of Koh
 n–Sham density functional theory (DFT)\, one of the most demanding HPC-i
 ntensive workloads in computational materials science\, on which these alg
 orithms are shown to substantially accelerate the time-to-solution.\n\nWe 
 first develop matrix-free algorithms for FE-discretized Helmholtz matrix
 –multivector products on multi-node CPU and GPU architectures\, where ex
 isting implementations are not designed for operator actions on large bloc
 ks of vectors. To this end\, we design batched evaluations with hardware-t
 ailored batch sizes to improve data locality\, together with even–odd de
 composition and SIMD vectorization on CPUs and kernel fusion\, constant-me
 mory strategies and register blocking on GPUs\, overlapping compute with d
 ata movement. For the operator action on 1024 vectors\, the implementation
  is up to 2.8× faster than the cell-matrix baseline on a single V100 node
  of OLCF Summit and 2.6× on a single Intel Xeon node of PARAM Pravega\, a
 nd up to 17× and 6.5× faster than the single-vector deal.II matrix-free 
 implementation and libCEED–MFEM\, respectively. Embedded in a Chebyshev-
 filtered subspace iteration (ChFSI) eigensolver for the Helmholtz operator
 \, the speedups over the cell-matrix approach reach 4.0× and 2.2× on a s
 ingle CPU and GPU node\, and 3.7× on ~3000 Xeon cores of Pravega and 1.4
 × on ~25 V100 GPUs of Summit. This framework also underpinned the DFT ele
 ctrostatics Poisson solves in the ACM Gordon Bell Prize-winning SC'23 stud
 y of quasicrystals and interacting extended defects\, which scaled to 64\,
 000 AMD MI250X GCDs of the OLCF Frontier exascale system.\n\nBuilding on t
 hese ideas\, we develop a matrix-free algorithm for the FE-discretized Koh
 n–Sham DFT eigenproblem on distributed CPU architectures. The underlying
  local and nonlocal operator actions on multivectors are recast as structu
 red tensor contractions over one-dimensional FE basis functions and quadra
 ture data. A unified batched layout is proposed that handles the action of
  both real- and complex-valued operators on multivectors while maximizing 
 cache reuse and SIMD utilization across AVX2\, AVX-512\, and SVE\, complem
 ented by even–odd decomposition and mixed-precision intrinsics. Matrix
 –multivector product benchmarks on OLCF Frontier (AMD EPYC)\, PARAM Prav
 ega (Intel Xeon)\, and RIKEN Fugaku (Fujitsu A64FX) show speedups of up to
  4–6× on representative material systems over the cell-matrix baseline 
 of the DFT-FE library\, with further end-to-end gains from a residual-base
 d ChFSI (R-ChFSI) eigensolver that tolerates inexact matrix–vector produ
 cts without sacrificing robustness.\n\nTaking this further to GPU architec
 tures\, we fuse the entire discretized Kohn–Sham Hamiltonian operator ac
 tion into a single kernel that combines a register–shared-memory hybrid 
 sum factorization\, constant-memory shape-function matrices\, even–odd d
 ecomposition\, and a precision-aware batched layout\, distributed across r
 anks by the Morton (Z-order) numbering for locality. FE constraints arisin
 g from boundary conditions or from hanging nodes on non-conforming meshes 
 are handled inline to avoid atomic operations\, and a reduced-precision (B
 F16) halo exchange is overlapped with computation without affecting the se
 lf-consistent field (SCF) convergence of the underlying DFT nonlinear eige
 nproblem. TF32 tensor cores and A100 asynchronous copies (cp.async) furthe
 r improve compute throughput and memory bandwidth. On the NVIDIA A100 GPUs
  of ALCF Polaris\, the filtering step in R-ChFSI eigensolver achieves spee
 dups up to 3.9× on representative benchmarks\, sustained in strong scalin
 g to 128 A100 GPUs.\n\nWe now turn to the linear systems arising in FE dis
 cretizations. When the mesh retains a global tensor-product structure\, it
 erative solution can be avoided altogether. We develop a distributed-memor
 y GPU fast-diagonalization (FD) solver that inverts the assembled FE Lapla
 cian exactly in a single\, non-iterative solve at O(N^(4/3)) FLOPs\, where
  N is the number of degrees of freedom\, supporting Dirichlet\, periodic\,
  and semi-periodic boundary conditions on rectilinear meshes. For the sing
 ular fully periodic case\, we deflate the constant mode rather than pinnin
 g a node\, keeping the tensor structure intact and avoiding the ill-condit
 ioning that pinning introduces. With O(N/P) compute and memory on each of 
 P ranks\, the solver scales efficiently on distributed GPUs. Against a con
 jugate-gradient (CG) solver with Jacobi preconditioner\, the proposed FD s
 olver is 125–140× faster on uniform meshes. On an atomistic-scale batte
 ry electrode–electrolyte interface (LLZO) discretized on an FE mesh (12.
 3 million DoF\, periodic in two directions\, Dirichlet in the third) and s
 olved to machine precision on 64 GPUs of Polaris\, the speedup is about 50
 0×\, rising to 4600× on strongly graded meshes. This advantage widens as
  the mesh becomes more graded\, because the cost and accuracy of the solve
 r are independent of the mesh-induced condition number.\n\nOn general mesh
 es\, where the tensor-product structure is lost and iterative solution is 
 unavoidable\, we develop a GPU-accelerated matrix-free preconditioned CG s
 olver. It fuses the Laplace kernel with the CG iteration\, uses tree-reduc
 tion inner products and a Chebyshev–Jacobi preconditioner free of global
  reductions. On 32 nodes (384 Intel Data Center Max GPUs) of the ALCF Auro
 ra exascale system\, it attains 5.86× higher throughput than the deal.II 
 matrix-free implementation. Since the iteration count of this single-level
  preconditioner grows with FE polynomial order\, we extend it to adaptivel
 y refined meshes by developing a fully device-native p-multigrid precondit
 ioner whose iteration count is nearly independent of the polynomial order.
  We further test this preconditioner on the variable-coefficient generaliz
 ed Poisson operators arising in the continuum solvation models used for el
 ectrochemical systems in DFT\, for which no fast direct solver exists. On 
 a mesh that models the Trp-cage protein (304 atoms\, 21.4 million DoF) in 
 implicit solvent\, it reduces CG iterations from 2583 to 147\, a 17.6× re
 duction.\n\nOverall\, this thesis develops a set of hardware-aware algorit
 hms for accelerating the two classes of computational problems that arise 
 in FE discretizations of PDEs\, namely eigenproblems and linear systems. I
 mplemented within the DFT-FE library and validated against production base
 lines at desired accuracy\, these methods reduce the time-to-solution of l
 arge-scale simulations from single-node CPUs to exascale GPU machines\, in
  several cases by orders of magnitude. The kernels\, data layouts\, and so
 lver designs make no assumptions specific to DFT\, and therefore carry ove
 r directly to higher-order FE discretizations in other scientific areas on
  modern HPC architectures.\n\n\n\nALL ARE WELCOME
CATEGORIES:Events,Ph.D. Thesis Colloquium
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