X-ray computed tomography (XCT)is used regularly at synchrotron light sources to study the internal morphology of materials at high resolution. However, experimental constraints, such as radiation sensitivity, can result in noisy or undersampled measurements. Further, depending on the resolution, sample size and data acquisition rates, the resulting noisy dataset can be terabyte-scale. Advanced iterative reconstruction techniques can produce high-quality images from noisy measurements, but their computational requirements have made their use exception rather than the rule. We propose here a novel memory-centric approach that avoids redundant computations at the expense of additional memory complexity. We develop a system, MemXCT, that uses an optimized SpMV implementation with two-level pseudo-Hilbert ordering and multi-stage input buffering. We evaluate MemXCT on various supercomputer architectures incolving KNL and GPU. MemXCT can reconstruct a large (11K×11K) mouse brain tomogram in ∼10 seconds using 4096 KNL nodes (256K cores), the largest iterative reconstruction achieved in near-real time.