arXiv subjects

Vishwas Rao

Publications and source records attributed to Vishwas Rao.

At least 19 recordsLinked to original sources

arXiv2026

Towards exascale fully relativistic pseudopotential density functional theory calculations enabled by mixed-precision computation and compressed-communication using residual based subspace iteration

Materials exhibiting noncollinear magnetism or strong spin-orbit-coupling underpin many spintronic and topological applications, but their simulations require complex two-component spinors and costs substantially more than scalar density functional theory (DFT). We present a GPU-centric exascale finite-element DFT framework with noncollinear magnetism and spin-orbit-coupling, combining (i) adaptive higher-order finite-element discretization, (ii) a matrix-free Poisson solver, (iii) residual-based Chebyshev filtered subspace iteration (R-ChFSI) for sparse generalized eigenproblems, (iv) R-ChFSI-enabled mixed precision computation with block floating-point compressed MPI communication, and (v) communication-efficient band partitioning. R-ChFSI permits inexact matrix-multivector products and $4\times$--$6.4\times$ compression of communicated data relative to FP64 while preserving double-precision robustness and reducing computation and data movement. Results demonstrate strong scaling on Aurora and Frontier, with up to $3.4\times$ faster Chebyshev filtering. On Aurora, total self-consistent-field (SCF) solve wall time reduces by $2\times$. An 800-node (9,600-GPU) calculation demonstrates fully relativistic pseudopotential DFT for $\sim$80,000 electrons in under 7 minutes per SCF iteration.

cond-mat.mtrl-sci↗

arXiv2026

High-Performance Star-M SVD for Big Data Compression

In the era of big data, effectively compressing large datasets while performing complex mathematical operations is crucial. Tensor-based decomposition methods have shown superior compression capabilities with minimal loss of accuracy compared to traditional matrix methods. Under the star-M tensor framework, tensors can be decomposed in a matrix-mimetic way, including using the star-M SVD. This tensor SVD has optimality guarantees and has shown exceptional performance on specific types of data, but software implementations have been mostly limited to productivity-oriented languages. In this work, we present our development of a shared-memory parallel, high-performance solution designed to efficiently implement the underlying algorithms. This software will enable optimal compression of extensive scientific datasets, paving the way for enhanced data analysis and insights.

cs.DC↗

arXiv2026

Rate-Distortion Bounds for Heterogeneous Random Fields on Finite Lattices

Since Shannon's foundational work, rate-distortion theory has defined the fundamental limits of lossy compression. Classical results, derived for memoryless and stationary ergodic sources in the asymptotic regime, have shaped both transform and predictive coding architectures, as well as practical standards such as JPEG. Finite-blocklength refinements, initiated by the non-asymptotic achievability and converse bounds of Kostina and Verdu, provide precise characterizations under excess-distortion probability constraints, but primarily for memoryless or statistically homogeneous models. In contrast, error-bounded practical lossy compressors for scientific computing, such as SZ, ZFP, MGARD, and SPERR, are designed for finite, high-dimensional, spatially correlated, and statistically heterogeneous random fields. These compressors partition data into fixed-size tiles that are processed independently, making tile size a central architectural constraint. Structural heterogeneity, finite lattice effects, and tiling constraints are not addressed by existing finite-blocklength analyses. This paper introduces a finite-blocklength rate-distortion framework for heterogeneous random fields on finite lattices, explicitly accounting for the tile-based architectures used in high-performance scientific compressors. The field is modeled as piecewise homogeneous with regionwise stationary second-order statistics, and tiling constraints are incorporated directly into the source model. Under an excess-distortion probability criterion, we establish non-asymptotic achievability, converse bounds and derive a second-order expansion that quantifies the impact of spatial correlation, region geometry, heterogeneity, and tile size on the rate and dispersion.

cs.IT↗

arXiv2025

Parametric Hierarchical Matrix Approximations to Kernel Matrices

Kernel matrices are ubiquitous in computational mathematics, often arising from applications in machine learning and scientific computing. In two or three spatial or feature dimensions, such problems can be approximated efficiently by a class of matrices known as hierarchical matrices. A hierarchical matrix consists of a hierarchy of small near-field blocks (or sub-matrices) stored in a dense format and large far-field blocks approximated by low-rank matrices. Standard methods for forming hierarchical matrices do not account for the fact that kernel matrices depend on specific hyperparameters; for example, in the context of Gaussian processes, hyperparameters must be optimized over a fixed parameter space. We introduce a new class of hierarchical matrices, namely, parametric (parameter-dependent) hierarchical matrices. Members of this new class are parametric $\mathcal{H}$-matrices and parametric $\mathcal{H}^{2}$-matrices. The construction of a parametric hierarchical matrix follows an offline-online paradigm. In the offline stage, the near-field and far-field blocks are approximated by using polynomial approximation and tensor compression. In the online stage, for a particular hyperparameter, the parametric hierarchical matrix is instantiated efficiently as a standard hierarchical matrix. The asymptotic costs for storage and computation in the offline stage are comparable to the corresponding standard approaches of forming a hierarchical matrix. However, the online stage of our approach requires no new kernel evaluations, and the far-field blocks can be computed more efficiently than standard approaches. {Numerical experiments show over $100\times$ speedups compared with existing techniques.}

math.NA↗

arXiv2025

Latent Twins

Over the past decade, scientific machine learning has transformed the development of mathematical and computational frameworks for analyzing, modeling, and predicting complex systems. From inverse problems to numerical PDEs, dynamical systems, and model reduction, these advances have pushed the boundaries of what can be simulated. Yet they have often progressed in parallel, with representation learning and algorithmic solution methods evolving largely as separate pipelines. With \emph{Latent Twins}, we propose a unifying mathematical framework that creates a hidden surrogate in latent space for the underlying equations. Whereas digital twins mirror physical systems in the digital world, Latent Twins mirror mathematical systems in a learned latent space governed by operators. Through this lens, classical modeling, inversion, model reduction, and operator approximation all emerge as special cases of a single principle. We establish the fundamental approximation properties of Latent Twins for both ODEs and PDEs and demonstrate the framework across three representative settings: (i) canonical ODEs, capturing diverse dynamical regimes; (ii) a PDE benchmark using the shallow-water equations, contrasting Latent Twin simulations with DeepONet and forecasts with a 4D-Var baseline; and (iii) a challenging real-data geopotential reanalysis dataset, reconstructing and forecasting from sparse, noisy observations. Latent Twins provide a compact, interpretable surrogate for solution operators that evaluate across arbitrary time gaps in a single-shot, while remaining compatible with scientific pipelines such as assimilation, control, and uncertainty quantification. Looking forward, this framework offers scalable, theory-grounded surrogates that bridge data-driven representation learning and classical scientific modeling across disciplines.

cs.LG↗

arXiv2025

Scientific Applications Leveraging Randomized Linear Algebra

This report showcases the role of, and future directions for, the field of Randomized Numerical Linear Algebra (RNLA) in a selection of scientific applications. These applications span the domains of imaging, genomics and dynamical systems, and are thematically connected by needing to perform linear algebra routines on large-scale matrices (with up to quantillions of entries). At such scales, the linear algebra routines face typical bottlenecks: memory constraints, data access latencies, and substantial floating-point operation costs. RNLA routines are discussed at a high-level to demonstrate how these routines are able to solve the challenges faced by traditional linear algebra routines, and, consequently, address the computational problem posed in the underlying application. For each application, RNLA's open challenges and possible future directions are also presented, which broadly fall into the categories: creating structure-aware RNLA algorithms; co-designing RNLA algorithms with hardware and mixed-precision considerations; and advancing modular, composable software infrastructure. Ultimately, this report serves two purposes: it invites domain scientists to engage with RNLA; and it offers a guide for future RNLA research grounded in real applications.

math.NA↗

arXiv2025

Structured Column Subset Selection for Bayesian Optimal Experimental Design

We consider optimal experimental design (OED) for Bayesian inverse problems, where the experimental design variables have a certain multiway structure. Given $d$ different experimental variables with $m_i$ choices per design variable $1 \le i\le d$, the goal is to select $k_i \le m_i$ experiments per design variable. Previous work has related OED to the column subset selection problem by mapping the design variables to the columns of a matrix $\mathbf{A}$. However, this approach is applicable only to the case $d=1$ in which the columns can be selected independently. We develop an extension to the case where the design variables have a multi-way structure. Our approach is to map the matrix $\mathbf{A}$ to a tensor and perform column subset selection on mode unfoldings of the tensor. We develop an algorithmic framework with three different algorithmic templates, and randomized variants of these algorithms. We analyze the computational cost of all the proposed algorithms and also develop greedy versions to facilitate comparisons. Numerical experiments on four different applications -- time-dependent inverse problems, seismic tomography, X-ray tomography, and flow reconstruction -- demonstrate the effectiveness and scalability of our methods for structured experimental design in Bayesian inverse problems.

math.NA↗

arXiv2025

Improving the Predictability of the Madden-Julian Oscillation at Subseasonal Scales with Gaussian Process Models

The Madden--Julian Oscillation (MJO) is an influential climate phenomenon that plays a vital role in modulating global weather patterns. In spite of the improvement in MJO predictions made by machine learning algorithms, such as neural networks, most of them cannot provide the uncertainty levels in the MJO forecasts directly. To address this problem, we develop a nonparametric strategy based on Gaussian process (GP) models. We calibrate GPs using empirical correlations and we propose a posteriori covariance correction. Numerical experiments demonstrate that our model has better prediction skills than the ANN models for the first five lead days. Additionally, our posteriori covariance correction extends the probabilistic coverage by more than three weeks.

math.NA↗

arXiv2025

Weakly-Constrained 4D Var for Downscaling with Uncertainty using Data-Driven Surrogate Models

Dynamic downscaling typically involves using numerical weather prediction (NWP) solvers to refine coarse data to higher spatial resolutions. Data-driven models such as FourCastNet have emerged as a promising alternative to the traditional NWP models for forecasting. Once these models are trained, they are capable of delivering forecasts in a few seconds, thousands of times faster compared to classical NWP models. However, as the lead times, and, therefore, their forecast window, increase, these models show instability in that they tend to diverge from reality. In this paper, we propose to use data assimilation approaches to stabilize them when used for downscaling tasks. Data assimilation uses information from three different sources, namely an imperfect computational model based on partial differential equations (PDE), from noisy observations, and from an uncertainty-reflecting prior. In this work, when carrying out dynamic downscaling, we replace the computationally expensive PDE-based NWP models with FourCastNet in a ``weak-constrained 4DVar framework" that accounts for the implied model errors. We demonstrate the efficacy of this approach for a hurricane-tracking problem; moreover, the 4DVar framework naturally allows the expression and quantification of uncertainty. We demonstrate, using ERA5 data, that our approach performs better than the ensemble Kalman filter (EnKF) and the unstabilized FourCastNet model, both in terms of forecast accuracy and forecast uncertainty.

physics.ao-ph↗

arXiv2025

Recovering sparse DFT from missing signals via interior point method on GPU

We propose a method to recover the sparse discrete Fourier transform (DFT) of a signal that is both noisy and potentially incomplete, with missing values. The problem is formulated as a penalized least-squares minimization based on the inverse discrete Fourier transform (IDFT) with an $\ell_1$-penalty term, reformulated to be solvable using a primal-dual interior point method (IPM). Although Krylov methods are not typically used to solve Karush-Kuhn-Tucker (KKT) systems arising in IPMs due to their ill-conditioning, we employ a tailored preconditioner and establish new asymptotic bounds on the condition number of preconditioned KKT matrices. Thanks to this dedicated preconditioner -- and the fact that FFT and IFFT operate as linear operators without requiring explicit matrix materialization -- KKT systems can be solved efficiently at large scales in a matrix-free manner. Numerical results from a Julia implementation leveraging GPU-accelerated interior point methods, Krylov methods, and FFT toolkits demonstrate the scalability of our approach on problems with hundreds of millions of variables, inclusive of real data obtained from the diffuse scattering from a slightly disordered Molybdenum Vanadium Dioxide crystal.

math.OC↗

arXiv2025

Optimal sensor placement under model uncertainty in the weak-constraint 4D-Var framework

In data assimilation, the model may be subject to uncertainties and errors. The weak-constraint data assimilation framework enables incorporating model uncertainty in the dynamics of the governing equations. We propose a new framework for near-optimal sensor placement in the weak-constrained setting. This is achieved by first deriving a design criterion based on the expected information gain, which involves the Kullback-Leibler divergence from the forecast prior to the posterior distribution. An explicit formula for this criterion is provided, assuming that the model error and background are independent and Gaussian and the dynamics are linear. We discuss algorithmic approaches to efficiently evaluate this criterion through randomized approximations. To provide further insight and flexibility in computations, we also provide alternative expressions for the criteria. We provide an algorithm to find near-optimal experimental designs using column subset selection, including a randomized algorithm that avoids computing the adjoint of the forward operator. Through numerical experiments in one and two spatial dimensions, we show the effectiveness of our proposed methods.

math.NA↗

arXiv2024

Scalable Multi-Period AC Optimal Power Flow Utilizing GPUs with High Memory Capacities

This paper demonstrates the scalability of open-source GPU-accelerated nonlinear programming (NLP) frameworks -- ExaModels.jl and MadNLP.jl -- for solving multi-period alternating current (AC) optimal power flow (OPF) problems on GPUs with high memory capacities (e.g., NVIDIA GH200 with 480 GB of unified memory). There has been a growing interest in solving multi-period AC OPF problems, as the increasingly fluctuating electricity market requires operation planning over multiple periods. These problems, formerly deemed intractable, are now becoming technologically feasible to solve thanks to the advent of high-memory GPU hardware and accelerated NLP tools. This study evaluates the capability of these tools to tackle previously unsolvable multi-period AC OPF instances. Our numerical experiments, run on an NVIDIA GH200, demonstrate that we can solve a multi-period OPF instance with more than 10 million variables up to $10^{-4}$ precision in less than 10 minutes. These results demonstrate the efficacy of the GPU-accelerated NLP frameworks for the solution of extreme-scale multi-period OPF. We provide ExaModelsPower.jl, an open-source modeling tool for multi-period AC OPF models for GPUs.

math.OC↗

arXiv2024

Bayesian D-Optimal Experimental Designs via Column Subset Selection

This paper tackles optimal sensor placement for Bayesian linear inverse problems, a popular version of the more general Optimal Experimental Design (OED) problem, using the D-optimality criterion. This is done by establishing connections between sensor placement and Column Subset Selection Problem (CSSP), which is a well-studied problem in Numerical Linear Algebra (NLA). In particular, we use the Golub-Klema-Stewart (GKS) approach which involves computing the truncated Singular Value Decomposition (SVD) followed by a pivoted QR factorization on the right singular vectors. The algorithms are further accelerated by using randomization to compute the low-rank approximation as well as for sampling the indices. The resulting algorithms are robust, computationally efficient, amenable to parallelization, require virtually no parameter tuning, and come with strong theoretical guarantees. One of the proposed algorithms is also adjoint-free which is beneficial in situations, where the adjoint is expensive to evaluate or is not available. Additionally, we develop a method for data completion without solving the inverse problem. Numerical experiments on model inverse problems involving the heat equation and seismic tomography in two spatial dimensions demonstrate the performance of our approaches.

math.NA↗

arXiv2024

Randomized Preconditioned Solvers for Strong Constraint 4D-Var Data Assimilation

The Strong Constraint 4D Variational (SC-4DVAR) data assimilation method is widely used in climate and weather applications. SC-4DVAR involves solving a minimization problem to compute the maximum a posteriori estimate, which we tackle using the Gauss-Newton method. The computation of the descent direction is expensive since it involves the solution of a large-scale and potentially ill-conditioned linear system, solved using the preconditioned conjugate gradient (PCG) method. To address this cost, we efficiently construct scalable preconditioners using three different randomization techniques, which all rely on a certain low-rank structure involving the Gauss-Newton Hessian. The proposed techniques come with theoretical guarantees on the condition number, and at the same time, are amenable to parallelization. We also develop an adaptive approach to estimate the sketch size and choose between the reuse or recomputation of the preconditioner. We demonstrate the performance and effectiveness of our methodology on two representative model problems--the Burgers and barotropic vorticity equation--showing a drastic reduction in both the number of PCG iterations and the number of Gauss-Newton Hessian products after including the preconditioner construction cost.

math.NA↗

arXiv2023

Credible Intervals for Probability of Failure with Gaussian Processes

Estimating the probability of failure for expensive simulations is a central task in reliability analysis for structural design, power grid design, and safety certification, among other areas. This work derives credible intervals on the probability of failure by modeling the simulation as a realization of a Gaussian process surrogate. These intervals are governed by the pointwise binary classification error of the surrogate and are compatible with the broad class of adaptive sampling schemes proposed in the literature. We further propose a novel batch sampling scheme that suggests multiple evaluation points per iteration, enabling parallel simulation on HPC systems. The method is empirically validated using our scalable, open-source implementation on a variety of test problems including a Tsunami model where failure is quantified in terms of maximum wave height.

stat.ME↗

arXiv2022

CAMERA: A Method for Cost-aware, Adaptive, Multifidelity, Efficient Reliability Analysis

Estimating probability of failure in aerospace systems is a critical requirement for flight certification and qualification. Failure probability estimation involves resolving tails of probability distribution, and Monte Carlo sampling methods are intractable when expensive high-fidelity simulations have to be queried. We propose a method to use models of multiple fidelities that trade accuracy for computational efficiency. Specifically, we propose the use of multifidelity Gaussian process models to efficiently fuse models at multiple fidelity, thereby offering a cheap surrogate model that emulates the original model at all fidelities. Furthermore, we propose a novel sequential \emph{acquisition function}-based experiment design framework that can automatically select samples from appropriate fidelity models to make predictions about quantities of interest in the highest fidelity. We use our proposed approach in an importance sampling setting and demonstrate our method on the failure level set estimation and probability estimation on synthetic test functions as well as two real-world applications, namely, the reliability analysis of a gas turbine engine blade using a finite element method and a transonic aerodynamic wing test case using Reynolds-averaged Navier--Stokes equations. We demonstrate that our method predicts the failure boundary and probability more accurately and computationally efficiently while using varying fidelity models compared with using just a single expensive high-fidelity model.

math.NA↗

arXiv2022

Learning the Evolution of Correlated Stochastic Power System Dynamics

A machine learning technique is proposed for quantifying uncertainty in power system dynamics with spatiotemporally correlated stochastic forcing. We learn one-dimensional linear partial differential equations for the probability density functions of real-valued quantities of interest. The method is suitable for high-dimensional systems and helps to alleviate the curse of dimensionality.

cs.LG↗

arXiv2022

Order-Disorder Transitions in (Ca$_x$Sr$_{1-x}$)$_3$Rh$_4$Sn$_{13}$

The classification of structural phase transitions as displacive or order-disorder in character is usually based on spectroscopic data above the transition. We use single crystal x-ray diffraction to investigate structural correlations in the quasi-skutterudites, (Ca$_x$Sr$_{1-x}$)$_3$Rh$_4$Sn$_{13}$, which have a quantum phase transition at $x\sim0.9$. Three-dimensional pair distribution functions show that the amplitudes of local atomic displacements are temperature-independent below the transition and persist to well above the transition, a signature of order-disorder behavior. The implications for the associated electronic transitions are discussed.

cond-mat.str-el↗