arXiv subjects

C. T. Kelley

Publications and source records attributed to C. T. Kelley.

15 recordsLinked to original sources

arXiv2026

Complexity of Projected Gradient Methods for Strongly Convex Optimization with Hölder Continuous Gradient Terms

This paper studies the complexity of projected gradient descent methods for a class of strongly convex constrained optimization problems where the objective function is expressed as a summation of $m$ component functions, each possessing a gradient that is Hölder continuous with an exponent $α_i \in (0, 1]$. Under this formulation, the gradient of the objective function may fail to be globally Hölder continuous, thereby rendering existing complexity results inapplicable to this class of problems. Our theoretical analysis reveals that, in this setting, the complexity of projected gradient methods is determined by $\hatα = \min_{i \in \{1, \dotsc, m\}} α_i$. We first prove that, with an appropriately fixed stepsize, the complexity bound for finding an approximate minimizer with a distance to the true minimizer less than $\varepsilon$ is $O (\log (\varepsilon^{-1}) \varepsilon^{2 (\hatα - 1) / (1 + \hatα)})$, which extends the well-known complexity result for $\hatα = 1$. Next we show that the complexity bound can be improved to $O (\log (\varepsilon^{-1}) \varepsilon^{2 (\hatα - 1) / (1 + 3 \hatα)})$ if the stepsize is updated by the universal scheme. We illustrate our complexity results by numerical examples arising from elliptic equations with a non-Lipschitz term.

math.OC↗

arXiv2025

Using MultiPrecisonArrays.jl: Iterative Refinement in Julia

MultiPrecisionArrays.jl is a Julia package. This package provides data structures and solvers for several variants of iterative refinement. It will become much more useful when half precision (aka Float16) is fully supported in LAPACK/BLAS. For now, its only general-purpose application is classical iterative refinement with double precision equations and single precision factorizations.

math.NA↗

arXiv2025

A New Complexity Result for Strongly Convex Optimization with Locally $α$-H{ö}lder Continuous Gradients

In this paper, we present a new complexity result for the gradient descent method with an appropriately fixed stepsize for minimizing a strongly convex function with locally $α$-H{ö}lder continuous gradients ($0 < α\leq 1$). The complexity bound for finding an approximate minimizer with a distance to the true minimizer less than $\varepsilon$ is $O(\log (\varepsilon^{-1}) \varepsilon^{2 α- 2})$, which extends the well-known complexity result for $α= 1$.

math.OC↗

arXiv2025

A Batch Power Iteration Approach for the Iterative Quasi-Monte Carlo Method Using a Randomized-Halton Sequence

The Iterative Quasi-Monte Carlo (iQMC) method is a recently developed hybrid method for neutron transport simulations. iQMC replaces standard quadrature techniques used in deterministic linear solvers with Quasi-Monte Carlo simulation for accurate and efficient solutions to the neutron transport equation. Previous iQMC studies utilized a fixed-seed approach wherein particles were reset to the same initial position and direction of travel at the start of every transport sweep. While the QMC samples offered greatly improved uniformity compared to pseudo-random samples, the fixed-seed approach meant that some regions of the problem were under-sampled and resulted in errors similar to ray effects observed in discrete ordinates methods. This work explores using randomized-Quasi Monte Carlo techniques (RQMC) to generate unique sets of QMC samples for each transport sweep and gain a much-improved sampling of the phase space. The use of RQMC introduces some stochastic noise to iQMC's iterative process, which was previously absent. To compensate, we adopt a ``batch'' approach similar to typical Monte Carlo k-eigenvalue problems, where the iQMC source is converged over $N_\text{inactive}$ batches, then results from $N_\text{active}$ batches are recorded and used to calculate the average and standard deviation of the solution. The RQMC batch method was implemented in the Monte Carlo Dynamic Code (MC/DC) and is shown to be a large improvement over the fixed-seed method. The batch method was able to provide iteratively stable and more accurate solutions with nearly two orders of magnitude reduction in the number of particle histories per batch. Notably, despite introducing some stochastic noise to the solution, the RQMC batch approach converges both the k-effective and mean scalar flux error at the theoretical QMC convergence rate of $O(N^{-1})$.

physics.comp-ph↗

arXiv2024

Interprecision transfers in iterative refinement

We make the interprecision transfers explicit in an algorithmic description of iterative refinement and obtain new insights into the algorithm. One example is the classic variant of iterative refinement where the matrix and the factorization are stored in a working precision and the residual is evaluated in a higher precision. In that case we make the observation that this algorithm will solve a promoted form of the original problem and thereby characterize the limiting behavior in a novel way and obtain a different version of the classic convergence analysis. We also discuss two approaches for interprecision transfer in the triangular solves.

math.NA↗

arXiv2024

Mitigating Spatial Error in the iterative-Quasi-Monte Carlo (iQMC) Method for Neutron Transport Simulations with Linear Discontinuous Source Tilting and Effective Scattering and Fission Rate Tallies

The iterative Quasi-Monte Carlo (iQMC) method is a recently proposed method for multigroup neutron transport simulations. iQMC can be viewed as a hybrid between deterministic iterative techniques, Monte Carlo simulation, and Quasi-Monte Carlo techniques. iQMC holds several algorithmic characteristics that make it desirable for high performance computing environments including a $O(N^{-1})$ convergence scheme, ray tracing transport sweep, and highly parallelizable nature similar to analog Monte Carlo. While there are many potential advantages of using iQMC there are also inherent disadvantages, namely the spatial discretization error introduced from the use of a mesh across the domain. This work introduces two significant modifications to iQMC to help reduce the spatial discretization error. The first is an effective source transport sweep, whereby the source strength is updated on-the-fly via an additional tally. This version of the transport sweep is essentially agnostic to the mesh, material, and geometry. The second is the addition of a history-based linear discontinuous source tilting method. Traditionally, iQMC utilizes a piecewise-constant source in each cell of the mesh. However, through the proposed source tilting technique iQMC can utilize a piecewise-linear source in each cell and reduce spatial error without refining the mesh. Numerical results are presented from the 2D C5G7 and Takeda-1 k-eigenvalue benchmark problems. Results show that the history-based source tilting significantly reduces error in global tallies and the eigenvalue solution in both benchmarks. Through the effective source transport sweep and linear source tilting iQMC was able to converge the eigenvalue from the 2D C5G7 problem to less than $0.04\%$ error on a uniform Cartesian mesh with only $204\times204$ cells.

physics.comp-ph↗

arXiv2023

Newton's Method in Three Precisions

We describe a three precision variant of Newton's method for nonlinear equations. We evaluate the nonlinear residual in double precision, store the Jacobian matrix in single precision, and solve the equation for the Newton step with iterative refinement with a factorization in half precision. We analyze the method as an inexact Newton method. This analysis shows that, except for very poorly conditioned Jacobians, the number of nonlinear iterations needed is the same that one would get if one stored and factored the Jacobian in double precision. In many ill-conditioned cases one can use the low precision factorization as a preconditioner for a GMRES iteration. That approach can recover fast convergence of the nonlinear iteration. We present an example to illustrate the results.

math.NA↗

arXiv2023

Reducing Spatial Discretization Error with Linear Discontinuous Source Tilting in Iterative Quasi-Monte Carlo for Neutron Transport

Recently, iterative Quasi-Monte Carlo (iQMC) was introduced as a new method of neutron transport which combines deterministic iterative methods and quasi-Monte Carlo simulation for more efficient solutions to the neutron transport equation. Previous iQMC results utilized a uniform Cartesian grid with a piecewise-constant source. Similar to "teleportation error" in Implicit Monte Carlo (IMC) methods, the spatial discretization and piecewise-constant source can lead to a significant spatial error that limits convergence of the overall method. Taking concepts from IMC, we have developed a history-based discontinuous piecewise-linear source tilting scheme to reduce spatial error in iQMC. The source tilting method is described below and afterward we present results from a fixed-source 2D reactor-like problem adapted from the Takeda-1 Benchmark problem.

physics.comp-ph↗

arXiv2023

iQMC: Iterative Quasi-Monte Carlo for k-Eigenvalue Neutron Transport Simulations

The Iterative Quasi-Monte Carlo method, or iQMC, replaces standard quadrature techniques used in deterministic linear solvers with Quasi-Monte Carlo simulation for more accurate and efficient solutions to the neutron transport equation. This work explores employing iQMC in the Monte-Carlo Dynamic Code (MCDC) to solve k-eigenvalue problems for neutron transport with both the standard power iteration and the generalized Davidson method, a Krylov Subspace method. Results are verified with the 3-D, 2-group, Takeda-1 Benchmark problem.

physics.comp-ph↗

arXiv2022

Adaptive Basis Sets for Practical Quantum Computing

Electronic structure calculations on small systems such as H$_2$, H$_2$O, LiH, and BeH$_2$ with chemical accuracy are still a challenge for the current generation of the noisy intermediate-scale quantum (NISQ) devices. One of the reasons is that due to the device limitations, only minimal basis sets are commonly applied in quantum chemical calculations, which allow one to keep the number of qubits employed in the calculations at minimum. However, the use of minimal basis sets leads to very large errors in the computed molecular energies as well as potential energy surface shapes. One way to increase the accuracy of electronic structure calculations is through the development of small basis sets better suited for quantum computing. In this work, we show that the use of adaptive basis sets, in which exponents and contraction coefficients depend on molecular structure, provide an easy way to dramatically improve the accuracy of quantum chemical calculations without the need to increase the basis set size and thus the number of qubits utilized in quantum circuits. As a proof of principle, we optimize an adaptive minimal basis set for quantum computing calculations on an H$_2$ molecule, in which exponents and contraction coefficients depend on the H-H distance, and apply it to the generation of H$_2$ potential energy surface on IBM-Q quantum devices. The adaptive minimal basis set reaches the accuracy of the double-zeta basis sets, thus allowing one to perform double-zeta quality calculations on quantum devices without the need to utilize twice as many qubits in simulations. This approach can be extended to other molecular systems and larger basis sets in a straightforward manner.

physics.chem-ph↗

arXiv2022

A Quasi-Monte Carlo Method with Krylov Linear Solvers for Multigroup Neutron Transport Simulations

In this work we investigate replacing standard quadrature techniques used in deterministic linear solvers with a fixed-seed Quasi-Monte Carlo calculation to obtain more accurate and efficient solutions to the neutron transport equation (NTE). Quasi-Monte Carlo (QMC) is the use of low-discrepancy sequences to sample the phase space in place of pseudo-random number generators used by traditional Monte Carlo (MC). QMC techniques decrease the variance in the stochastic transport sweep and therefore increase the accuracy of the iterative method. Historically, QMC has largely been ignored by the particle transport community because it breaks the Markovian assumption needed to model scattering in analog MC particle simulations. However, by using iterative methods the NTE can be modeled as a pure-absorption problem. This removes the need to explicitly model particle scattering and provides an application well-suited for QMC. To obtain solutions we experimented with three separate iterative solvers: the standard Source Iteration (SI) and two linear Krylov Solvers, GMRES and BiCGSTAB. The resulting hybrid iterative-QMC (iQMC) solver was assessed on three one-dimensional slab geometry problems. In each sample problem the Krylov Solvers achieve convergence with far fewer iterations (up to 8x) than the Source Iteration. Regardless of the linear solver used, the hybrid method achieved an approximate convergence rate of $O(N^{-1})$, as compared to the expected $O(N^{-1/2})$ of traditional MC simulation, across all test problems.

physics.comp-ph↗

arXiv2015

Parallel implementation of electronic structure eigensolver using a partitioned folded spectrum method

A parallel implementation of an eigensolver designed for electronic structure calculations is presented. The method is applicable to computational tasks that solve a sequence of eigenvalue problems where the solution for a particular iteration is similar but not identical to the solution from the previous iteration. Such problems occur frequently when performing electronic structure calculations in which the eigenvectors are solutions to the Kohn-Sham equations. The eigenvectors are represented in some type of basis but the problem sizes are normally too large for direct diagonalization in that basis. Instead a subspace diagonalization procedure is employed in which matrix elements of the Hamiltonian operator are generated and the eigenvalues and eigenvectors of the resulting reduced matrix are obtained using a standard eigensolver from a package such as LAPACK or SCALAPACK. While this method works well and is widely used, the standard eigensolvers scale poorly on massively parallel computer systems for the matrix sizes typical of electronic structure calculations. We present a new method that utilizes a partitioned folded spectrum scheme (PFSM) that takes into account the iterative nature of the problem and performs well on massively parallel systems. Test results for a range of problems are presented that demonstrate an equivalent level of accuracy when compared to the standard eigensolvers, while also executing up to an order of magnitude faster. Unlike O(N) methods, the technique works equally well for metals and systems with unoccupied orbitals as for insulators and semiconductors. Timing and accuracy results are presented for a range of systems, including a 512 atom diamond cell, a cluster of 13 C60 molecules, bulk copper, a 216 atom silicon cell with a vacancy, using 40 unoccupied states/atom, and a 4000 atom aluminum supercell.

physics.comp-ph↗

arXiv2006

Spatially Distributed Stochastic Systems: equation-free and equation-assisted preconditioned computation

Spatially distributed problems are often approximately modelled in terms of partial differential equations (PDEs) for appropriate coarse-grained quantities (e.g. concentrations). The derivation of accurate such PDEs starting from finer scale, atomistic models, and using suitable averaging, is often a challenging task; approximate PDEs are typically obtained through mathematical closure procedures (e.g. mean-field approximations). In this paper, we show how such approximate macroscopic PDEs can be exploited in constructing preconditioners to accelerate stochastic simulations for spatially distributed particle-based process models. We illustrate how such preconditioning can improve the convergence of equation-free coarse-grained methods based on coarse timesteppers. Our model problem is a stochastic reaction-diffusion model capable of exhibiting Turing instabilities.

q-bio.QM↗

arXiv2006

Condition Estimates for Pseudo-Arclength Continuation

We bound the condition number of the Jacobian in pseudo arclength continuation problems, and we quantify the effect of this condition number on the linear system solution in a Newton GMRES solve. In pseudo arclength continuation one repeatedly solves systems of nonlinear equations $F(u(s),λ(s))=0$ for a real-valued function $u$ and a real parameter $λ$, given different values of the arclength $s$. It is known that the Jacobian $F_x$ of $F$ with respect to $x=(u,λ)$ is nonsingular, if the path contains only regular points and simple fold singularities. We introduce a new characterization of simple folds in terms of the singular value decomposition, and we use it to derive a new bound for the norm of $F_x^{-1}$. We also show that the convergence rate of GMRES in a Newton step for $F(u(s),λ(s))=0$ is essentially the same as that of the original problem $G(u,λ)=0$. In particular we prove that the bounds on the degrees of the minimal polynomials of the Jacobians $F_x$ and $G_u$ differ by at most 2. We illustrate the effectiveness of our bounds with an example from radiative transfer theory.

math.NA↗

arXiv2004

Newton-Krylov solvers for time-steppers

We study how the Newton-GMRES iteration can enable dynamic simulators (time-steppers) to perform fixed-point and path-following computations.For a class of dissipative problems, whose dynamics are characterized by a slow manifold, the Jacobian matrices in such computations are compact perturbations of the identity. We examine the number of GMRES iterations required for each nonlinear iteration as a function of the dimension of the slow subspace and the time-stepper reporting horizon. In a path-following computation, only a small number (one or two) of additional GMRES iterations is required.

math.DS↗