arXiv subjects

Adrianna Gillman

Publications and source records attributed to Adrianna Gillman.

At least 19 recordsLinked to original sources

arXiv2026

Algorithm librla: A library of randomized linear algebra routines

The library \texttt{librla} is a randomized linear algebra library that is specifically designed for the intermediate-sized matrices (of dimension up to roughly 10,000) that arise in applications such as reduced order modeling, fast direct solvers, least squares solves and, in some settings, data compression. \texttt{librla} is the first software package that is both stable and efficient in several high-level languages: MATLAB, Python and Julia. It also provides increased functionality over existing software. Specifically, it allows the user to choose to create a factorization based on a fixed rank or a desired tolerance. The factorization options include QR, SVD and the interpolative decomposition. Additionally, the factorization can be generated either with access to the matrix or access to a matrix-vector multiplication routine. Numerical results compare the Python implementation with the available PyTorch and SciPy randomized factorizations. Performance of \texttt{librla} in the three languages is comparable.

cs.MS↗

arXiv2026

Generating Entangled Steady States in Multistable Open Quantum Systems via Initial State Control

Entanglement underpins the power of quantum technologies, yet it is fragile and typically destroyed by dissipation. Paradoxically, the same dissipation, when carefully engineered, can drive a system toward robust entangled steady states. However, this engineering task is nontrivial, as dissipative many-body systems are complex, particularly when they support multiple steady states. Here, we derive analytic expressions that predict how the steady state of a system evolving under a Lindblad equation depends on the initial state, without requiring integration of the dynamics. These results extend the frameworks developed in Refs. [Phys. Rev. A 89, 022118 (2014) and Phys. Rev. X 6, 041031 (2016)], showing that while the steady-state manifold is determined by the Liouvillian kernel, the weights within it depend on both the Liouvillian and the initial state. We identify a special class of Liouvillians for which the steady state depends only on the initial overlap with the kernel. Our framework provides analytical insight and a computationally efficient tool for predicting steady states in open quantum systems. As an application, we propose schemes to generate metrologically useful entangled steady states in spin ensembles via balanced collective decay.

quant-ph↗

arXiv2025

An efficient boundary integral equation solution technique for solving aperiodic scattering problems from two-dimensional, periodic boundaries

This manuscript presents an efficient boundary integral equation technique for solving two-dimensional Helmholtz problems defined in the half-plane bounded by an infinite, periodic curve with Neumann boundary conditions and an aperiodic point source. The technique is designed for boundaries where one period does not require a large number of discretization points to achieve high accuracy. The Floquet--Bloch transform turns the problem into evaluating a contour integral where the integrand is the solution of quasiperiodic boundary value problems. To approximate the integral, one must solve a collection of these problems. This manuscript uses a variant of the periodizing scheme by Cho and Barnett which alleviates the need for evaluating the quasiperiodic Green's function and is amenable to a large amount of precomputation that can be reused for all of the necessary solves. The solution technique is accelerated by the use of low rank linear algebra. The numerical results illustrate that the presented method is 20-30 faster than the technique utilizing the quasiperiodic Green's function for a stair-like geometry.

math.NA↗

arXiv2023

An iterative solver for the HPS discretization applied to three dimensional Helmholtz problems

This manuscript presents an efficient solver for the linear system that arises from the Hierarchical Poincaré-Steklov (HPS) discretization of three dimensional variable coefficient Helmholtz problems. Previous work on the HPS method has tied it with a direct solver. This work is the first efficient iterative solver for the linear system that results from the HPS discretization. The solution technique utilizes GMRES coupled with a locally homogenized block-Jacobi preconditioner. The local nature of the discretization and preconditioner naturally yield the matrix-free application of the linear system. Numerical results illustrate the performance of the solution technique. This includes an experiment where a problem approximately 100 wavelengths in each direction that requires more than a billion unknowns to achieve approximately 4 digits of accuracy takes less than 20 minutes to solve.

math.NA↗

arXiv2022

A fast direct solver for two dimensional quasi-periodic multilayered media scattering problems, Part II

This manuscript is the second in a series presenting fast direct solution techniques for solving two-dimensional wave scattering problems from quasi-periodic multilayered structures. The fast direct solvers presented in the series are for the linear system that results from the discretization of a robust integral formulation. The fast direct solver presented in this manuscript has a computational cost that scales linearly with respect to the number of discretization points on the interfaces and the number of layers. The latter is an improvement over the previous solver and makes the new solver more efficient especially for problems involving multiple incident angles and changes to the layered media. Numerical results illustrate the improved performance of the new solver over the previous one.

math.NA↗

arXiv2022

A fast direct solver for integral equations on locally refined boundary discretizations and its application to multiphase flow simulations

In transient simulations of particulate Stokes flow, to accurately capture the interaction between the constituent particles and the confining wall, the discretization of the wall often needs to be locally refined in the region approached by the particles. Consequently, standard fast direct solvers lose their efficiency since the linear system changes at each time step. This manuscript presents a new computational approach that avoids this issue by pre-constructing a fast direct solver for the wall ahead of time, computing a low-rank factorization to capture the changes due to the refinement, and solving the problem on the refined discretization via a Woodbury formula. Numerical results illustrate the efficiency of the solver in accelerating particulate Stokes simulations.

math.NA↗

arXiv2020

An alternative extended linear system for boundary value problems on locally perturbed geometries

This manuscript presents a new extended linear system for integral equation based techniques for solving boundary value problems on locally perturbed geometries. The new extended linear system is similar to a previously presented technique for which the authors have constructed a fast direct solver. The key features of the work presented in this paper are that the fast direct solver is more efficient for the new extended linear system and that problems involving specialized quadrature for weakly singular kernels can be easily handled. Numerical results illustrate the improved performance of the fast direct solver for the new extended system when compared to the fast direct solver for the original extended system.

math.NA↗

arXiv2020

A fast direct solver for two dimensional quasi-periodic multilayered media scattering problems

This manuscript presents a fast direct solution technique for solving two dimensional wave scattering problems from quasi-periodic multilayered structures. When the interface geometries are complex, the dominant term in the computational cost of creating the direct solver scales $O(NI)$ where $N$ is the number of discretization points on each interface and $I$ is the number of interfaces. The bulk of the precomputation can be re-used for any choice of incident wave. As a result, the direct solver can solve over 200 scattering problems involving an eleven layer geometry with complex interfaces 100 times faster than building a new fast direct solver from scratch for each new set of boundary data. An added benefit of the presented solver is that building an updated solver for a new geometry involving a replaced interface or a change in material property in one layer is inexpensive compared to building a new fast direct solver from scratch.

math.NA↗

arXiv2019

A parallel shared-memory implementation of a high-order accurate solution technique for variable coefficient Helmholtz problems

The recently developed Hierarchical Poincaré-Steklov (HPS) method is a high-order discretization technique that comes with a direct solver. Results from previous papers demonstrate the method's ability to solve Helmholtz problems to high accuracy without the so-called pollution effect. While the asymptotic scaling of the direct solver's computational cost is the same as the nested dissection method, serial implementations of the solution technique are not practical for large scale numerical simulations. This manuscript presents the first parallel implementation of the HPS method. Specifically, we introduce an approach for a shared memory implementation of the solution technique utilizing parallel linear algebra. This approach is the foundation for future large scale simulations on supercomputers and clusters with large memory nodes. Performance results on a desktop computer (resembling a large memory node) are presented.

math.NA↗

arXiv2018

An adaptive high order direct solution technique for elliptic boundary value problems

This manuscript presents an adaptive high order discretization technique for elliptic boundary value problems. The technique is applied to an updated version of the Hierarchical Poincaré-Steklov (HPS) method. Roughly speaking, the HPS method is based on local pseudospectral discretizations glued together with Poincaré-Steklov operators. The new version uses a modified tensor product basis which is more efficient and stable than previous versions. The adaptive technique exploits the tensor product nature of the basis functions to create a criterion for determining which parts of the domain require additional refinement. The resulting discretization achieves the user prescribed accuracy and comes with an efficient direct solver. The direct solver increases the range of applicability to time dependent problems where the cost of solving elliptic problems previously limited the use of implicit time stepping schemes.

math.NA↗

arXiv2017

A fast direct solver for boundary value problems on locally perturbed geometries

Many applications involve solving several boundary value problems on geometries that are local perturbations of an original geometry. The boundary integral equation for a problem on a locally perturbed geometry can be expressed as a low rank update to the original system. A fast direct solver for the new linear system is presented in this paper. The solution technique utilizes a precomputed fast direct solver for the original geometry to efficiently create the low rank factorization of the update matrix and to accelerate the application of the Sherman-Morrison formula. The method is ideally suited for problems where the local perturbation is the same but its placement on the boundary changes and problems where the local perturbation is a refined discretization on the same geometry. Numerical results illustrate that for fixed local perturbation the method is three times faster than building a new fast direct solver from scratch.

math.NA↗

arXiv2016

An accelerated Poisson solver based on multidomain spectral discretization

This paper presents a numerical method for variable coefficient elliptic PDEs with mostly smooth solutions on two dimensional domains. The PDE is discretized via a multi-domain spectral collocation method of high local order (order 30 and higher have been tested and work well). Local mesh refinement results in highly accurate solutions even in the presence of local irregular behavior due to corner singularities, localized loads, etc. The system of linear equations attained upon discretization is solved using a direct (as opposed to iterative) solver with $O(N^{1.5})$ complexity for the factorization stage and $O(N \log N)$ complexity for the solve. The scheme is ideally suited for executing the elliptic solve required when parabolic problems are discretized via time-implicit techniques. In situations where the geometry remains unchanged between time-steps, very fast execution speeds are obtained since the solution operator for each implicit solve can be pre-computed.

math.NA↗

arXiv2016

High resolution inverse scattering in two dimensions using recursive linearization

We describe a fast, stable algorithm for the solution of the inverse acoustic scattering problem in two dimensions. Given full aperture far field measurements of the scattered field for multiple angles of incidence, we use Chen's method of recursive linearization to reconstruct an unknown sound speed at resolutions of thousands of square wavelengths in a fully nonlinear regime. Despite the fact that the underlying optimization problem is formally ill-posed and non-convex, recursive linearization requires only the solution of a sequence of linear least squares problems at successively higher frequencies. By seeking a suitably band-limited approximation of the sound speed profile, each least squares calculation is well-conditioned and involves the solution of a large number of forward scattering problems, for which we employ a recently developed, spectrally accurate, fast direct solver. For the largest problems considered, involving 19,600 unknowns, approximately one million partial differential equations were solved, requiring approximately two days to compute using a parallel MATLAB implementation on a multi-core workstation.

math.NA↗

arXiv2016

An integral equation technique for scattering problems with mixed boundary conditions

This paper presents an integral formulation for Helmholtz problems with mixed boundary conditions. Unlike most integral equation techniques for mixed boundary value problems, the proposed method uses a global boundary charge density. As a result, Calderón identities can be utilized to avoid the use of hypersingular integral operators. More importantly, the formulation avoids spurious resonances. Numerical results illustrate the performance of the proposed solution technique.

math.NA↗

arXiv2015

A fast algorithm for simulating multiphase flows through periodic geometries of arbitrary shape

This paper presents a new boundary integral equation (BIE) method for simulating particulate and multiphase flows through periodic channels of arbitrary smooth shape in two dimensions. The authors consider a particular system---multiple vesicles suspended in a periodic channel of arbitrary shape---to describe the numerical method and test its performance. Rather than relying on the periodic Green's function as classical BIE methods do, the method combines the free-space Green's function with a small auxiliary basis, and imposes periodicity as an extra linear condition. As a result, we can exploit existing free-space solver libraries, quadratures, and fast algorithms, and handle a large number of vesicles in a geometrically complex channel. Spectral accuracy in space is achieved using the periodic trapezoid rule and product quadratures, while a first-order semi-implicit scheme evolves particles by treating the vesicle-channel interactions explicitly. New constraint-correction formulas are introduced that preserve reduced areas of vesicles, independent of the number of time steps taken. By using two types of fast algorithms, (i) the fast multipole method (FMM) for the computation of the vesicle-vesicle and the vesicle-channel hydrodynamic interaction, and (ii) a fast direct solver for the BIE on the fixed channel geometry, the computational cost is reduced to $O(N)$ per time step where $N$ is the spatial discretization size. Moreover, the direct solver inverts the wall BIE operator at $t = 0$, stores its compressed representation and applies it at every time step to evolve the vesicle positions, leading to dramatic cost savings compared to classical approaches. Numerical experiments illustrate that a simulation with $N=128, 000$ can be evolved in less than a minute per time step on a laptop.

math.NA↗

arXiv2013

A high-order accurate accelerated direct solver for acoustic scattering from surfaces

We describe an accelerated direct solver for the integral equations which model acoustic scattering from curved surfaces. Surfaces are specified via a collection of smooth parameterizations given on triangles, a setting which generalizes the typical one of triangulated surfaces, and the integral equations are discretized via a high-order Nystrom method. This allows for rapid convergence in cases in which high-order surface information is available. The high-order discretization technique is coupled with a direct solver based on the recursive construction of scattering matrices. The result is a solver which often attains $O(N^{1.5})$ complexity in the number of discretization nodes $N$ and which is resistant to many of the pathologies which stymie iterative solvers in the numerical simulation of scattering. The performance of the algorithm is illustrated with numerical experiments which involve the simulation of scattering from a variety of domains, including one consisting of a collection of 1000 ellipsoids with randomly oriented semiaxes arranged in a grid, and a domain whose boundary has 12 curved edges and 8 corner points.

math.NA↗

arXiv2013

A spectrally accurate direct solution technique for frequency-domain scattering problems with variable media

This paper presents a direct solution technique for the scattering of time-harmonic waves from a bounded region of the plane in which the wavenumber varies smoothly in space.The method constructs the interior Dirichlet-to-Neumann (DtN) map for the bounded region via bottom-up recursive merges of (discretization of) certain boundary operators on a quadtree of boxes.These operators take the form of impedance-to-impedance (ItI) maps. Since ItI maps are unitary, this formulation is inherently numerically stable, and is immune to problems of artificial internal resonances. The ItI maps on the smallest (leaf) boxes are built by spectral collocation on tensor-product grids of Chebyshev nodes. At the top level the DtN map is recovered from the ItI map and coupled to a boundary integral formulation of the free space exterior problem, to give a provably second kind equation.Numerical results indicate that the scheme can solve challenging problems 70 wavelengths on a side to 9-digit accuracy with 4 million unknowns, in under 5 minutes on a desktop workstation. Each additional solve corresponding to a different incident wave (right-hand side) then requires only 0.04 seconds.

math.NA↗

arXiv2013

An O(N) algorithm for constructing the solution operator to 2D elliptic boundary value problems in the absence of body loads

The large sparse linear systems arising from the finite element or finite difference discretization of elliptic PDEs can be solved directly via, e.g., nested dissection or multifrontal methods. Such techniques reorder the nodes in the grid to reduce the asymptotic complexity of Gaussian elimination from $O(N^{2})$ to $O(N^{1.5})$ for typical problems in two dimensions. It has recently been demonstrated that the complexity can be further reduced to O(N) by exploiting structure in the dense matrices that arise in such computations (using, e.g., $\mathcal{H}$-matrix arithmetic). This paper demonstrates that such \textit{accelerated} nested dissection techniques become particularly effective for boundary value problems without body loads when the solution is sought for several different sets of boundary data, and the solution is required only near the boundary (as happens, e.g., in the computational modeling of scattering problems, or in engineering design of linearly elastic solids.

math.NA↗