arXiv subjects

Daniel Appelö

Publications and source records attributed to Daniel Appelö.

At least 19 recordsLinked to original sources

arXiv2026

A Multi-Frequency Helmholtz Solver Based on the WaveHoltz Algorithm

We develop and analyze a new approach for simultaneously computing multiple solutions to the Helmholtz equation for different frequencies and different forcing functions. The new Multi-Frequency WaveHoltz (MFWH) algorithm is an extension of the original WaveHoltz method and both are based on time-filtering solutions to an associated wave equation. With MFWH, the different Helmholtz solutions are computed simultaneously by solving a single wave equation combined with multiple time filters. The MFWH algorithm defines a fixed-point iteration which can be accelerated with Krylov methods such as GMRES. The solution of the wave equation can be efficiently solved with either explicit time-stepping or implicit time-stepping using as few as five time-steps per period. When combined with an $O(N)$ solver for the implicit equations, such a multigrid, the scheme has an $O(N)$ solution cost when the frequencies are fixed and the number of grid points $N$ increases. High-order accurate approximations in space are used together with second-order accurate approximations in time. We show how to remove time discretization errors so that the MFWH solutions converge to the corresponding solutions to the discretized Helmholtz problems. Numerical results are given using second-order accurate and fourth-accurate discretizations to confirm the convergence theory.

math.NA↗

arXiv2026

Filon Methods for Highly Oscillatory Controlled Quantum Systems

Fast and accurate classical simulation of quantum systems is a central challenge in the design and control of quantum computers, but the highly oscillatory dynamics of these systems severely limit the efficiency of standard numerical methods. To address this, we adapt Filon quadrature for oscillatory integrals into two numerical methods, called Filon and Controlled Filon, for solving linear systems of ODEs with highly oscillatory solutions. We tailor both methods for efficient implementation in controlled quantum systems, and the Controlled Filon method additionally accounts for the oscillatory structure of the control pulses. We show by numerical experiments that these methods significantly reduce the computational cost of accurately simulating systems of superconducting transmon qubits by decreasing the number of timesteps needed to reach a given level of precision, with only a modest increase in the cost per timestep. For a realistic simulation of the dynamics of a CNOT gate, the Controlled Filon method is the most efficient method tested at every target accuracy, outperforming the best Hermite method by up to 6x and the Hermite method of the same order by up to 500x.

quant-ph↗

arXiv2026

LR-WaveHoltz: A Low-Rank Helmholtz Solver

We propose a low-rank method for solving the Helmholtz equation. Our approach is based on the WaveHoltz method, which computes Helmholtz solutions by applying a time-domain filter to the solution of a related wave equation. The wave equation is discretized by high-order multiblock summation-by-parts finite differences. In two dimensions we seek to compress the solution in matrix form, and in three dimensions using tensor trains. To control rank growth we use step-truncation during time stepping and a low-rank Anderson acceleration for the WaveHoltz fixed point iteration. We have carried out extensive numerical experiments demonstrating the convergence and efficacy of the iterative scheme for free- and half-space problems in two and three dimensions with constant and piecewise constant wave speeds.

math.NA↗

arXiv2026

The WaveHoltz Heterogeneous Multiscale Method

We consider the numerical solution of the wave equation in materials with rapidly varying coefficients, and time harmonic sources. For these problems, direct discretization is prohibitively costly, and instead multiscale methods are used. There are several multiscale methods that directly discretize in the frequency domain. In this work we instead start in the time-domain and combine a finite difference Heterogeneous Multiscale Method (HMM) for the wave equation with the WaveHoltz method. Each WaveHoltz iteration marches the wave equation towards the time-periodic Helmholtz solution. The advantages of the WaveHoltz method relative to traditional Helmholtz solvers carry over directly to the multiscale problems considered here. Since, in addition, the time-domain solver does not artificially impose boundary conditions on the micro-scale problems, no boundary errors from the micro-scale problems are present in the homogenized frequency domain solution.

math.NA↗

arXiv2026

A Sub-linear Low-Rank Solver for Poisson's Equation using Machine Learning Frameworks for GPU Acceleration

In this paper we explore a fast Poisson solver for problems with a solution that is known to be low-rank. We use an adaptive and warm started cross approximation called Cross-DEIM that iterates between index selection and and cross approximation to generate a low-rank solution. This paper focuses on leveraging a modern machine learning framework, PyTorch, as a general purpose array language to implement low-rank solvers based on Cross-DEIM. PyTorch enables native access to GPUs and accelerators but with a user-friendly high-level interface. We investigate statistical leverage scores for the index selection for the cross approximation due to the cost associated with the pivoted algorithms used with the discrete empirical interpolation methods (DEIM and QDEIM) which are historically preferred. The cross approximation is naturally paired with a Discrete Sine Transform (DST) Poisson solver. This allows the Fast Fourier Transform (FFT) to be evaluated in batches along dimensions independently without any global transpose even in higher dimensions. We present performance results running on a A100 GPU and AMD EPYC CPU demonstrating the usefulness of the approach that enables problems sizes that previously were not feasible.

cs.PF↗

arXiv2026

Completely Positive and Trace Preserving Schemes with Tensor Train Compression for the Lindblad Equation

We propose a family of low-rank, completely positive and trace preserving schemes for the Lindblad equation, a common model for open quantum systems. Low-rank representation is employed at two levels: the density matrix is factorized into the product of tall-skinny matrices, and the columns of these matrices are further represented using the tensor train (TT) format, also know as matrix product states (MPS). This two-level low-rank format fits naturally into our existing Kraus is King scheme (arXiv:2409.08898v2 [math.NA]) for the Lindblad equation, whose underlying operations are arithmetic on the columns of the tall-skinny matrices. We show how these operations can be performed efficiently in the TT/MPS format, with particular emphasis on density matrix rank-truncation. We conclude with extensive numerical experiments demonstrating the convergence of this scheme and its efficiency in simulating systems with up to $10^{19}$ degrees of freedom using only modest compute resources.

math.NA↗

arXiv2025

A new cross approximation for Tucker tensors and its application in Tucker-Anderson Acceleration

This paper proposes two new algorithms related to the Tucker tensor format. The first method is a new cross approximation for Tucker tensors, which we call Cross$^2$-DEIM. Cross$^2$-DEIM is an iterative method that uses a fiber sampling strategy, sampling $O(r)$ fibers in each mode, where $r$ denotes the target rank. The fibers are selected based on the discrete empirical interpolation method (DEIM). Cross$^2$-DEIM resemblances the Fiber Sampling Tucker Decomposition (FSTD)2 approximation, and has favorable computational scaling compared to existing methods in the literature. We demonstrate good performance of Cross$^2$-DEIM in terms of iteration count and intermediate memory. First we design a fast direct Poisson solver based on Cross$^2$-DEIM and the fast Fourier transform. This solver can be used as a stand alone or as a preconditioner for low-rank solvers for elliptic problems. The second method is a low-rank solver for nonlinear tensor equation in Tucker format by Anderson acceleration (AA), which we call Tucker-AA. Tucker-AA is an extension of low-rank AA (lrAA) proposed in our prior work for low-rank solution to nonlinear matrix equation. We apply Cross$^2$-DEIM with warm-start in Tucker-AA to deal with the nonlinearity in the equation. We apply low-rank operations in AA, and by an appropriate rank truncation strategy, we are able to control the intermediate rank growth. We demonstrated the performance for Tucker-AA for approximate solutions nonlinear PDEs in 3D.

math.NA↗

arXiv2025

The Hermite-Taylor Correction Function Method for Embedded Boundary and Maxwell's Interface Problems

We propose a novel Hermite-Taylor correction function method to handle embedded boundary and interface conditions for Maxwell's equations. The Hermite-Taylor method evolves the electromagnetic fields and their derivatives through order $m$ in each Cartesian coordinate. This makes the development of a systematic approach to enforce boundary and interface conditions difficult. Here we use the correction function method to update the numerical solution where the Hermite-Taylor method cannot be applied directly. Time derivatives of boundary and interface conditions, converted into spatial derivatives, are enforced to obtain a stable method and relax the time-step size restriction of the Hermite-Taylor correction function method. The proposed high-order method offers a flexible systematic approach to handle embedded boundary and interface problems, including problems with discontinuous solutions at the interface. This method is also easily adaptable to other first order hyperbolic systems.

math.NA↗

arXiv2025

A $P$-Adaptive Hermite Method for Nonlinear Dispersive Maxwell's Equations

In this work, we introduce a novel Hermite method to handle Maxwell's equations for nonlinear dispersive media. The proposed method achieves high-order accuracy and is free of any nonlinear algebraic solver, requiring solving instead small local linear systems for which the dimension is independent of the order. The implementation of order adaptive algorithms is straightforward in this setting, making the resulting p-adaptive Hermite method appealing for the simulations of soliton-like wave propagation.

math.NA↗

arXiv2025

Robust Implicit Adaptive Low Rank Time-Stepping Methods for Matrix Differential Equations

In this work, we develop implicit rank-adaptive schemes for time-dependent matrix differential equations. The dynamic low rank approximation (DLRA) is a well-known technique to capture the dynamic low rank structure based on Dirac-Frenkel time-dependent variational principle. In recent years, it has attracted a lot of attention due to its wide applicability. Our schemes are inspired by the three-step procedure used in the rank adaptive version of the unconventional robust integrator (the so called BUG integrator) for DLRA. First, a prediction (basis update) step is made computing the approximate column and row spaces at the next time level. Second, a Galerkin evolution step is invoked using a base implicit solve for the small core matrix. Finally, a truncation is made according to a prescribed error threshold. Since the DLRA is evolving the differential equation projected on to the tangent space of the low rank manifold, the error estimate of the BUG integrator contains the tangent projection (modeling) error which cannot be easily controlled by mesh refinement. This can cause convergence issue for equations with cross terms. To address this issue, we propose a simple modification, consisting of merging the row and column spaces from the explicit step truncation method together with the BUG spaces in the prediction step. In addition, we propose an adaptive strategy where the BUG spaces are only computed if the residual for the solution obtained from the prediction space by explicit step truncation method, is too large. We prove stability and estimate the local truncation error of the schemes under assumptions. We benchmark the schemes in several tests, such as anisotropic diffusion, solid body rotation and the combination of the two, to show robust convergence properties.

math.NA↗

arXiv2024

High Order Accurate Hermite Schemes on Curvilinear Grids with Compatibility Boundary Conditions

High order accurate Hermite methods for the wave equation on curvilinear domains are presented. Boundaries are treated using centered compatibility conditions rather than more standard one-sided approximations. Both first-order-in-time (FOT) and second-order-in-time (SOT) Hermite schemes are developed. Hermite methods use the solution and multiple derivatives as unknowns and achieve space-time orders of accuracy $2m-1$ (FOT) and $2m$ (SOT) for methods using $(m+1)^d$ degree of freedom per node in $d$ dimensions. The compatibility boundary conditions (CBCs) are based on taking time derivatives of the boundary conditions and using the governing equations to replace the time derivatives with spatial derivatives. These resulting constraint equations augment the Hermite scheme on the boundary. The solvability of the equations resulting from the compatibility conditions are analyzed. Numerical examples demonstrate the accuracy and stability of the new schemes in two dimensions.

math.NA↗

arXiv2024

Fast and high-order approximation of parabolic equations using hierarchical direct solvers and implicit Runge-Kutta methods

An additive Runge-Kutta method is used for the time stepping, which integrates the linear stiff terms by an explicit singly diagonally implicit Runge-Kutta (ESDIRK) method and the nonlinear terms by an explicit Runge-Kutta (ERK) method. In each time step, the implicit solve is performed by the recently developed Hierarchical Poincaré-Steklov (HPS) method. This is a fast direct solver for elliptic equations that decomposes the space domain into a hierarchical tree of subdomains and builds spectral collocation solvers locally on the subdomains. These ideas are naturally combined in the presented method since the singly diagonal coefficient in ESDIRK and a fixed time-step ensures that the coefficient matrix in the implicit solve of HPS remains the same for all time stages. This means that the precomputed inverse can be efficiently reused, leading to a scheme with complexity (in two dimensions) $\mathcal{O}(N^{1.5})$ for the precomputation where the solution operator to the elliptic problems is built, and then $\mathcal{O}(N \log N)$ for the solve in each time step. The stability of the method is proved for first order in time and any order in space, and numerical evidence substantiates a claim of stability for a much broader class of time discretization methods. Numerical experiments supporting the accuracy of efficiency of the method in one and two dimensions are presented.

math.NA↗

arXiv2023

Deterministic and Bayesian Characterization of Quantum Computing Devices

Motivated by the noisy and fluctuating behavior of current quantum computing devices, this paper presents a data-driven characterization approach for estimating transition frequencies and decay times in a Lindbladian dynamical model of a superconducting quantum device. The data includes parity events in the transition frequency between the first and second excited states. A simple but effective mathematical model, based upon averaging solutions of two Lindbladian models, is demonstrated to accurately capture the experimental observations. A deterministic point estimate of the device parameters is first performed to minimize the misfit between data and Lindbladian simulations. These estimates are used to make an informed choice of prior distributions for the subsequent Bayesian inference. An additive Gaussian noise model is developed for the likelihood function, which includes two hyper-parameters to capture the noise structure of the data. The outcome of the Bayesian inference are posterior probability distributions of the transition frequencies, which for example can be utilized to design risk neutral optimal control pulses. The applicability of our approach is demonstrated on experimental data from the Quantum Device and Integration Testbed (QuDIT) at Lawrence Livermore National Laboratory, using a tantalum-based superconducting transmon device.

quant-ph↗

arXiv2023

The Hermite-Taylor Correction Function Method for Maxwell's Equations

The Hermite-Taylor method, introduced in 2005 by Goodrich, Hagstrom and Lorenz, is highly efficient and accurate when applied to linear hyperbolic systems on periodic domains. Unfortunately its widespread use has been prevented by the lack of a systematic approach to implementing boundary conditions. In this paper we present the Hermite-Taylor Correction Function method, which provides exactly such a systematic approach for handing boundary conditions. Here we focus on Maxwell's equations but note that the method is easily extended to other hyperbolic problems.

math.NA↗

arXiv2022

El-WaveHoltz: A Time-Domain Iterative Solver for Time-Harmonic Elastic Waves

We consider the application of the WaveHoltz iteration to time-harmonic elastic wave equations with energy conserving boundary conditions. The original WaveHoltz iteration for acoustic Helmholtz problems is a fixed-point iteration that filters the solution of the wave equation with time-harmonic forcing and boundary data. As in the original WaveHoltz method, we reformulate the fixed point iteration as a positive definite linear system of equations that is iteratively solved by a Krylov method. We present two time-stepping schemes, one explicit and one (novel) implicit, which completely remove time discretization error from the WaveHoltz solution by performing a simple modification of the initial data and time-stepping scheme. Numerical experiments indicate an iteration scaling similar to that of the original WaveHoltz method, and that the convergence rate is dictated by the shortest (shear) wave speed of the problem. We additionally show that the implicit scheme can be advantageous in practice for meshes with disparate element sizes.

math.NA↗

arXiv2022

Extensions and Analysis of an Iterative Solution of the Helmholtz Equation via the Wave Equation

In this paper we extend analysis of the WaveHoltz iteration -- a time-domain iterative method for the solution of the Helmholtz equation. We expand the previous analysis of energy conserving problems and prove convergence of the WaveHoltz iteration for problems with impedance boundary conditions in a single spatial dimension. We then consider interior Dirichlet/Neumann problems with damping in any spatial dimension, and show that for a sufficient level of damping the WaveHoltz iteration converges in a number of iteration independent of the frequency. Finally, we present a discrete analysis of the WaveHoltz iteration for a family of higher order time-stepping schemes. We show that the fixed-point of the discrete WaveHoltz iteration converges to the discrete Helmholtz solution with the order of the time-stepper chosen. We present numerical examples and demonstrate that it is possible to completely remove time discretization error from the WaveHoltz solution through careful analysis of the discrete iteration together with updated quadrature formulas.

math.NA↗

arXiv2022

An Energy-Based Discontinuous Galerkin Method with Tame CFL Numbers for the Wave Equation

We extend and analyze the energy-based discontinuous Galerkin method for second order wave equations on staggered and structured meshes. By combining spatial staggering with local time-stepping near boundaries, the method overcomes the typical numerical stiffness associated with high order piecewise polynomial approximations. In one space dimension with periodic boundary conditions and suitably chosen numerical fluxes, we prove bounds on the spatial operators that establish stability for CFL numbers $c \frac {Δt}{h} < C$ independent of order when stability-enhanced explicit time-stepping schemes of matching order are used. For problems on bounded domains and in higher dimensions we demonstrate numerically that one can march explicitly with large time steps at high order temporal and spatial accuracy.

math.NA↗

arXiv2022

Universal AMG Accelerated Embedded Boundary Method Without Small Cell Stiffness

We develop a universally applicable embedded boundary finite difference method, which results in a symmetric positive definite linear system and does not suffer from small cell stiffness. Our discretization is efficient for the wave, heat and Poisson's equation with Dirichlet boundary conditions. When the system needs to be inverted we can use the conjugate gradient method, accelerated by algebraic multigrid techniques. A series of numerical tests for the wave, heat and Poisson's equation and applications to shape optimization problems verify the accuracy, stability, and efficiency of our method. Our fast computational techniques can be extended to moving boundary problems (e.g. Stefan problem), to the Navier-Stokes equations, and to the Grad-Shafranov equations for which problems are posed on domains with complex geometry and fast simulations are very important.

math.NA↗