An official website of the United States government
Inference, optimization, and inverse problems are but three examples of mathematical operations that require the repeated solution of a complex system of mathematical equations. To this end, surrogates are often used to approximate the output of these large computer simulations, providing fast and cheap approximation solutions. Statistical emulators are surrogates that, in addition to predicting the mean behavior of the system, provide an estimate of the error in that prediction. Classical Gaussian stochastic process emulators predict scalar outputs based on a modest number of input parameters. Making predictions across a space-time field of input variables is not feasible using classical Gaussian process methods. Parallel partial emulation is a new statistical emulator methodology that predicts a field of outputs based on the input parameters. Parallel partial emulation is constructed as a Gaussian process in parameter space, but no correlation among space or time points is assumed. Thus the computational work of parallel partial emulation scales as the cube of the number of input parameters (as traditional Gaussian Process emulation) and linearly with a space-time grid. The numerical methods used in numerical simulations are often designed to exploit properties of the equations tobe solved. For example, modern solvers for hyperbolic conservation laws satisfy conservation at each time step, insuring overall conservation of the physical variables. Similarly, symplectic methods are used to solve Hamiltonian problems in physics. It is of interest, then, to study whether parallel partial emulation predictions inherit properties possessed by the simulation outputs. Does an emulated solution of a conservation law preserve the conserved quantities? Does an emulator of a Hamiltonian system preserve the energy? This paper investigates the properties of emulator predictions, in the context of systems of partial differential equations. We study conservation properties for three different kinds ofequations-conservation laws, reaction-diffusion systems, and a Hamiltonian system.We also investigate the effective convergence, in parameter space, of the predicted solution of a highly nonlinear system modeling shape memory alloys.
more »
« less
Fast multi-source nanophotonic simulations using augmented partial factorization
Abstract Numerical solutions of Maxwell’s equations are indispensable for nanophotonics and electromagnetics but are constrained when it comes to large systems, especially multi-channel ones such as disordered media, aperiodic metasurfaces and densely packed photonic circuits where the many inputs require many large-scale simulations. Conventionally, before extracting the quantities of interest, Maxwell’s equations are first solved on every element of a discretization basis set that contains much more information than is typically needed. Furthermore, such simulations are often performed one input at a time, which can be slow and repetitive. Here we propose to bypass the full-basis solutions and directly compute the quantities of interest while also eliminating the repetition over inputs. We do so by augmenting the Maxwell operator with all the input source profiles and all the output projection profiles, followed by a single partial factorization that yields the entire generalized scattering matrix via the Schur complement, with no approximation beyond discretization. This method applies to any linear partial differential equation. Benchmarks show that this approach is 1,000–30,000,000 times faster than existing methods for two-dimensional systems with about 10,000,000 variables. As examples, we demonstrate simulations of entangled photon backscattering from disorder and high-numerical-aperture metalenses that are thousands of wavelengths wide.
more »
« less
- Award ID(s):
- 2146021
- PAR ID:
- 10385731
- Publisher / Repository:
- Nature Publishing Group
- Date Published:
- Journal Name:
- Nature Computational Science
- Volume:
- 2
- Issue:
- 12
- ISSN:
- 2662-8457
- Page Range / eLocation ID:
- p. 815-822
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
null (Ed.)This work derives a residual-based a posteriori error estimator for reduced models learned with non-intrusive model reduction from data of high-dimensional systems governed by linear parabolic partial differential equations with control inputs. It is shown that quantities that are necessary for the error estimator can be either obtained exactly as the solutions of least-squares problems in a non-intrusive way from data such as initial conditions, control inputs, and high-dimensional solution trajectories or bounded in a probabilistic sense. The computational procedure follows an offline/online decomposition. In the offline (training) phase, the high-dimensional system is judiciously solved in a black-box fashion to generate data and to set up the error estimator. In the online phase, the estimator is used to bound the error of the reduced-model predictions for new initial conditions and new control inputs without recourse to the high-dimensional system. Numerical results demonstrate the workflow of the proposed approach from data to reduced models to certified predictions.more » « less
-
null (Ed.)In this paper, we propose a local discontinuous Galerkin (LDG) method for nonlinear and possibly degenerate parabolic stochastic partial differential equations, which is a high-order numerical scheme. It extends the discontinuous Galerkin (DG) method for purely hyperbolic equations to parabolic equations and shares with the DG method its advantage and flexibility. We prove the L 2 -stability of the numerical scheme for fully nonlinear equations. Optimal error estimates ( O ( h (k+1) )) for smooth solutions of semi-linear stochastic equations is shown if polynomials of degree k are used. We use an explicit derivative-free order 1.5 time discretization scheme to solve the matrix-valued stochastic ordinary differential equations derived from the spatial discretization. Numerical examples are given to display the performance of the LDG method.more » « less
-
null (Ed.)Domain decomposition methods are widely used for the numerical solution of partial differential equations on high performance computers. We develop an adjoint-based a posteriori error analysis for both multiplicative and additive overlapping Schwarz domain decomposition methods. The numerical error in a user-specified functional of the solution (quantity of interest) is decomposed into contributions that arise as a result of the finite iteration between the subdomains and from the spatial discretization. The spatial discretization contribution is further decomposed into contributions arising from each subdomain. This decomposition of the numerical error is used to construct a two stage solution strategy that efficiently reduces the error in the quantity of interest by adjusting the relative contributions to the error.more » « less
-
The numerical solution of partial differential equations (PDEs) is challenging because of the need to resolve spatiotemporal features over wide length- and timescales. Often, it is computationally intractable to resolve the finest features in the solution. The only recourse is to use approximate coarse-grained representations, which aim to accurately represent long-wavelength dynamics while properly accounting for unresolved small-scale physics. Deriving such coarse-grained equations is notoriously difficult and often ad hoc. Here we introduce data-driven discretization, a method for learning optimized approximations to PDEs based on actual solutions to the known underlying equations. Our approach uses neural networks to estimate spatial derivatives, which are optimized end to end to best satisfy the equations on a low-resolution grid. The resulting numerical methods are remarkably accurate, allowing us to integrate in time a collection of nonlinear equations in 1 spatial dimension at resolutions 4× to 8× coarser than is possible with standard finite-difference methods.more » « less
