Kevin Carlberg

Meta · Facebook Reality Labs

This work proposes a method for model reduction of finite-volume models that guarantees the resulting reduced-order model is conservative, thereby preserving the structure intrinsic to finite-volume discretizations. The proposed reduced-order models associate with optimization problems characterized by a minimum-residual objective function and nonl...

Nearly all model-reduction techniques project the governing equations onto a linear subspace of the original state space. Such subspaces are typically computed using methods such as balanced truncation, rational interpolation, the reduced-basis method, and (balanced) POD. Unfortunately, restricting the state to evolve in a linear subspace imposes a...

In many applications, projection-based reduced-order models (ROMs) have demonstrated the ability to provide rapid approximate solutions to high-fidelity full-order models (FOMs). However, there is no a priori assurance that these approximate solutions are accurate; their accuracy depends on the ability of the low-dimensional trial basis to represen...

This work proposes a machine-learning framework for modeling the error incurred by approximate solutions to parameterized dynamical systems. In particular, we extend the machine-learning error models (MLEM) framework proposed in Ref. 15 to dynamical systems. The proposed Time-Series Machine-Learning Error Modeling (T-MLEM) method constructs a regre...

This work introduces the network uncertainty quantification (NetUQ) method for performing uncertainty propagation in systems composed of interconnected components. The method assumes the existence of a collection of components, each of which is characterized by exogenous-input random variables, endogenous-input random variables, output random varia...

This work proposes a model-reduction approach for the material point method on nonlinear manifolds. Our technique approximates the kinematics by approximating the deformation map using an implicit neural representation that restricts deformation trajectories to reside on a low-dimensional manifold. By explicitly approximating the deformation map, i...

The excessive runtime of high-fidelity partial differential equation (PDE) solvers makes them unsuitable for time-critical applications. We propose to accelerate PDE solvers using reduced-order modeling (ROM). Whereas prior ROM approaches reduce the dimensionality of discretized vector fields, our continuous reduced-order modeling (CROM) approach b...

In this paper, we introduce a methodology for improving the accuracy and efficiency of reduced‐order models (ROMs) constructed using the least‐squares Petrov{Galerkin (LSPG) projection method through the introduction of preconditioning. Unlike prior related work, which focuses on preconditioning the linear systems arising within the ROM numerical s...

This paper introduces a methodology for improving the accuracy and efficiency of reduced order models (ROMs) constructed using the least-squares Petrov-Galerkin (LSPG) projection method through the introduction of preconditioning. Unlike prior related work, which focuses on preconditioning the linear systems arising within the ROM numerical solutio...

A novel domain-decomposition least-squares Petrov–Galerkin (DD-LSPG) model-reduction method applicable to parameterized systems of nonlinear algebraic equations (e.g., arising from discretizing a parameterized partial-differential-equations problem) is proposed. In contrast with previous works, we adopt an algebraically non-overlapping decompositio...

This work proposes a model-reduction approach for the material point method on nonlinear manifolds. The technique approximates the $\textit{kinematics}$ by approximating the deformation map in a manner that restricts deformation trajectories to reside on a low-dimensional manifold expressed from the extrinsic view via a parameterization function. B...

While reduced-order models (ROMs) have demonstrated success in many applications across computational science, challenges remain when applied both to extreme-scale models due to the prohibitive cost of generating requisite training data, and to decomposable systems due to many-query problems often requiring repeated reconfigurations of system compo...

This work proposes a machine-learning framework for modeling the error incurred by approximate solutions to parameterized dynamical systems. In particular, we extend the machine-learning error models (MLEM) framework proposed in Ref. Freno and Carlberg (2019) to dynamical systems. The proposed Time-Series Machine-Learning Error Modeling (T-MLEM) me...

In many applications, projection-based reduced-order models (ROMs) have demonstrated the ability to provide rapid approximate solutions to high-fidelity full-order models (FOMs). However, there is no a priori assurance that these approximate solutions are accurate; their accuracy depends on the ability of the low-dimensional trial basis to represen...

This work introduces Pressio, an open-source project aimed at enabling leading-edge projection-based reduced order models (ROMs) for large-scale nonlinear dynamical systems in science and engineering. Pressio provides model-reduction methods that can reduce both the number of spatial and temporal degrees of freedom for any dynamical system expressi...

Nearly all model-reduction techniques project the governing equations onto a linear subspace of the original state space. Such subspaces are typically computed using methods such as balanced truncation, rational interpolation, the reduced-basis method, and (balanced) proper orthogonal decomposition (POD). Unfortunately, restricting the state to evo...

This work proposes a windowed least-squares (WLS) approach for model-reduction of dynamical systems. The proposed approach sequentially minimizes the time-continuous full-order-model residual within a low-dimensional space-time trial subspace over time windows. The approach comprises a generalization of existing model reduction approaches, as parti...

This work proposes a windowed least-squares (WLS) approach for model-reduction of dynamical systems. The proposed approach sequentially minimizes the time-continuous full-order-model residual within a low-dimensional space-time trial subspace over time windows. The approach comprises a generalization of existing model reduction approaches, as parti...

Numerical simulations of the cardiovascular system are affected by uncertainties arising from a substantial lack of data related to the boundary conditions and the physical parameters of the mathematical models. Quantifying the impact of this uncertainty on the numerical results along the circulatory network is challenged by the complexity of both...

This work proposes an approach for latent dynamics learning that exactly enforces physical conservation laws. The method comprises two steps. First, we compute a low-dimensional embedding of the high-dimensional dynamical-system state using deep convolutional autoencoders. This defines a low-dimensional nonlinear manifold on which the state is subs...

Data I/O poses a significant bottleneck in large-scale CFD simulations; thus, practitioners would like to significantly reduce the number of times the solution is saved to disk, yet retain the ability to recover any field quantity (at any time instance) a posteriori. The objective of this work is therefore to accurately recover missing CFD data a p...

Nearly all model-reduction techniques project the governing equations onto a linear subspace of the original state space. Such subspaces are typically computed using methods such as balanced truncation, rational interpolation, the reduced-basis method, and (balanced) POD. Unfortunately, restricting the state to evolve in a linear trial subspace imp...

In many applications, projection-based reduced-order models (ROMs) have demonstrated the ability to provide rapid approximate solutions to high-fidelity full-order models (FOMs). However, there is no a priori assurance that these approximate solutions are accurate; their accuracy depends on the ability of the low-dimensional trial basis to represen...

Physics-based modeling and simulation has become indispensable in aerospace applications ranging from aircraft design to uncertainty quantification. However, as designs are pushed to extreme operating conditions and decisions are driven by increasingly fine-grained tradeoffs in noise, safety, cost, and efficiency, greater demands are being placed o...

This work proposes a machine-learning framework for constructing statistical models of errors incurred by approximate solutions to parameterized systems of nonlinear equations. These approximate solutions may arise from early termination of an iterative method, a lower-fidelity model, or a projection-based reduced-order model, for example. The prop...

This work proposes a technique for constructing a statistical closure model for reduced-order models (ROMs) applied to stationary systems modeled as parameterized systems of algebraic equations. The proposed technique extends the reduced-order-model error surrogates (ROMES) method to closure modeling. The original ROMES method applied Gaussian-proc...

This work proposes a technique for constructing a statistical closure model for reduced-order models (ROMs) applied to stationary systems modeled as parameterized systems of algebraic equations. The proposed technique extends the reduced-order-model error surrogates (ROMES) method [13] to closure modeling. The original ROMES method applied Gaussian...

This work proposes a space-time least-squares Petrov-Galerkin (ST-LSPG) projection method for model reduction of nonlinear dynamical systems. In contrast to typical nonlinear model-reduction methods that first apply (Petrov-)Galerkin projection in the spatial dimension and subsequently apply time integration to numerically resolve the resulting low...

Data I/O poses a significant bottleneck in large-scale CFD simulations; thus, practitioners would like to significantly reduce the number of times the solution is saved to disk, yet retain the ability to recover any field quantity (at any time instance) a posteriori. The objective of this work is therefore to accurately recover missing CFD data a p...

This work introduces a new method to efficiently solve optimization problems constrained by partial differential equations (PDEs) with uncertain coefficients. The method leverages two sources of inexactness that trade accuracy for speed: (1) stochastic collocation based on dimension-adaptive sparse grids (SGs), which approximates the stochastic obj...

This work proposes a machine-learning framework for constructing statistical models of errors incurred by approximate solutions to parameterized systems of nonlinear equations. These approximate solutions may arise from early termination of an iterative method, a lower-fidelity model, or a projection-based reduced-order model, for example. The prop...

Summary of contributions to nonlinear model reduction.

This work proposes a method for model reduction of finite-volume models that guarantees the resulting reduced-order model is conservative, thereby preserving the structure intrinsic to finite-volume discretizations. The proposed reduced-order models associate with optimization problems characterized by a minimum-residual objective function and nonl...

Radiation heat transfer is an important phenomenon in many physical systems of practical interest. When participating media is important, the radiative transfer equation (RTE) must be solved for the radiative intensity as a function of location, time, direction, and wavelength. In many heat-transfer applications, a quasi-steady assumption is valid,...

A machine learning-based framework for modeling the error introduced by surrogate models of parameterized dynamical systems is proposed. The framework entails the use of high-dimensional regression techniques (eg, random forests, and LASSO) to map a large set of inexpensively computed "error indicators" (ie, features) produced by the surrogate mode...

This work proposes a structure-preserving model reduction method for marginally stable linear time-invariant (LTI) systems. In contrast to Lyapunov-stability-based approaches for stable and anti-stable systems---which ensure the poles of the reduced system remain in the open left-half and right-half planes, respectively---the proposed method preser...

This work proposes a space-time least-squares Petrov-Galerkin (ST-LSPG) projection method for model reduction of nonlinear dynamical systems. In contrast to typical nonlinear model-reduction methods that first apply (Petrov-)Galerkin projection in the spatial dimension and subsequently apply time integration to numerically resolve the resulting low...

Structure-preserving nonlinear model reduction for finite-volume models

Least-squares Petrov--Galerkin (LSPG) model-reduction techniques such as the Gauss--Newton with Approximated Tensors (GNAT) method have shown promise, as they have generated stable, accurate solutions for large-scale turbulent, compressible flow problems where standard Galerkin techniques have failed. However, there has been limited comparative ana...

This work proposes a machine-learning-based framework for estimating the error introduced by surrogate models of parameterized dynamical systems. The framework applies high-dimensional regression techniques (e.g., random forests, LASSO) to map a large set of inexpensively-computed 'error indicators' (i.e., features) produced by the surrogate model...

We consider the numerical solution of parameterized linear systems where the system matrix, the solution, and the right-hand side are parameterized by a set of uncertain input parameters. We explore spectral methods in which the solutions are approximated in a chosen finite-dimensional subspace. It has been shown that the stochastic Galerkin projec...

We consider the numerical solution of parameterized linear systems where the system matrix, the solution, and the right-hand side are parameterized by a set of uncertain input parameters. We explore spectral methods in which the solutions are approximated in a chosen finite-dimensional subspace. It has been shown that the stochastic Galerkin projec...

This work proposes a data-driven method for enabling the efficient, stable time-parallel numerical solution of systems of ordinary differential equations (ODEs). The method assumes that low-dimensional bases that accurately capture the time evolution of the state are available. The method adopts the parareal framework for time parallelism, which is...

This work proposes a data-driven method for enabling the efficient, stable time-parallel numerical solution of systems of ordinary differential equations (ODEs). The method assumes that low-dimensional bases that accurately capture the time evolution of the state are available. The method adopts the parareal framework for time parallelism, which is...

Radiation heat transfer is an important phenomenon in many physical systems of practical interest. When participating media is important, the radiative transfer equation (RTE) must be solved for the radiative intensity as a function of location, time, direction, and wavelength. In many heat transfer applications, a quasi-steady assumption is valid....

Data-driven time parallelism

This talk presents a new Krylov-subspace-recycling method for efficiently solving sequences of linear systems of equations characterized by varying right-hand sides and symmetric-positive-definite matrices. As opposed to typical truncation strategies used in recycling such as deflation, we propose a truncation method based on a technique from nonli...

This work presents a new Krylov-subspace-recycling method for efficiently solving sequences of linear systems of equations characterized by varying right-hand sides and symmetric-positive-definite matrices. As opposed to typical truncation strategies used in recycling such as deflation, we propose a truncation method inspired by goal-oriented prope...

This work presents a method to adaptively refine reduced-order models a posteriori without requiring additional full-order-model solves. The technique is analogous to mesh-adaptive $h$-refinement: it enriches the reduced-basis space by `splitting' selected basis vectors into several vectors with disjoint support. The splitting scheme is defined by...

Implicit numerical integration of nonlinear ODEs requires solving a system of nonlinear equations at each time step. Each of these systems is often solved by a Newton-like method, which incurs a sequence of linear-system solves. Most model-reduction techniques for nonlinear ODEs exploit knowledge of system's spatial behavior to reduce the computati...

LSPG v Galerkin analysis

This work proposes a model-reduction methodology that preserves Lagrangian structure and achieves computational efficiency in the presence of high-order nonlinearities and arbitrary parameter dependence. As such, the resulting reduced-order model retains key properties such as energy conservation and symplectic time-evolution maps. We focus on para...

This work presents a technique for statistically modeling errors introduced by reduced-order models. The method employs Gaussian-process regression to construct a mapping from a small number of computationally inexpensive “error indicators” to a distribution over the true error. The variance of this distribution can be interpreted as the (epistemic...

Adaptive h-refinement

The ROMES method

This work proposes a model-reduction methodology that preserves Lagrangian structure (equivalently Hamiltonian structure) and achieves computational efficiency in the presence of high-order nonlinearities and arbitrary parameter dependence. As such, the resulting reduced-order model retains key properties such as energy conservation and symplectic...

The Gauss--Newton with approximated tensors (GNAT) method is a nonlinear model reduction method that operates on fully discretized computational models. It achieves dimension reduction by a Petrov--Galerkin projection associated with residual minimization; it delivers computational efficency by a hyper-reduction procedure based on the `gappy POD' t...

This work proposes a model-reduction methodology that both preserves Lagrangian structure and leads to computationally inexpensive models, even in the presence of high-order nonlinearities. We focus on parameterized simple mechanical systems under Rayleigh damping and external forces, as structural-dynamics models often fit this description. The pr...

The goal of this work is to accurately evaluate large-scale, nonlinear, finite-volume-based fluid dynam- ics models at low computational cost. To accomplish this objective, this work employs the Gauss– Newton with approximated tensors (GNAT) nonlinear model reduction method originally presented in Ref. 1. This technique decreases the system dimensi...

A novel model reduction technique for static systems is presented. The method is developed using a goal-oriented framework, and it extends the concept of snapshots for proper orthogonal decomposition (POD) to include (sensitivity) derivatives of the state with respect to system input parameters. The resulting reduced-order model generates accurate...

A Petrov–Galerkin projection method is proposed for reducing the dimension of a discrete non-linear static or dynamic computational model in view of enabling its processing in real time. The right reduced-order basis is chosen to be invariant and is constructed using the Proper Orthogonal Decomposition method. The left reduced-order basis is select...

This document presents the results of a testbed developed for the comparison of model reduction techniques on large-scale linear and nonlinear, static and dynamic systems. The current capabilities of the testbed allow the user to compare eight model reduction methods on three linear, dynamical systems and three reduction techniques on five nonlinea...

This paper applies the Gappy proper orthogonal decomposition method, a recently-developed quantitative methodology for reconstructing unknown data, to archaeological problems and highlights the benefits of the method for quantitative analysis within the field. There are three main advantages of the method over polynomial regression, which is most c...

A rigorous method for interpolating a set of parameterized linear structural dynamics reduced-order models (ROMs) is presented. By design, this method does not operate on the underlying set of parameterized full-order models. Hence, it is amenable to an online real-time implementation. It is based on mapping appropriately the ROM data onto a tangen...

We present an adaptive proper orthogonal decomposition (POD)-Krylov reduced-order model (ROM) for structural optimization. At each step of the optimization loop, we compute approximate solutions to the structural state and sensitivity equations using a novel POD-augmented conjugate gradient (CG) algorithm. This algorithm consists of three stages. I...

A rigorous method for interpolating a set of parameterized linear structural dynamics reduced-order models (ROMs) is presented. By design, this method does not operate on the underlying set of parameterized full-order models. Hence, it is amenable to a real-time and on-line implementation. It is based on mapping appropriately the ROM data onto a ta...

Reduced basis methods are powerful tools that can significantly speed up computationally expensive analyses in a variety of “many-query” and real-time applications, including de- sign optimization. Unfortunately, these techniques produce reduced-order models (ROMs) that are costly to construct and are not always robust in the parameter space. Furth...