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Popular Bayes filters often apply linearization techniques, such as Taylor expansion or stochastic linear regression, to enable the use of the Kalman filter structure, but this can lead to large errors in strongly nonlinear systems. The…
In the context of Gaussian process regression with functional inputs, it is common to treat the input as a vector. The parameter space becomes prohibitively complex as the number of functional points increases, effectively becoming a…
Most nonlinear partial differential equation (PDE) solvers require the Jacobian matrix associated to the differential operator. In PETSc, this is typically achieved by either an analytic derivation or numerical approximation method such as…
We develop efficient numerical integration methods for computing an integral whose integrand is a product of a smooth function and the Gaussian function with a small standard deviation. Traditional numerical integration methods applied to…
Ordinary differential equations (ODEs) are commonly used to model dynamic behavior of a system. Because many parameters are unknown and have to be estimated from the observed data, there is growing interest in statistics to develop…
This work investigates finite differences and the use of interpolation models to obtain approximations to the first and second derivatives of a function. Here, it is shown that if a particular set of points is used in the interpolation…
Differentiable programming is a fresh programming paradigm which composes parameterized algorithmic components and trains them using automatic differentiation (AD). The concept emerges from deep learning but is not only limited to training…
It is common to model a deterministic response function, such as the output of a computer experiment, as a Gaussian process with a Mat\'ern covariance kernel. The smoothness parameter of a Mat\'ern kernel determines many important…
In this paper, we critically examine the prevalent practice of using additive mixtures of Mat\'ern kernels in single-output Gaussian process (GP) models and explore the properties of multiplicative mixtures of Mat\'ern kernels for…
Automatic Differentiation (AD) is instrumental for science and industry. It is a tool to evaluate the derivative of a function specified through a computer program. The range of AD application domain spans from Machine Learning to Robotics…
The Bayesian smoothing equations are generally intractable for systems described by nonlinear stochastic differential equations and discrete-time measurements. Gaussian approximations are a computationally efficient way to approximate the…
We present a systematic approach for calculating higher-order derivatives of smooth functions on a uniform grid using Pad\'e approximants. We illustrate our findings by deriving higher-order approximations using traditional second-order…
We consider minimizing finite-sum and expectation objective functions via Hessian-averaging based subsampled Newton methods. These methods allow for gradient inexactness and have fixed per-iteration Hessian approximation costs. The recent…
This paper presents an analysis of properties of two hybrid discretization methods for Gaussian derivatives, based on convolutions with either the normalized sampled Gaussian kernel or the integrated Gaussian kernel followed by central…
Graphical models describe associations between variables through the notion of conditional independence. Gaussian graphical models are a widely used class of such models where the relationships are formalized by non-null entries of the…
Fast prototyping of new SE(3) estimation objectives remains awkward in practice. Modern Lie-group frameworks -- GTSAM, manif, Sophus, SymForce, Ceres -- target first-order workloads through different code-generation and…
In multivariate time series, the estimation of the covariance matrix of the observation innovations plays an important role in forecasting as it enables the computation of the standardized forecast error vectors as well as it enables the…
Thermodynamic and flash equilibrium calculations are the cornerstones of simulation process calculations. The iterative approach, a widely used nonlinear problem-solving technique, relies on derivative calculations throughout the procedure…
We review the current state of automatic differentiation (AD) for array programming in machine learning (ML), including the different approaches such as operator overloading (OO) and source transformation (ST) used for AD, graph-based…
In this work, the uncertainty associated with the finite element discretization error is modeled following the Bayesian paradigm. First, a continuous formulation is derived, where a Gaussian process prior over the solution space is updated…