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We consider the $\mathcal{H}^2$-formatted compression and computational estimation of covariance functions on a compact set in $\mathbb{R}^d$. The classical sample covariance or Monte Carlo estimator is prohibitively expensive for many…
We develop a new spatial semidiscrete multiscale method based upon the edge multiscale methods to solve semilinear parabolic problems with heterogeneous coefficients and smooth initial data. This method allows for a cheap spatial…
This article introduces and analyzes a new explicit, easily implementable, and full discrete accelerated exponential Euler-type approximation scheme for additive space-time white noise driven stochastic partial differential equations…
In the stochastic gradient descent (SGD) for sequential simulations such as the neural stochastic differential equations, the Multilevel Monte Carlo (MLMC) method is known to offer better theoretical computational complexity compared to the…
The order of convergence of the Monte Carlo method is 1/2 which means that we need quadruple samples to decrease the error in half in the numerical simulation. Multilevel Monte Carlo methods reach the same order of error by spending less…
We present a sequential Monte Carlo sampler algorithm for the Bayesian analysis of generalised linear mixed models (GLMMs). These models support a variety of interesting regression-type analyses, but performing inference is often extremely…
In this paper, we investigate the problem of strong approximation of the solutions of stochastic differential equations (SDEs) when the drift coefficient is given in integral form. We investigate its upper error bounds, in terms of the…
By formulating the inverse problem of partial differential equations (PDEs) as a statistical inference problem, the Bayesian approach provides a general framework for quantifying uncertainties. In the inverse problem of PDEs, parameters are…
On the one hand, the explicit Euler scheme fails to converge strongly to the exact solution of a stochastic differential equation (SDE) with a superlinearly growing and globally one-sided Lipschitz continuous drift coefficient. On the other…
Approximate inference in probabilistic graphical models (PGMs) can be grouped into deterministic methods and Monte-Carlo-based methods. The former can often provide accurate and rapid inferences, but are typically associated with biases…
In this paper the numerical solution of non-autonomous semilinear stochastic evolution equations driven by an additive Wiener noise is investigated. We introduce a novel fully discrete numerical approximation that combines a standard…
The use of sparse precision (inverse covariance) matrices has become popular because they allow for efficient algorithms for joint inference in high-dimensional models. Many applications require the computation of certain elements of the…
Modeling physical phenomena like heat transport and diffusion is crucially dependent on the numerical solution of partial differential equations (PDEs). A PDE solver finds the solution given coefficients and a boundary condition, whereas an…
This work analyzes the overall computational complexity of the stochastic Galerkin finite element method (SGFEM) for approximating the solution of parameterized elliptic partial differential equations with both affine and non-affine random…
Gaussian process (GP) models form a core part of probabilistic machine learning. Considerable research effort has been made into attacking three issues with GP models: how to compute efficiently when the number of data is large; how to…
We propose an adaptive importance sampling scheme for Gaussian approximations of intractable posteriors. Optimization-based approximations like variational inference can be too inaccurate while existing Monte Carlo methods can be too slow.…
Hamiltonian Monte Carlo is a widely used algorithm for sampling from posterior distributions of complex Bayesian models. It can efficiently explore high-dimensional parameter spaces guided by simulated Hamiltonian flows. However, the…
We consider the numerical solution of scalar, nonlinear degenerate convection-diffusion problems with random diffusion coefficient and with random flux functions. Building on recent results on the existence, uniqueness and continuous…
In many problems, complex non-Gaussian and/or nonlinear models are required to accurately describe a physical system of interest. In such cases, Monte Carlo algorithms are remarkably flexible and extremely powerful approaches to solve such…
We derive computable error estimates for finite element approximations of linear elliptic partial differential equations (PDE) with rough stochastic coefficients. In this setting, the exact solutions contain high frequency content that…