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Ordinary differential equations (ODEs) are fundamental tools for modeling complex dynamic systems across scientific disciplines. However, parameter estimation in ODE models is challenging due to the multimodal nature of the likelihood…
The variational quantum Monte Carlo (VQMC) method received significant attention in the recent past because of its ability to overcome the curse of dimensionality inherent in many-body quantum systems. Close parallels exist between VQMC and…
This paper develops a probabilistic numerical method for solution of partial differential equations (PDEs) and studies application of that method to PDE-constrained inverse problems. This approach enables the solution of challenging inverse…
In this paper, we apply quasi-Monte Carlo (QMC) methods with an initial preintegration step to estimate cumulative distribution functions and probability density functions in uncertainty quantification (UQ). The distribution and density…
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…
Discrete time stochastic optimal control problems and Markov decision processes (MDPs), respectively, serve as fundamental models for problems that involve sequential decision making under uncertainty and as such constitute the theoretical…
Reliable uncertainty estimation is crucial for machine learning models, especially in safety-critical domains. While exact Bayesian inference offers a principled approach, it is often computationally infeasible for deep neural networks.…
Quantum-mechanical methods are widely used for understanding molecular interactions throughout biology, chemistry, and materials science. Quantum diffusion Monte Carlo (DMC) and coupled cluster with single, double, and perturbative triple…
The multilevel Monte Carlo (MLMC) method has proven to be an effective variance-reduction statistical method for Uncertainty quantification in PDE models. It combines approximations at different levels of accuracy using a hierarchy of…
Dynamical mean-field theory (DMFT) is one of the most widely used theoretical methods for electronic structure calculations, providing self-consistent solutions even in low-temperature regimes, which are exact in the limit of infinite…
Bayesian statistical inverse problems are often solved with Markov chain Monte Carlo (MCMC)-type schemes. When the problems are governed by large-scale discrete nonlinear partial differential equations (PDEs), they are computationally…
While multilevel Monte Carlo (MLMC) methods for the numerical approximation of partial differential equations with random coefficients enjoy great popularity, combinations with spatial adaptivity seem to be rare. We present an adaptive MLMC…
The deep Kolmogorov method is a simple and popular deep learning based method for approximating solutions of partial differential equations (PDEs) of the Kolmogorov type. In this work we provide an error analysis for the deep Kolmogorov…
The theoretical development of quasi-Monte Carlo (QMC) methods for uncertainty quantification of partial differential equations (PDEs) is typically centered around simplified model problems such as elliptic PDEs subject to homogeneous zero…
We propose a generic Markov Chain Monte Carlo (MCMC) algorithm to speed up computations for datasets with many observations. A key feature of our approach is the use of the highly efficient difference estimator from the survey sampling…
Inverse problems involving partial differential equations (PDEs) are widely used in science and engineering. Although such problems are generally ill-posed, different regularisation approaches have been developed to ameliorate this problem.…
Quantum Monte Carlo (QMC) methods are some of the most accurate methods for simulating correlated electronic systems. We investigate the compatibility, strengths and weaknesses of two such methods, namely, diffusion Monte Carlo (DMC) and…
In Quasi-Monte Carlo integration, the integration error is believed to be generally smaller than in classical Monte Carlo with the same number of integration points. Using an appropriate definition of an ensemble of quasi-randompoint sets,…
Quasi-Monte Carlo (QMC) is a powerful method for evaluating high-dimensional integrals. However, its use is typically limited to distributions where direct sampling is straightforward, such as the uniform distribution on the unit hypercube…
In this paper, we consider the numerical solution of a nonlinear Schrodinger equation with spatial random potential. The randomly shifted quasi-Monte Carlo (QMC) lattice rule combined with the time-splitting pseudospectral discretization is…