arXiv subjects

Jiaxing Liang

Publications and source records attributed to Jiaxing Liang.

4 recordsLinked to original sources

arXiv2026

Recursive rounding of sample size estimation for multi-fidelity Monte Carlo

In multifidelity Monte Carlo (MFMC), optimal sample allocations are typically derived from a continuous relaxation of a variance minimization problem, with integer solutions obtained through post hoc rounding. Such rounding procedures may fail to fully exploit the available cost budget, particularly under tight cost budgets or when model costs vary significantly. In this work, we reformulate the MFMC allocation problem as a variance-constrained cost minimization problem that is equivalent to the standard budget-constrained formulation at the continuous level. This reformulation admits a recursive structure that enables the construction of an integer allocation strategy based on Bellman's principle of optimality. Since the resulting MFMC allocation has a mathematical structure similar to the optimal multilevel Monte Carlo (MLMC) allocation, the proposed strategy naturally extends to MLMC. The resulting algorithm constructs integer-valued sample allocations that more closely follow the continuous variance--cost tradeoff while using the prescribed variance tolerance more efficiently. Numerical experiments demonstrate that the proposed approach satisfies the prescribed variance tolerance with less computational overhead than standard rounding strategies.

math.NA↗

arXiv2026

Surrogate-based multilevel Monte Carlo methods for uncertainty quantification in the Grad-Shafranov free boundary problem

We explore a hybrid technique to quantify the variability in the numerical solutions to a free boundary problem associated with magnetic equilibrium in axisymmetric fusion reactors amidst parameter uncertainties. The method aims at reducing computational costs by integrating a surrogate model into a multilevel Monte Carlo method. The resulting surrogate-enhanced multilevel Monte Carlo methods reduce the cost of simulation by factors as large as $10^4$ compared to standard Monte Carlo simulations involving direct numerical solutions of the associated Grad-Shafranov partial differential equation. Accuracy assessments also show that surrogate-based sampling closely aligns with the results of direct computation, confirming its effectiveness in capturing the behavior of plasma boundary and geometric descriptors.

physics.comp-ph↗

arXiv2023

Multilevel Monte Carlo methods for the Grad-Shafranov free boundary problem

The equilibrium configuration of a plasma in an axially symmetric reactor is described mathematically by a free boundary problem associated with the celebrated Grad--Shafranov equation. The presence of uncertainty in the model parameters introduces the need to quantify the variability in the predictions. This is often done by computing a large number of model solutions on a computational grid for an ensemble of parameter values and then obtaining estimates for the statistical properties of solutions. In this study, we explore the savings that can be obtained using multilevel Monte Carlo methods, which reduce costs by performing the bulk of the computations on a sequence of spatial grids that are coarser than the one that would typically be used for a simple Monte Carlo simulation. We examine this approach using both a set of uniformly refined grids and a set of adaptively refined grids guided by a discrete error estimator. Numerical experiments show that multilevel methods dramatically reduce the cost of simulation, with cost reductions typically on the order of 60 or more and possibly as large as 200. Adaptive gridding results in more accurate computation of geometric quantities such as x-points associated with the model.

physics.comp-ph↗

arXiv2021

Surrogate Approximation of the Grad-Shafranov Free Boundary Problem via Stochastic Collocation on Sparse Grids

In magnetic confinement fusion devices, the equilibrium configuration of a plasma is determined by the balance between the hydrostatic pressure in the fluid and the magnetic forces generated by an array of external coils and the plasma itself. The location of the plasma is not known a priori and must be obtained as the solution to a free boundary problem. The partial differential equation that determines the behavior of the combined magnetic field depends on a set of physical parameters (location of the coils, intensity of the electric currents going through them, magnetic permeability, etc.) that are subject to uncertainty and variability. The confinement region is in turn a function of these stochastic parameters as well. In this work, we consider variations on the current intensities running through the external coils as the dominant source of uncertainty. This leads to a parameter space of dimension equal to the number of coils in the reactor. With the aid of a surrogate function built on a sparse grid in parameter space, a Monte Carlo strategy is used to explore the effect that stochasticity in the parameters has on important features of the plasma boundary such as the location of the x-point, the strike points, and shaping attributes such as triangularity and elongation. The use of the surrogate function reduces the time required for the Monte Carlo simulations by factors that range between 7 and over 30.

math.NA↗