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Vaibhav Mehandiratta

Publications and source records attributed to Vaibhav Mehandiratta.

5 recordsLinked to original sources
arXiv2026

Stochastic Galerkin Method for Fractional Boundary Value Problems: Convergence Analysis and Numerical Treatment

We study two-point fractional boundary value problems with uncertain input data, where randomness may enter through the coefficients and boundary conditions. To quantify the resulting uncertainty in the solution, we employ the generalized polynomial chaos (gPC) framework and develop a stochastic Galerkin formulation of the problem. A particular focus of this work is the convergence analysis of the resulting approximation. Rather than imposing assumptions directly on the stochastic coefficients appearing in the gPC representation, we introduce minimal regularity assumptions on the input data and use them to establish the properties required for the convergence analysis. Based on these results, we prove the convergence of the stochastic Galerkin approximation to the corresponding gPC solution. Numerical experiments are presented to illustrate the theoretical findings and to investigate the influence of random coefficients and boundary conditions on the statistical behavior of the solution.

math.NA↗
arXiv2026

QGPINNs: A Physics-Informed Neural Network Framework for Nonlocal Differential Equations on Quantum Graphs

We propose QGPINNs, a physics-informed neural network framework developed in PyTorch for the numerical solution of nonlocal differential equations on quantum graphs. The framework is designed as a general computational implementation in which the solution on each edge of the graph is approximated by a neural network, while a unified graph-based loss function enforces the governing equations together with initial, boundary, and vertex transmission conditions. In particular, the formulation incorporates standard continuity and Kirchhoff-Neumann vertex conditions and Dirichlet boundary conditions into the learning process to couple the local edge-wise neural approximations into a global solution on the graph. The framework is developed for two representative classes of nonlinear models: multi-order fractional elliptic problems and time-fractional evolution equations on quantum graphs. To improve accuracy and training stability, QGPINNs integrates several graph-adapted learning strategies, including soft and hard constraint enforcement, dynamic loss balancing, Fourier feature embeddings, and a learnable singularity-capturing feature for weakly singular solutions arising in the considered problems. The framework also extends naturally to inverse problems, including the identification of the orders of fractional operators and physical parameters from noisy observational data. We validate the accuracy, computational efficiency, and physical consistency of the proposed framework through numerical experiments on benchmark graph structures and real-world networks, including the IEEE 14-bus system and an open-channel agricultural drainage network.

cs.LG↗
arXiv2025

Linear cost and exponentially convergent approximation of Gaussian Matérn processes on intervals

The computational cost for inference and prediction of statistical models based on Gaussian processes with Matérn covariance functions scales cubicly with the number of observations, limiting their applicability to large data sets. The cost can be reduced in certain special cases, but there are currently no generally applicable exact methods with linear cost. Several approximate methods have been introduced to reduce the cost, but most of these lack theoretical guarantees for the accuracy. We consider Gaussian processes on bounded intervals with Matérn covariance functions and for the first time develop a generally applicable method with linear cost and with a covariance error that decreases exponentially fast in the order $m$ of the proposed approximation. The method is based on an optimal rational approximation of the spectral density and results in an approximation that can be represented as a sum of $m$ independent Gaussian Markov processes, which facilitates easy usage in general software for statistical inference, enabling its efficient implementation in general statistical inference software packages. Besides the theoretical justifications, we demonstrate the accuracy empirically through carefully designed simulation studies which show that the method outperforms all state-of-the-art alternatives in terms of accuracy for a fixed computational cost in statistical tasks such as Gaussian process regression.

math.ST↗
arXiv2023

High order approximation to Caputo derivative on graded mesh and time-fractional diffusion equation for non-smooth solutions

In this paper, a high-order approximation to Caputo-type time-fractional diffusion equations involving an initial-time singularity of the solution is proposed. At first, we employ a numerical algorithm based on the Lagrange polynomial interpolation to approximate the Caputo derivative on the non-uniform mesh. Then truncation error rate and the optimal grading constant of the approximation on a graded mesh are obtained as $\min\{4-α,rα\}$ and $\frac{4-α}α$, respectively, where $α\in(0,1)$ is the order of fractional derivative and $r\geq 1$ is the mesh grading parameter. Using this new approximation, a difference scheme for the Caputo-type time-fractional diffusion equation on graded temporal mesh is formulated. The scheme proves to be uniquely solvable for general $r$. Then we derive the unconditional stability of the scheme on uniform mesh. The convergence of the scheme, in particular for $r=1$, is analyzed for non-smooth solutions and concluded for smooth solutions. Finally, the accuracy of the scheme is verified by analyzing the error through a few numerical examples.

math.NA↗