arXiv subjects

Aryan Saxena

Publications and source records attributed to Aryan Saxena.

3 recordsLinked to original sources

arXiv2026

Reduced Order Modeling of One-Dimensional Conservative PDEs via the Cumulative Distribution Transform

We propose a reduced order modeling (ROM) framework for 1D conservative PDEs based on the cumulative distribution transform (CDT). The CDT maps nonnegative, equal-mass states into a Hilbert space in which 1D Wasserstein distances become weighted $L^2$ distances and translations become affine shifts. This makes the transform especially suited for transport-dominated dynamics, where Eulerian linear-subspace ROMs often suffer from slow decay of Kolmogorov widths. We study this phenomenon for scalar conservative dynamics by analyzing the solution manifold in CDT coordinates. For linear transport, the transformed solution manifold is contained in the 2-dimensional space spanned by the transformed initial datum and the constant function, and has zero Kolmogorov $2$-width. For nonlinear hyperbolic conservation laws, we prove two complementary types of estimates: robust $O(n^{-1})$ bounds that rely only on the conservative transport structure and remain meaningful after shock formation, and sharper $O(n^{-2})$ bounds in smooth pre-shock regimes. For conservative advection-diffusion, we show that the CDT trajectory remains within distance $O(\sqrt{DT})$ of the pure-transport plane, and we also obtain sharper $O(D^2T^2)$ estimates under additional regularity or away from initial layers. In both cases, the zero 2-width behavior of linear transport is recovered as the diffusion coefficient tends to zero. Motivated by these estimates, we develop a CDT-POD numerical scheme: snapshots are mapped to CDT space, Proper Orthogonal Decomposition (POD) is performed in transformed coordinates, and the inverse CDT is used to reconstruct physical states. Numerical experiments for several transport-dominated dynamics show that CDT-POD can capture solution manifolds with substantially fewer modes than Eulerian POD.

math.AP↗

arXiv2026

A Discrete Cumulative Distribution Transform via Optimal Transport

This paper develops a fully discrete cumulative distribution transform (CDT) for atomic probability measures on the real line. The transform is defined through monotone quantile maps and admits explicit linear-time algorithms for both forward transformation and inverse reconstruction based solely on cumulative mass matching. Unlike the classical continuous setting, deterministic transport between atomic measures cannot generally split masses, so exact reconstruction may fail at finite resolution. We establish a precise cumulative-mass compatibility criterion for exact finite-resolution recovery and prove weak convergence of reconstructed measures under reference refinement. Several structural properties of the discrete CDT are derived, including translation, composition, and scaling laws, and the framework is extended to a discrete signed cumulative distribution transform with thresholded stabilization near zero crossings. By avoiding continuous interpolation, the proposed framework provides a simple fixed-reference transport representation for discrete data. Numerical examples illustrate translation linearization, compatibility-controlled reconstruction, refinement consistency, and stabilization of the signed transform.

math.ST↗

arXiv2026

Additive Noise, Shift Recovery, and Signed Signals in the Cumulative Distribution Transform

The cumulative distribution transform (CDT) is a quantile-based transport representation that exactly linearizes one-dimensional translations of positive densities. We study how this structure behaves under additive perturbations and how it can be exploited for shift recovery. Under a local nondegeneracy condition, we derive a first-order expansion showing that additive noise in physical space induces a nonlocal perturbation in CDT space through the primitive of the noise, weighted by the reciprocal density. This yields an explicit description of transform-domain sensitivity and shows, in particular, that perturbations are amplified in low-density regions. When the physical-space perturbation is modeled as a centered Gaussian random field, the induced first-order CDT perturbation is again Gaussian, with an explicit covariance kernel. We then use this structure to study recovery in CDT coordinates. In the known-template setting, the transport shift is obtained by projection onto the constant mode, giving an explicit estimator together with exactness in the noiseless case and a stability bound under perturbations. In the unknown-template setting, multiple observations permit joint recovery of the shifts and a common template up to the natural constant-mode gauge, leading to a simple de-shift--and--average procedure. We also consider a signed-signal analogue based on the signed cumulative distribution transform (SCDT), where shifts are estimated numerically by feature matching and unknown templates are recovered by alternating alignment and averaging. Numerical experiments validate the perturbation analysis and illustrate effective recovery for both density-valued and signed signals.

eess.SP↗