arXiv subjects

Phuong M. Nguyen

Publications and source records attributed to Phuong M. Nguyen.

10 recordsLinked to original sources

arXiv2026

Forward-Time Black-Scholes Reconstruction via Regularized Legendre Reduction

We study a forward-time formulation of the Black-Scholes equation with state-dependent volatility. In contrast to the classical terminal-value pricing problem, where the option payoff is prescribed at maturity and the price is computed backward in time, the present problem prescribes the current option-price profile and seeks to recover the option-price profile at the expiration date T. This formulation is ill-posed, since the equation evolves in the unstable direction of the parabolic operator and high-frequency perturbations in the initial data may be strongly amplified. To address this difficulty, we introduce a price-dimensional reduction based on shifted Legendre polynomials. The original Black-Scholes equation is projected onto a finite-dimensional Legendre basis in the asset-price variable, leading to a system of ordinary differential equations in time for the expansion coefficients. This reduction acts as a spectral cutoff and also relaxes the degeneracy caused by the factor S^2 at the zero-price boundary. The main reconstruction method is a dimension-reduced Legendre--Tikhonov method. We prove existence, uniqueness, data stability, and convergence for each fixed truncation level. We also include a reduced PINN solver as a secondary computational comparison after the Legendre reduction. Numerical experiments with smooth, butterfly-spread, and European put payoffs show that the Legendre--Tikhonov method recovers the terminal option-price profile from noisy initial data, while the reduced PINN solver provides a useful additional benchmark. Comparisons with the conventional physical-space quasi-reversibility method demonstrate the stabilizing effect of the Legendre reduction.

q-fin.CP↗

arXiv2026

A Carleman contraction method for inverse initial data recovery in the Navier-Stokes equations with unknown body force

We solve an inverse initial data problem for the incompressible Navier-Stokes system. The objective is to recover the initial velocity and pressure from lateral boundary observations, without assuming that the time-independent body force is known. To eliminate this unknown force, we differentiate the momentum equation with respect to time and then apply a Legendre polynomial-exponential time-dimensional reduction. This procedure yields a coupled system of elliptic equations for the expansion coefficients. We then construct a contractive map for this reduced system on a suitable admissible set equipped with a Carleman-weighted norm. Its fixed point yields an approximate solution of the time-dimensional reduction model, and the contraction property gives rise to a globally convergent Picard iteration. Finally, we present a numerical algorithm based on this framework and numerical experiments showing accurate reconstructions of the initial velocity and pressure from synthetic boundary data.

math.AP↗

arXiv2025

Inverse scattering without phase: Carleman convexification and phase retrieval via the Wentzel--Kramers--Brillouin approximation

This paper addresses the challenging and interesting inverse problem of reconstructing the spatially varying dielectric constant of a medium from phaseless backscattering measurements generated by single-point illumination. The underlying mathematical model is governed by the three-dimensional Helmholtz equation, and the available data consist solely of the magnitude of the scattered wave field. To address the nonlinearity and severe ill-posedness of this phaseless inverse scattering problem, we introduce a robust, globally convergent numerical framework combining several key regularization strategies. Our method first employs a phase retrieval step based on the Wentzel--Kramers--Brillouin (WKB) ansatz, where the lost phase information is reconstructed by solving a nonlinear optimization problem. Subsequently, we implement a Fourier-based dimension reduction technique, transforming the original problem into a more stable system of elliptic equations with Cauchy boundary conditions. To solve this resulting system reliably, we apply the Carleman convexification approach, constructing a strictly convex weighted cost functional whose global minimizer provides an accurate approximation of the true solution. Numerical simulations using synthetic data with high noise levels demonstrate the effectiveness and robustness of the proposed method, confirming its capability to accurately recover both the geometric location and contrast of hidden scatterers.

math.NA↗

arXiv2024

A global approach for the inverse scattering problem using a Carleman contraction map

This paper addresses the inverse scattering problem in the domain Omega. The input data, measured outside Omega, involve the waves generated by the interaction of plane waves with various directions and unknown scatterers fully occluded inside Omega. The output of this problem is the spatially dielectric constant of these scatterers. Our approach to solving this problem consists of two primary stages. Initially, we eliminate the unknown dielectric constant from the governing equation, resulting in a system of partial differential equations. Subsequently, we develop the Carleman contraction mapping method to effectively tackle this system. It is noteworthy to highlight this method's robustness. It does not request a precise initial guess of the true solution, and its computational cost is not expensive. Some numerical examples are presented.

math.NA↗

arXiv2023

Numerical differentiation by the polynomial-exponential basis

Our objective is to calculate the derivatives of data corrupted by noise. This is a challenging task as even small amounts of noise can result in significant errors in the computation. This is mainly due to the randomness of the noise, which can result in high-frequency fluctuations. To overcome this challenge, we suggest an approach that involves approximating the data by eliminating high-frequency terms from the Fourier expansion of the given data with respect to the polynomial-exponential basis. This truncation method helps to regularize the issue, while the use of the polynomial-exponential basis ensures accuracy in the computation. We demonstrate the effectiveness of our approach through numerical examples in one and two dimensions.

math.NA↗

arXiv2021

The MBHBM$^{\star}$ Project -- II. Molecular Gas Kinematics in the Lenticular Galaxy NGC 3593 Reveal a Supermassive Black Hole

As part of the Measuring Black Holes in Below Milky Way-mass (M$^\star$) galaxies (MBHBM$^\star$) Project, we present a dynamical measurement of the supermassive black hole (SMBH) mass in the nearby lenticular galaxy NGC 3593, using cold molecular gas $^{12}$CO(2-1) emission observed at an angular resolution of $\approx0''.3$ ($\approx10$ pc) with the Atacama Large Millimeter/submillimeter Array (ALMA). Our ALMA observations reveal a circumnuclear molecular gas disc (CND) elongated along the galaxy major axis and rotating around the SMBH. Using dynamical modelling, the molecular gas kinematics allow us to infer a SMBH mass $M_{\rm BH}=2.40_{-1.05}^{+1.87}\times10^6$ M$_\odot$ (only statistical uncertainties at the $3σ$ level). We also detect a massive core of cold molecular gas (CMC) of mass $M_{\rm CMC}=(5.4\pm1.2)\times10^6$ M$_\odot$ and effective (half-mass) radius $r_{\rm CMC,e}=11.2\pm2.8$ pc, co-spatial with a nuclear star cluster (NSC) of mass $M_{\rm NSC}=(1.67\pm0.48)\times10^7$ M$_\odot$ and effective radius $r_{\rm NSC,e}=5.0\pm1.0$~pc (or $0''.15\pm0''.03$). The mass profiles of the CMC and NSC are well described by Sérsic functions with indices $1-1.4$. Our $M_{\rm BH}$ and $M_{\rm NSC}$ estimates for NGC 3593 agree well with the recently compiled $M_{\rm BH}$-$M_{\rm NSC}$ scaling relation. Although the $M_{\rm NSC}$ uncertainty is twice the inferred $M_{\rm BH}$, the rapid central rise of the rotation velocities of the CND (as the radius decreases) clearly suggests a SMBH. Indeed, our dynamical models show that even if $M_{\rm NSC}$ is at the upper end of its allowed range, the evidence for a black hole does not vanish, but remains with a lower limit of $M_{\rm BH}>3\times10^5$ M$_\odot$.

astro-ph.GA↗

arXiv2021

Black hole mass measurement using ALMA observations of [CI] and CO emissions in the Seyfert 1 galaxy NGC7469

We present a supermassive black hole (SMBH) mass measurement in the Seyfert 1 galaxy NGC7469 using Atacama Large Millimeter/submillimeter Array (ALMA) observations of the atomic-${\rm [CI]}$(1-0) and molecular-$^{12}$CO(1-0) emission lines at the spatial resolution of $\approx0.3$" (or $\approx$ 100 pc). These emissions reveal that NGC7469 hosts a circumnuclear gas disc (CND) with a ring-like structure and a two-arm/bi-symmetric spiral pattern within it, surrounded by a starbursting ring. The CND has a relatively low $σ/V\approx0.35$ ($r\sim0.5$") and $\sim0.19$ ($r>0.5"$), suggesting that the gas is dynamically settled and suitable for dynamically deriving the mass of its central source. As is expected from X-ray dominated region (XDR) effects that dramatically increase an atomic carbon abundance by dissociating CO molecules, we suggest that the atomic [CI](1-0) emission is a better probe of SMBH masses than CO emission in AGNs. Our dynamical model using the ${\rm [CI]}$(1-0) kinematics yields a $M_{\rm BH}=1.78^{+2.69}_{-1.10}\times10^7$M$_\odot$ and $M/L_{\rm F547M}=2.25^{+0.40}_{-0.43}$ (M$_\odot$/L$_\odot$). The model using the CO(1-0) kinematics also gives a consistent $M_{\rm BH}$ with a larger uncertainty, up to an order of magnitude, i.e.\ $M_{\rm BH}=1.60^{+11.52}_{-1.45}\times10^7$M$_\odot$. This newly dynamical $M_{\rm BH}$ is $\approx$ 2 times higher than the mass determined from the reverberation mapped (RM) method using emissions arising in the unresolved broad-line region (BLR). Given this new $M_{\rm BH}$, we are able to constrain the specific RM dimensionless scaling factor of $f=7.2^{+4.2}_{-3.4}$ for the AGN BLR in NGC7469. The gas within the unresolved BLR thus has a Keplerian virial velocity component and the inclination of $i\approx11.0^\circ$$_{-2.5}^{+2.2}$, confirming its face-on orientation in a Seyfert 1 AGN by assuming a geometrically thin BLR model.

astro-ph.GA↗

arXiv2019

The MBHBM* Project -- I: Measurement of the Central Black Hole Mass in Dwarf Galaxy NGC 3504 Using Molecular Gas Kinematics

We present the first measurement of the mass of a supermassive black hole (SMBH) in the nearby double-barred spiral galaxy NGC 3504 as part of the Measuring Black Holes Below the Milky Way ($M_{\star}$) mass galaxies (MBHBM$_{\star}$) Project. Our analysis is based on Atacama Large Millimeter/submillimeter Array (ALMA) Cycle-5 observations of the ${\rm ^{12}CO(2-1)}$ emission line. NGC 3504 has a circumnuclear gas disk (CND), which has a relatively high-velocity dispersion of 30 \kms. Our dynamical models of the CND yield a \Mbh~of $M_{\rm BH}=1.02^{+0.18}_{-0.15}\times10^7$\Msun and a mass-to-light ratio in $H$-band of \ml$_{\rm H}=0.66^{+1.44}_{-0.65}$ (\Msun/\Lsun). This black hole (BH) mass is consistent with BH--galaxy scaling relations. We also detect a central deficiency in the ${\rm ^{12}CO(2-1)}$ integrated intensity map with a diameter of 2.7 pc at the putative position of the SMBH. However, this hole is filled by a dense gas tracer ${\rm CS(5-4)}$ that peaks at the galaxy center found in one of the three low-velocity-resolution continuum spectral correlators. The ${\rm CS(5-4)}$ line has the same kinematics with the ${\rm ^{12}CO(2-1)}$ line within the CND, suggesting that it is also an alternative transition for measuring the central \Mbh~in NGC 3504 probably more accurately than the current commonly used of ${\rm ^{12}CO(2-1)}$ due to its centralization.

astro-ph.GA↗

arXiv2019

Improved Dynamical Constraints on the Masses of the Central Black Holes in Nearby Low-mass Early-type Galactic Nuclei And the First Black Hole Determination for NGC 205

We improve the dynamical black hole (BH) mass estimates in three nearby low-mass early-type galaxies--NGC 205, NGC 5102, and NGC 5206. We use new \hst/STIS spectroscopy to fit the star formation histories of the nuclei in these galaxies, and use these measurements to create local color--mass-to-light ratio (\ml) relations. We then create new mass models from \hst~imaging and combined with adaptive optics kinematics, we use Jeans dynamical models to constrain their BH masses. The masses of the central BHs in NGC 5102 and NGC 5206 are both below one million solar masses and are consistent with our previous estimates, $9.12_{-1.53}^{+1.84}\times10^5$\Msun~and $6.31_{-2.74}^{+1.06}\times10^5$\Msun~(3$σ$ errors), respectively. However, for NGC 205, the improved models suggest the presence of a BH for the first time, with a best-fit mass of $6.8_{-6.7}^{+95.6}\times10^3$\Msun~(3$σ$ errors). This is the least massive central BH mass in a galaxy detected using any method. We discuss the possible systematic errors of this measurement in detail. Using this BH mass, the existing upper limits of both X-ray, and radio emissions in the nucleus of NGC 205 suggest an accretion rate $\lesssim$$10^{-5}$ of the Eddington rate. We also discuss the color--\mleff~relations in our nuclei and find that the slopes of these vary significantly between nuclei. Nuclei with significant young stellar populations have steeper color--\mleff~relations than some previously published galaxy color--\mleff~relations.

astro-ph.GA↗

arXiv2018

Nearby Early-Type Galactic Nuclei at High Resolution: Dynamical Black Hole and Nuclear Star Cluster Mass Measurements

We present a detailed study of the nuclear star clusters (NSCs) and massive black holes (BHs) of four of the nearest low-mass early-type galaxies: M32, NGC205, NGC5012, and NGC5206. We measure dynamical masses of both the BHs and NSCs in these galaxies using Gemini/NIFS or VLT/SINFONI stellar kinematics, Hubble Space Telescope (HST) imaging, and Jeans Anisotropic Models. We detect massive BHs in M32, NGC5102, and NGC5206, while in NGC205, we find only an upper limit. These BH mass estimates are consistent with previous measurements in M32 and NGC205, while those in NGC5102 and NGC5206 are estimated for the first time, and both found to be $<$$10^6~M_{\odot}$. This adds to just a handful of galaxies with dynamically measured sub-million $M_{\odot}$ central BHs. Combining these BH detections with our recent work on NGC404's BH, we find that 80\% (4/5) of nearby, low-mass ($10^9-10^{10}~M_{\odot}$; $σ_{\star}\sim20-70$ km/s) early-type galaxies host BHs. Such a high occupation fraction suggests the BH seeds formed in the early epoch of cosmic assembly likely resulted in abundant seeds, favoring a low-mass seed mechanism of the remnants, most likely from the first generation of massive stars. We find dynamical masses of the NSCs ranging from $2-80\times10^6~M_{\odot}$ and compare these masses to scaling relations for NSCs based primarily on photometric mass estimates. Color gradients suggest younger stellar populations lie at the centers of the NSCs in three of the four galaxies (NGC205, NGC5102, and NGC5206), while the morphology of two are complex and are best-fit with multiple morphological components (NGC5102 and NGC5206). The NSC kinematics show they are rotating, especially in M32 and NGC5102 ($V/σ_{\star}\sim0.7$).

astro-ph.GA↗