# Echo Location: Distances to Galactic Supernovae From ASAS-SN Light Echoes
and 3D Dust Maps

Kyle D. Neumann ,1 Michael A. Tucker ,2,3 Christopher S. Kochanek,2,3 Benjamin
J. Shappee ,4 and K. Z. Stanek 2,3
1 Department of Astronomy & Astrophysics, Pennsylvania State University, 525
Davey Lab, University Park, PA 16802, USA
2 Department of Astronomy, The Ohio State University, 140 West 18th Avenue,
Columbus, OH 43210, USA
3 Center for Cosmology and AstroParticle Physics (CCAPP), The Ohio State
University, 191 W. Woodruff Ave., Columbus, OH 43210, USA
4 Institute for Astronomy, University of Hawai’i, 2680 Woodlawn Drive,
Honolulu, HI 96822, USA E-mail<EMAIL_ADDRESS>0000-0002-2701-8433
0000-0002-2471-8442 0000-0003-4631-1149 0009-0001-1470-8400

(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Light echoes occur when light from a luminous transient is scattered by dust
back into our line of sight with a time delay due to the extra propagation
distance. We introduce a novel approach to estimating the distance to a source
by combining light echoes with recent three-dimensional dust maps. We identify
light echoes from the historical supernovae Cassiopeia A and Tycho’s SN in
nearly a decade of imaging from the All-Sky Automated Survey for Supernovae
(ASAS-SN). Using these light echoes, we find distances of $3.6\pm 0.1$ kpc and
$3.2^{+0.1}_{-0.2}$ kpc to Cas A and Tycho, respectively, which are generally
consistent with previous estimates but are more precise. These distance
uncertainties are primarily dominated by the low distance resolution of the 3D
dust maps, which will likely improve in the future. The candidate single
degenerate explosion donor stars B and G in Tycho are clearly foreground
stars. Finally, the inferred reddening towards each SN agrees well with the
intervening H i column density estimates from X-ray analyses of the remnants.

###### keywords:

supernovae: general – supernovae: individual: Cassiopeia A, SN 1572 (Tycho’s
supernova) – ISM: dust, extinction – ISM: supernova remnants

††pubyear: 2024††pagerange: Echo Location: Distances to Galactic Supernovae
From ASAS-SN Light Echoes and 3D Dust Maps–Echo Location: Distances to
Galactic Supernovae From ASAS-SN Light Echoes and 3D Dust Maps

##  1 Introduction

Scattered light echoes (hereafter light echoes or echoes) occur when optical
light from a luminous transient scatters off interstellar dust towards the
observer. Dust heated by absorption of the radiation can also produce a
thermal echo in the mid-infrared (e.g., Krause et al. 2005). The most famous
local examples of scattered light echoes are SN 1987A (Crotts, 1988) and V838
Mon (Bond et al., 2003; Tylenda, 2004), but Rest et al. (2012) also detected
echoes from the Great Eruption of $\eta$ Carinae, Rest et al. (2008, hereafter
R08) detected echoes from the Galactic supernovae (SNe) Cassiopeia A
(hereafter Cas A) and SN 1572 (hereafter Tycho or Tycho’s SN), and Rest et al.
(2005) detected echoes from several SNe in the Magellanic Clouds. Light echoes
can be resolved for SNe in nearby galaxies (e.g., SN 1993J, Sugerman & Crotts
2002; SN 2014J, Crotts 2015), but the small angular sizes are typically only
resolvable from space.

Krause et al. (2008a) classified Cas A as a Type IIb SN using spectroscopy of
its echoes and Rest et al. (2011) subsequently used spectra of multiple echoes
along different sight lines to show that Cas A had distinct explosion
asymmetries. Similarly, light echo spectra from Tycho’s SN (Krause et al.,
2008b) confirmed its classification as a thermonuclear (Type Ia) explosion
based on historical records of its light curve (Ruiz-Lapuente, 2004). The
spectra of echoes from $\eta$ Carinae provide a unique probe of this enigmatic
transient (Rest et al. 2012, Smith et al. 2018). This shows how light echoes
can provide unique probes of historical events long after they have faded.

Fig. 1 shows the geometry of a light echo. Chevalier (1986), Sugerman (2003),
and Tylenda (2004) provide the basic models for the appearance of a light echo
for various dust geometries, and Rest et al. (2005) expands upon these models
for Galactic SNe. The location and motion of echoes determine the distance
between the reflecting dust and the source but not the distance to the source
or the dust (Bond & Sparks, 2009). With the addition of polarization
measurements, it is possible to determine the overall distance to the source
because the maximum polarization constrains the scattering angle to provide
the extra geometric constraint needed to make a distance estimate (Sparks,
1994). Versions of this method have been used to estimate the distance to V838
Mon (Sparks et al., 2008) and SN 1987A (e.g., Cikota et al., 2023).

Distances to supernova remnants (SNR) are generally quite uncertain. Only two
have direct parallaxes to a pulsar remnant (Vela, Dodson et al. 2003, and the
Crab, Chatterjee et al. 2009) and three are interacting binaries with
parallaxes to the companion stars (see Kochanek, 2021). Dispersion measure
models can be used for pulsars but are only accurate to $\sim 25\%$ (e.g.,
Taylor & Cordes 1993). H i (or CO) radio absorption velocities combined with a
Galactic kinematic model can provide approximate distances for SNRs towards
the inner galaxy (e.g., first by Ilovaisky & Lequeux 1972) and the correlation
between radio diameter and surface brightness (the $\Sigma$-$D$ relation) also
provides approximate distances (e.g., Case & Bhattacharya 1998). Kinematic
models of the SNR expansion, matching proper motions and velocities or simply
the expansion rate and size, depend on models for the three-dimensional shape
of the expansion or the deceleration history (e.g., Chevalier et al. 1980,
Reed et al. 1995, Hayato et al. 2010, Alarie et al. 2014 for Cas A and Tycho).
All of these non-parallex methods have significant distance uncertainties
leading to substantial uncertainties in remnant masses and energetics. This
also leads to difficulties in identifying companion stars unbound in the
explosion (e.g., Ruiz-Lapuente et al., 2004a; Kerzendorf et al., 2013; Shappee
et al., 2013, for Tycho). One new approach is to use wide-field, multi-fiber
spectrographs to determine the distance at which stars show high velocity Ca
ii or Na i absorption lines from the SNR (Kochanek et al. 2024).

Recent years have seen the rapid development of three-dimensional (3D) maps of
the Galactic dust distribution. While there are older models (e.g., Drimmel et
al., 2003; Berry et al., 2012; Green et al., 2015), the availability of Gaia
(Gaia Collaboration et al., 2016; Lindegren et al., 2018, 2021) parallax
measurements revolutionized our ability to map the Milky Way dust distribution
on scales of kpc, leading to many new models (e.g., Green et al., 2019;
Dharmawardena et al., 2024; Edenhofer et al., 2024). Given a 3D dust map,
there are two ways to estimate the distance to a SNR. The first approach is to
find the distance where the extinction matches an estimate of the extinction
of the SNR, usually from the X-ray gas absorption column density (e.g., Wang
et al. 2020) and assuming a gas-to-dust ratio. The second approach, which we
introduce here, is through dust echoes, where knowing the distance to the dust
supplies the missing information needed to determine the distance from an echo
(see Fig. 1). This second approach has the advantage that multiple,
independent distance measurements can be used if there are multiple, well-
separated echoes.

Figure 1: The geometry of a light echo.

Using wide-field imaging from the All-Sky Automated Survey for Supernovae
(ASAS-SN111http://www.astronomy.ohio-state.edu/~assassin/; Shappee et al.
2014) from 2014 through 2021, we identify light echoes from two galactic SNe,
Cas A and Tycho, and use them to estimate their distances. In our fiducial
analysis, we combine the ASAS-SN echoes with the Tycho and Cas A echoes from
R08. We use the Bayestar19 3D dust maps (Green et al. 2019). While the
Dharmawardena et al. (2024) maps provide better distance resolution, their
distance range ($<2.8$ kpc) is too restricted for these SNRs. We did find that
the Green et al. (2019) and Dharmawardena et al. (2024) maps agree well inside
this distance limit for our lines of sight. In Section 2, we describe how we
find light echoes in the ASAS-SN data. In Section 3, we use these 3D dust maps
and light echo geometry to constrain the distances to these SN. In Section 4,
we summarize our results.

##  2 The Tycho and Cas A Echoes In ASAS-SN

ASAS-SN observes the entire sky with 20 telescopes on 5 mounts (since
late-2017) using a tiling of fields. A field is observed using three 90 second
exposures, initially in the $V$-band through late-2018 and now in the $g$-band
beginning in late-2017 (Shappee et al. 2014, Kochanek et al. 2017, Hart et al.
2023). There are now $\sim 4000$ images available per field collected over
roughly 10 years. To search for echoes, we coadd data in “observing year” bins
(i.e., divided by solar conjunctions) and then use image subtraction to look
for changes. We start by assessing the quality of the images in a given ASAS-
SN field. Images flagged as low-quality by visual inspection, such as those
obtained through moderate cloud cover, are immediately removed. Images with
large residuals after kernel convolution are also removed as this indicates
the photometric quality of the observation is poor. Then, we discard the 33%
of the remaining images with the highest sky backgrounds. These images are
typically obtained during nights with high Moon illumination and/or high-
altitude cloud cover. The 33% threshold is somewhat arbitrary, but
experimentation found that discarding $\approx 25-35\%$ of the images produces
the best results. After applying these quality cuts, the median number of
images per field is $\approx 200$ in $V$-band and $\approx 350$ in $g$-band.

Next, the images are coadded by observing season. We tested different methods
for scaling the images during the stacking process (e.g., median, mode, sky
value) but in the end, we simply averaged all images to create the yearly
stack. This worked well because (1) all ASAS-SN images are obtained with the
same exposure time ($90$ s) and (2) the lower-quality images were already
discarded in the initial quality assessment. The final $V$\- and $g$-band
yearly stacked images of the region around Tycho’s SN and Cas A have $\sim 60$
and $\sim 90$ images, respectively, with median surface brightness limits of
$V\approx 22$ mag arcsec-2 and $g\approx 24$ mag arcsec-2.

Finally, the yearly stacks are pair subtracted using the ISIS image
subtraction software (Alard, 2000). All image pairs are subtracted, so 4
stacked images labeled $1-4$ produce 6 subtraction pairs ($1-2$, $1-3$,
$\ldots$, $4-3$). This carries several benefits relative to only subtracting
sequential stacks (e.g., $1-2$, $2-3$, $3-4$). Galactic light echoes move tens
of arcseconds per year (R08), so only subtracting sequential stacks risks
self-subtraction of any light echo signal if the angular motion is less than
the PSF FWHM ($\approx$ 2 pixels = 16″). The pair subtraction also helps to
separate true light echoes from subtraction artifacts through their time
evolution. Subtraction artifacts around bright sources are reduced with an
iterative $5\sigma$ clipping routine. Finally, the subtracted images are
filtered using a $5\times 5$ pixel median filter ($\approx 40\arcsec\times
40\arcsec$) to highlight the spatially resolved light echoes and to suppress
any remaining subtraction artifacts (see Fig. 2).

We visually inspected the subtracted images for light echoes, focusing on the
region around Cas A and Tycho’s SN and flagging those that show coherence
across multiple subtractions. Li et al. (2022) attempted to find light echoes
using a machine-learning approach, but with only limited success, in large
part due to the lack of good training data. For each identified echo, we fit a
linear segment to the echo in each subtraction (see Figs. 2 and 3), and we
average these to calculate their central point and position angle (P.A.). This
technique gives us a position with an accuracy of $\sim 20\arcsec$. Nearby
echoes with coherent motion away from the SNR are labeled as part of the same
echo “group” (or simply group). Not all images of a given echo will be used in
the subsequent analysis. Some image stacks have fewer images and noisier
subtractions or a subtraction artifact interfering with the analysis. We only
keep echoes with detections in a least two of the subtracted images. Table 2
lists the coordinates, P.A.s, and mean times of observations for the 57
identified echoes.

Figure 2: Echoes from Cas A. The left panel shows the difference image for
epochs separated by two years, and the right panel shows all the detected echo
groups in this field. The images are very spatially distorted – the red cross
is 0.1°by 0.1°with masking of saturated and bad pixels. The lines show the
position and orientation of the echo and color codes the age, showing that the
echoes are expanding outwards from Cas A as expected. Dotted boxes show how
the echoes are grouped to deal with the spatial correlations of the dust maps.
The blue line is the light echo identified by R08. Dashed red lines extend
from each echo group to the geometric center of the Cas A remnant. The offset
from the red cross to the geometric center of Cas A is 7.6°. Figure 3: Echoes
from Tycho’s SN. The zoomed-in image shows the $g$-band Group 2 in images
separated by two years where the pink line is the older detection, and the
green line is the later detection. These images are significantly distorted –
the red cross is 0.1° by 0.1°. The lines show the position and orientation of
the echo and the color codes the age, showing that the echoes are expanding
outwards from Tycho as expected. Echo groups are identified with boxes and the
blue line is the light echo in this field identified by R08. A dashed red line
extends from the center of each group to the center of Tycho. The geometric
center of Tycho is 6.4°from the red cross.

##  3 Analysis

Creating a dust echo depends on having dust to scatter the light at the
distance required by the geometry of Fig. 1. The constraint equation for an
echo (e.g., Tylenda 2004) is

$c\Delta t=(Z-z)+(D-d)$ (1)

where

$\Delta t=t_{\mathrm{o}}-t_{\mathrm{e}}$

is the difference between the time of observation ($t_{\mathrm{o}}$) and
explosion ($t_{\mathrm{e}}$). The time delay ($c\Delta t$) equals the
difference in length between the direct and the dust-scattered paths from the
source. If $\theta$ is the observed angle between the echo and the SN, then
the distance between the observer and the dust is

$D=\frac{c\Delta t(c\Delta t+2x)}{2(c\Delta t+x-x\cos{\theta})}.$ (2)

For a given source distance, $x$, and explosion time, $t_{e}$, the
uncertainties in the estimate of $D$ are negligible since they only arise from
the uncertainties in $t_{o}$, which is known to a fraction of a year, and
$\theta$, which is measured to an accuracy of $\sim 20\arcsec$ out of an
overall angle of $\theta\simeq 7^{\circ}$ (e.g., Figs. 2 and 3).

Thus, given the echo’s location and the time of observation, the observed echo
can only be produced for a given source distance and time of explosion if
there is dust at a distance $D$ along the line of sight to the echo. The
amplitude of the echo is largely determined by the density of dust at that
distance with secondary effects from the scattering angle which we will
ignore, so we need the dust density as a function of distance
$P_{j}(D_{j}|x,t_{e})$. We fit the Bayestar19 integral dust distribution with
monotonic splines (scipy.interpolate.PchipInterpolator, Virtanen et al., 2020)
and then use the derivative of the monotonic splines with respect to the
distance $D_{j}$ to define $P_{j}(D_{j}|x,t_{e})$. Since Bayestar19 supplies 5
posterior distributions for each sight line, we use the average of the 5
density estimates. Fig. 4 shows these distributions for two Tycho echoes with
very different dust distributions. In one case, there is a well-defined dust
sheet at a distance of 2-3 kpc which will strongly constrain the dust
distance, $D_{j}$. In the second case, the distribution of dust along the line
of sight is much broader and will provide little constraint on the dust
distance. Because the distance resolution of the Bayestar19 dust maps is
limited, even the thin dust sheet is significantly smeared out in distance.
This makes it essentially impossible to usefully constrain the explosion time
and limits the precision of the distance estimate from any single echo.

Since the angular resolution of the Bayestar19 dust maps is $\sim 5\arcmin$
near Cas A and Tycho, we will derive essentially the same dust density
distribution for echoes with smaller separations. If we multiply the distance
posteriors of such echoes, then we are treating their dust distributions as
independent measurements, yet they are not. To avoid this, we work in terms of
echo groups consisting of all echoes with separations less than $6\arcmin$. In
practice, this means groups form a time sequence of echoes expanding away from
the SNR like those seen in Figs. 2 and 3. For the echoes $j$ in group $i$ we
use the mean of their dust density distributions

$\bar{P}_{i}(D_{i}|x,t_{e})=\left\langle P_{i}(D_{j}|x,t_{e})\right\rangle.$
(3)

as a single dust density distribution for the group. This should be a
conservative approach to handling the correlations in the dust maps.

The dust distributions, $\bar{P}_{i}(D_{i}|x,t_{e})$, for each group $i$
should be uncorrelated, so we can multiply their distance posteriors. Using
Bayes theorem, our posterior for the distance and age is then

$P(x,t_{e}|\hbox{data})\propto P(x)P(t_{e})\prod_{i}\int
dD_{i}\bar{P}_{i}(D_{i}|x,t_{e})$ (4)

where $P(x)$ and $P(t_{e})$ are the priors for the SN distance and explosion
time, respectively. Bayesian probability distributions are always normalized
to unity, which means that the normalization of the
$\bar{P}_{i}(D_{i}|x,t_{e})$ dust density distributions drops out of the final
posteriors. We use a uniform SN distance prior, $P(x)$, from 1.75 kpc to 6.25
kpc for both Cas A and Tycho. This prior spans their previous distance
estimates (see below). We used Gaussian priors for the time of explosion
$P(t_{e})$ with the parameters defined in Sections 3.1 and 3.2. Using a
uniform prior for the time of explosion does not affect the distance estimate.
As noted earlier, the low distance resolution of the dust maps means that we
are unable to constrain the time of explosion, so the posteriors for the time
of explosion are basically just the priors. Fortunately, the distance
posterior is essentially uncorrelated with the time of explosion.

As a simple sanity check on the results, we can compare the extinction at the
inferred distance to the SNR with the H i column density $N(H)$ inferred from
X-ray observations of the SNR. The $E(g-r)$ extinction of the Bayestar19 maps
can be converted to $E(B-V)=1.11\times E(g-r)$ (see, e.g., Appendix B in Green
et al., 2014), and the visual extinction is related to the column density by

$\hbox{$N(\rm H)$}=8.3\times
10^{21}~{}\mathrm{cm}^{-2}~{}\mathrm{mag}^{-1}\times E(B-V)$ (5)

(e.g., Liszt, 2014a, b) with a scatter of $\approx 10\%$ (e.g., Lenz et al.,
2017; Nguyen et al., 2018; Zuo et al., 2021).

Figure 4: Examples of a well (_left_) and a poorly (_right_) localized dust
sheet for two Tycho echoes. The upper panels show the integral distributions
and the supplied statistical samplings, and the bottom panels show the
resulting estimates of the dust density $P(D)$ (colors) and their mean
(black).

###  3.1 Cassiopeia A

Cas A is the remnant of a Type IIb SN (Krause et al., 2008a), and it was the
most recent galactic SN that may have been observed. If the explosion was
observed, it would have been the sixth magnitude object 3 Cassiopeiae observed
by John Flamsteed in 1680 August 16, but its separation of 12′ from Cas A
makes the identification uncertain (Ashworth, 1980; Hughes, 1980). Studies of
the proper motions of the ejecta by Thorstensen et al. (2001) and Fesen et al.
(2006) estimate explosion times of 1671 - 1680 and $1681\pm 19$ years,
respectively, which are consistent with Flamsteed’s observation. Based on
this, we adopt a Gaussian prior for $P(t_{e})$, centered on 1676 CE with a
dispersion of 5 yrs. Using a flat prior spanning 1670 - 1680 produces
identical results.

Fig. 2 shows an example of a Cas A light echo at an offset of approximately
$7.6\degr$ from the geometric center of
$\alpha(J2000)=23^{h}23^{m}27\aas@@fstack{s}77,\delta(J2000)=+58\degr
48^{\prime}49\aas@@fstack{\prime\prime}4$ for Cas A found by Thorstensen et
al. (2001). This geometric center lies $6\aas@@fstack{\prime\prime}6\pm
1\aas@@fstack{\prime\prime}5$ from the neutron star formed in the SN. The
total light echo is approximately 1° in length, and we broke it into 5
separate echo groups (Groups 1 to 5) where Group 3 and Group 5 are the only
segments clearly visible in the $g$-band data. This light echo appears to be a
continuation of the light echo located in Field No. 3824 of R08, as shown in
Fig. 2. From 2014 to 2021, we found 35 echoes in 5 echo groups, and we analyze
them along with 6 echoes identified by R08 as a complimentary sample. These 6
echoes will be treated as 6 individual echo groups consisting of a single echo
each.

Fig. 5 shows an example of inferring the Cas A distance from a single echo,
Echo 4 in Group 5. Using Eq. 2, the known time of observation, and the offset
of the echo from the SNR, we estimate the dust density along the line of
sight, $P_{i}(D_{i}|x,t_{e})$ (Eq. 3). We integrate over the dust distance
weighted by the density (Eq. 4) to get a probability distribution for the SN
distance and explosion time, $P_{i}(x,t_{e}|\mathrm{data})$. We then
marginalize the distribution over the time axis to get the posterior
probability distribution for the SN distance, $P_{i}(x|\mathrm{data})$. As
seen in Fig. 5, a single thin dust sheet dominates this line of sight, leading
to a well-constrained distance of $3.7_{-0.4}^{+0.2}$ kpc with a single echo.
As expected, the posterior for the explosion time is essentially identical to
the prior due to the uncertainties associated with dust distances. We also see
that the distance is uncorrelated with the explosion time.

Figure 5: An example of estimating the distance to Cas A for a single echo
(Echo 4 of Group 5). The left panels show the integrated (_top_) and
differential (_bottom_) dust distributions. The _upper right_ panel shows the
2D probability distribution for the Cas A distance and its explosion year. The
yellow line shows the adopted explosion time of 1680 August 16, when John
Flamsteed may have observed Cas A (Ashworth, 1980). The _lower right_ panel is
the posterior distribution marginalized along time to show the resulting
distance posterior. The solid vertical line is the distance of maximum
likelihood, and the dotted lines are the 1$\sigma$ uncertainty.

We did the full calculations for the $V$-band ASAS-SN echo groups, the
$g$-band ASAS-SN echo groups, the echoes identified by R08, and the entire
joint echo group sample with the results listed in Table 1 and shown in Fig.
6. We use the distance at the maximum of the posterior distribution for the
central value and the $1\sigma$ uncertainty calculated from the 16% to 84%
range of the integral posterior probability distributions. All four results
are mutually consistent, with a final joint distance estimate of $3.6\pm 0.1$
kpc. Fig. 7 shows the 2D and time-integrated posterior distributions for this
joint sample. Fig. 6 also shows other distance estimates for Cas A. The best
existing estimates come from kinematic models of the proper motions and radial
velocities of emission line gas in the SNR, with Reed et al. (1995) finding
$3.4^{+0.3}_{-0.1}$ kpc and Alarie et al. (2014) finding $3.33\pm 0.10$ kpc.
Our result is consistent with the former and $2\sigma$ inconsistent with the
latter. At base, these kinematic models compare the proper motion of the
expansion $\mu$ to the line of sight expansion velocity $v$ and determine the
distance by matching the two, $d=v/\mu$. While they find the remnant to be
fairly spherical, very modest asymmetries could explain the 9% distance
difference with Alarie et al. (2014). Even so, such close (9%) agreement
between two radically different methods is encouraging.

Figure 6: Our estimates for the distance to Cas A (_bottom_ , colored)
compared to the literature estimates (gray) with 1$\sigma$ uncertainties. Our
estimates include ASAS-SN $V$-band echoes (green), ASAS-SN $g$-band echoes
(blue), echoes from R08 (red), and the joint echo sample (black). The
literature distances estimates (_upper_) are from Reed et al. (1995) and
Alarie et al. (2014).

Figure 7: 2D (_upper_) and time-integrated (_lower_) posterior distribution
for joint samples of Cas A (_left_) and Tycho (_right_) distances and
explosion dates. The yellow dashed lines represent the adopted explosion time
of 1680 August 16 and 1572 November 21 as defined in Figs. 5 and 8 for Cas A
and Tycho, respectively. The solid vertical black lines are the distance of
maximum likelihood, and the dotted lines are the 1$\sigma$ uncertainty.

Fig. 10 shows the dust distribution along the line of sight to the center of
Cas A along with our distance estimate. We find that Cas A lies inside or just
behind a dust sheet with significant optical depth. If we sample this dust
distribution over our distance posterior, we find an extinction posterior of
$E(g-r)=1.7_{-0.6}^{+0.3}$. For comparison, Hwang & Laming (2012) measure
$N(H)\simeq 2\times 10^{22}\mathrm{~{}cm~{}}^{-2}$ (with no reported
uncertainties) from Chandra X-ray observations, or $E(g-r)\simeq 2.2\pm 0.3$
if we assume a 10% measurement uncertainty and add the 10% scatter in the
extinction to column density conversion. This is consistent with our result.
At least for this sight line, using the dust towards Cas A to estimate its
distance would give a very strong lower limit because of the dust sheet at
$\sim 3.5$ kpc, but a much weaker upper limit because the next increase in the
integrated extinction lies near 6 kpc and is a fairly weak jump.

Table 1: Distance Estimates Sample | Cas A (kpc) | Tycho (kpc)
---|---|---
$V$-Band | $3.6^{+0.3}_{-0.1}$ | $4.2^{+0.2}_{-0.9}$
$g$-Band | $3.5^{+0.1}_{-0.2}$ | $3.3^{+0.2}_{-0.2}$
Rest+ 08 | $3.7^{+0.2}_{-0.1}$ | $3.1^{+0.2}_{-0.2}$
Joint | $3.6^{+0.1}_{-0.1}$ | $3.2^{+0.1}_{-0.2}$

###  3.2 Tycho’s Supernova

Tycho’s SN was a Type Ia SN first observed on 1572 November 11 by Jerónimo
Muñoz and Tycho Brahe separately, as discussed by Ruiz-Lapuente (2004), and it
was observed by European and Asian astronomers for two years. Tycho (SN 1572)
and Kepler (SN 1604) were the most recent, clearly observed SN in the Galaxy
given the questions surrounding the possible Flamsteed observation of Cas A.
Spectroscopy of a light echo 400 years later conclusively showed that Tycho’s
SN was a Type Ia SN (Krause et al., 2008b). The direct observations provide a
light curve at peak (see Ruiz-Lapuente 2004), allowing us to use a very tight
Gaussian prior on the explosion time centered on the discovery date, 1572
November 11, and with a dispersion of 20 days. Using a flat prior on the
explosion time produces identical results.

Fig. 3 shows three echo groups associated with Tycho at an average offset of
approximately $6.4\degr$ from the geometric center of
$\alpha(J2000)=00^{h}25^{m}19\aas@@fstack{s}9,\delta(J2000)=+64\degr
08^{\prime}18\aas@@fstack{\prime\prime}2$ for Tycho found by Ruiz-Lapuente et
al. (2004b). Groups 1 and 2 are separated from Group 3 by approximately
$4\degr$. Group 3 may be associated with the echo located in field 4821 of
R08. Groups 1 and 2 are only seen in the $V$\- and $g$-bands, respectively,
while Group 3 is seen in both. We locate 22 echoes associated with 3 echo
groups over almost 7 years of observations and analyze them along with the 6
echoes identified by R08. Each of these 6 R08 echoes are treated as separate
echo groups consisting of a single echo.

Fig. 8 is an example of the results for Tycho Echo 13 of Group 2. For this
echo, there are two dust sheets along the line of sight, but the second sheet
lies close to the 6.25 kpc upper limit of our distance prior. Nonetheless, its
presence produces a second probability peak and leads to a posterior distance
estimate of $3.3_{-0.7}^{+1.1}$ kpc with a relatively large positive distance
uncertainty. This is an example of why having multiple groups helps. As we
combine echo groups, the probability associated with the true distance will be
reinforced, while the probability associated with such secondary peaks should
increasingly cancel provided the distances of any secondary dust sheets are
relatively uncorrelated. The explosion time posterior is again unchanged from
the prior, as expected, and there is no correlation of the distance estimate
with the explosion time. The distance estimates from the $V$-band echo groups,
$g$-band echo groups, R08 echoes, and the joint echo group sample are given in
Table 1 and shown in Fig. 9. The results from the echo samples are mutually
consistent, with a final distance estimate of $3.2_{-0.2}^{+0.1}$ kpc from the
joint sample. While all distance estimates are mutually consistent, the
$V$-band estimate has significantly larger uncertainties due to the broader
dust distributions over their echoes and fewer available images. Fig. 7 shows
the 2D and time-integrated posterior distributions for the joint sample.

Figure 8: An example of estimating the distance to Tycho’s SN for a single
echo (Echo 11 of Group 2). The left panels show the integral (_top_) and
differential (_bottom_) dust distributions. The _upper right_ panel is the 2D
probability distribution for the Tycho distance and explosion date where the
yellow line is 1572 November 21, the expected day of peak luminosity (Ruiz-
Lapuente, 2004). The _lower right_ panel is the projected probability
distribution for the SN distance based on this echo where the solid vertical
line is the distance of maximum likelihood and the dotted lines are the
1$\sigma$ uncertainty. Figure 9: Our estimates for the distance to Tycho’s SN
(_bottom_ , colored) compared to the literature estimates (gray) with
1$\sigma$ uncertainties. Our estimates include ASAS-SN $V$-band echoes
(green), ASAS-SN $g$-band echoes (blue), echoes from R08 (red), and joint echo
samples (black). The literature distance estimates (_upper_) are from
Chevalier et al. (1980), Albinson et al. (1986), Schwarz et al. (1995), Ruiz-
Lapuente (2004), Völk et al. (2008), Krause et al. (2008b), Hayato et al.
(2010).

Fig. 9 also shows other estimates of the distance to Tycho. The two 21 cm H i
absorption velocity distances of $1.7$-$3.7$ kpc (Albinson et al., 1986) and
$4.5\pm 0.5$ kpc (Schwarz et al., 1995), illustrate the problems with this
method, driven in part by the velocity perturbations of a spiral arm.
Chevalier et al. (1980) using a small number of proper motions and radial
velocities found a distance of $2.3\pm 0.5$ kpc. Using similar methods, Smith
et al. (1991) found a distance of $1.5$-$3.1$ kpc. Hayato et al. (2010)
combined X-ray proper motions from Katsuda et al. (2010) with their
measurements of X-ray line widths to find a distance of $4\pm 1$ kpc. Based on
the estimated peak brightness and light curve structure of the SN, Krause et
al. (2008b) estimated a distance of $3.5_{-0.9}^{+1.5}$ kpc and Ruiz-Lapuente
(2004) estimated a distance of $2.8\pm 0.4$ kpc. Finally, we note that Völk et
al. (2008) argue that the lack of a $\gamma$-ray detection of Tycho requires a
minimum distance of $3.3$ kpc. Our results are generally consistent with these
estimates if only because of their large uncertainties.

Fig. 10 also shows the dust distribution along the line of sight to the center
of Cas A along with our distance estimate. Unlike Cas A, the dust is almost
uniformly distributed in distance, and we find a posterior estimate of the
extinction towards the center of Tycho of $E(g-r)=0.82_{-0.02}^{+0.03}$. For
comparison, Cassam-Chenaï et al. (2007) find $N(H)\approx 7\times
10^{21}\mathrm{~{}cm~{}}^{-2}$ or $E(g-r)\simeq 0.8\pm 0.1$ from Chandra X-ray
observations making the same assumptions about the uncertainties as for Cas A.
This agrees extremely well with our estimate. At least for this sight line,
the dust towards the SN would provide a much weaker distance estimate because
the gradient of extinction with distance is so weak. The estimate of the
extinction from the column density would need the precision of the
uncertainties on the extinction posterior to produce a similar distance
uncertainty.

##  4 Discussion And Conclusions

We present new distance measurements to the historical Cas A and Tycho SNe by
combining light echoes and 3D dust maps, finding fiducial distances of $3.6\pm
0.1$ and $3.2_{-0.2}^{+0.1}$ kpc, respectively, that are most precise to date.
The largest systematic uncertainties arise from any miscalibrations in the
dust distance scale, but as these are anchored on Gaia parallaxes, they are
unlikely to be large. The uncertainties are driven by the low distance
resolution of the 3D dust maps and can be improved either by higher resolution
maps or the addition of more echoes sampling different line of sight dust
distributions. The extinctions predicted by our distances agree with those
expected from X-ray measurements of the H i column densities towards these two
SNRs. Our method can be applied to any eruptive or explosive Galactic event
luminous enough to produce resolved light echoes.

As noted in the introduction, Tycho has been the focus of many searches for a
surviving stellar binary companion to the white dwarf which exploded (e.g.,
Ruiz-Lapuente et al. 2004a, González Hernández et al. 2009, Kerzendorf et al.
2013, Kerzendorf et al. 2018). Finding such a companion would be the first
clear evidence of a single degenerate channel for Type Ia SNe, particularly
given the failure to find evidence for the hydrogen expected to be stripped
from the donor by the explosion in nebular phase spectra of large samples of
normal Type Ia SNe (e.g., Tucker et al. 2020). The distance uncertainties for
both the candidate stars and Tycho itself have made the search for an unbound
companion challenging. Gaia parallaxes have largely addressed the problem of
the distances to the stars (see Ruiz-Lapuente et al. 2019), and our echo
distance largely solves the problem of the distance to Tycho. Ruiz-Lapuente et
al. (2019) listed 15 stars with Gaia distances consistent with a distance of
$2.7\pm 1.0$ kpc that roughly spans the earlier measurements in Fig. 9. Only 7
of these stars have distances consistent with our SNR distance, largely due to
having larger parallax uncertainties. The occasionally favored candidate donor
stars B and G are both clearly foreground stars.

Figure 10: Average extinction as a function of distance at the explosion
locations for each SN from the Bayestar19 reddening maps (Green et al., 2019).
Solid lines represent the extinction functions, and dashed lines represent the
median distances found from the joint sample for each SN. Colored regions are
the $1\sigma$ uncertainty calculated from the extinction map.

Our ability to find light echoes with ASAS-SN slowly improves due to both the
slowly increasing integration time and the longer time baseline. With a
sufficiently long baseline (nearly 20 years), we can search for echoes in the
Magellanic Clouds where the long time scale is set by the expected time for an
echo to traverse ASAS-SN’s 16″point spread function (PSF). The main problem is
that older echoes have a lower surface brightness, and the high stellar
densities near the plane of the Galaxy saturate the PSF, limiting ASAS-SN’s
surface brightness sensitivity (at higher latitudes, its surface brightness
sensitivity is quite good, reaching 27 mag/arcsec2 in the g-band, Jennerjahn
et al. in prep). Other existing surveys like ATLAS (Tonry et al., 2018), ZTF
(Bellm et al., 2019), and the Dragonfly array (Abraham & van Dokkum, 2014)
should perform better at lower latitudes. The Vera C. Rubin Observatory
(Ivezić et al., 2019) can also contribute to such searches in the near future.
Effectively searching for echoes remains a problem, since they are painful to
search for by eye and initial attempts using machine learning seem not to be
revolutionary.

## Acknowledgements

CSK and KZS are supported by NSF grants AST-1907570, 2307385, and 2407206. The
Shappee group at the University of Hawai’i is supported with funds from NSF
(grants AST-1908952, AST-1911074, & AST-1920392) and NASA (grants HST-
GO-17087, 80NSSC24K0521, 80NSSC24K0490, 80NSSC24K0508, 80NSSC23K0058, &
80NSSC23K1431). We thank Las Cumbres Observatory and its staff for their
continued support of ASAS-SN. ASAS-SN is funded by Gordon and Betty Moore
Foundation grants GBMF5490 and GBMF10501 and the Alfred P. Sloan Foundation
grant G-2021-14192.

Software used: Astropy (Astropy Collaboration et al. 2022), Iraf (Tody, 1986),
Matplotlib (Hunter 2007), NumPy (Harris et al. 2020), SciPy (Virtanen et al.,
2020).

## Data Availability

All data used in this paper are either public or in the tables associated with
the paper.

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Table 2: Light Echoes

SNa | IDb | Filterc | Groupd | RAe | Dece | MJDf | Min MJDg | Max MJDg |  $\theta$h
---|---|---|---|---|---|---|---|---|---
Cas A | 1 | V | 2 | 00:20:04 | +61:58:61 | 57022 | 57008 | 57045 | 7.6508
Cas A | 2 | V | 3 | 00:19:24 | +61:52:61 | 57022 | 57008 | 57045 | 7.5450
Cas A | 3 | V | 4 | 00:19:44 | +61:44:61 | 57022 | 57008 | 57045 | 7.5445
$\ldots$ | $\ldots$ | $\ldots$ | $\ldots$ | $\ldots$ | $\ldots$ | $\ldots$ | $\ldots$ | $\ldots$ | $\ldots$
Tycho | 1 | V | 1 | 00:49:19 | +58:43:04 | 57026 | 57008 | 57045 | 6.126
Tycho | 2 | V | 3 | 01:06:38 | +59:21:11 | 57026 | 57008 | 57045 | 6.825
Tycho | 3 | V | 1 | 00:49:23 | +58:42:49 | 57326 | 57194 | 57413 | 6.134
$\ldots$ | $\ldots$ | $\ldots$ | $\ldots$ | $\ldots$ | $\ldots$ | $\ldots$ | $\ldots$ | $\ldots$ | $\ldots$

This table is available in its entirety in a machine-readable form in the
online journal. A portion is shown here for guidance regarding its form and
content.
a SN or SN remnant associated with the echo.
b Identification number associated with the echo per SN.
c Filter associated with each echo.
d Group number associated with each echo.
e Right ascension and declination for the J2000 epoch.
f Averaged Modified Julian Date of the observation.
g Minimum and maximum Modified Julian Date of observations
h Separation of the echo from the center of the SN in degrees.


# Meshfree method for solving the elliptic Monge-Amp$\grave{\textrm{e}}$re
equation with Dirichlet boundary**footnotemark: *

Zhiyong Liu  Qiuyan Xu∗<EMAIL_ADDRESS>School of Mathematics and
Statistics,
Ningxia University, Yinchuan, China, 750021

###### Abstract

We prove the convergence of meshfree method for solving the elliptic Monge-
Amp$\grave{\textrm{e}}$re equation with Dirichlet boundary on the bounded
domain. $L^{2}$ error is obtained based on the kernel-based trial spaces
generated by the compactly supported radial basis functions. We obtain the
convergence result when the testing discretization is finer than the trial
discretization. The convergence rate depend on the regularity of the solution,
the smoothness of the computing domain, and the approximation of scaled
kernel-based spaces. The presented convergence theory covers a wide range of
kernel-based trial spaces including stationary approximation and non-
stationary approximation. An extension to non-Dirichlet boundary condition is
in a forthcoming paper.

###### keywords:

Monge-Amp$\grave{{e}}$re equation; Meshfree method; Radial basis functions;
Convergence rate

t1t1footnotetext: The research is supported by the Natural Science Foundations
of China (No.12061057,12202219), and the Natural Science Foundations of
Ningxia Province (No.2023AAC05001,2023AAC03003).

##  1 Introduction

We consider the meshfree method for numerically solving the Monge-
Amp$\grave{\textrm{e}}$re equation with Dirichlet boundary condition:

$\displaystyle\det(D^{2}u(\textbf{x}))$ $\displaystyle=$ $\displaystyle
f(\textbf{x}),\quad\textrm{in}\ \Omega\subset\mathbb{R}^{d},$ (1.1)
$\displaystyle u(\textbf{x})$ $\displaystyle=$ $\displaystyle
g(\textbf{x}),\quad\textrm{on}\ \partial\Omega,$ (1.2)

here $\Omega$ is a strictly convex polygonal domain with a smooth boundary
$\partial\Omega$, $f$ is a strictly positive function, and $\det(D^{2}u)$ is
the determinant of the Hessian matrix $D^{2}u$. This equation is usually
augmented by the convexity constraint of $u$, which implies that the equation
is elliptic. The numerical solution of Monge-Amp$\grave{\textrm{e}}$re
equation is an very active topic of research. Many researchers developed kinds
of numerical methods and analyzed their convergence, such as the finite
difference methods [1, 2], finite element methods [3, 4], and spline methods
[5]. In order to design a certain type of numerical method which can avoid the
tedious mesh generation and the domain integration and can be capable of
dealing with any irregular distribution of nodes, some meshfree methods based
radial basis functions have been researched, which is started from 2013 in the
paper [6]. And from then on, a series of meshfree methods have been proposed
for solving the elliptic Monge-Amp$\grave{\textrm{e}}$re equation [7, 8, 9,
10]. Unfortunately, the convergence theory of meshfree method has not been
proven yet. The present paper will prove the convergence of meshfree method
for the Monge-Amp$\grave{\textrm{e}}$re equation. To simplify the proof, we
only consider the case of taking $d=2$:

$\displaystyle Fu:=\det(D^{2}u(x,y))$ $\displaystyle=$ $\displaystyle
f(x,y),\quad\textrm{in}\ \Omega\subset\mathbb{R}^{2},$ (1.3) $\displaystyle
u(x,y)$ $\displaystyle=$ $\displaystyle g(x,y),\quad\textrm{on}\
\partial\Omega.$ (1.4)

We will rely the B$\ddot{\textrm{o}}$hmer/Schaback’s nonlinear discretization
theory [11, 12] and the ideas of the latest paper [13], which discussed the
convergence of unsymmetric radial basis functions method for second order
quasilinear elliptic equations.

The paper is organized as follows. Section 2 is devoted to some notation and
an introduction of kernel-based trial spaces generated by compactly supported
radial basis functions. Section 3 introduces the well-posed theory of the
Monge-Amp$\grave{\textrm{e}}$re equation and its meshfree discretization.
Section 4 is the theoretical analysis of convergence.

##  2 Radial basis functions trial spaces

A translation-invariant kernel such as the strictly positive definite radial
function $\Phi(\textbf{x},\textbf{y})=\Phi(\textbf{x}-\textbf{y})$ will
generate a reproducing kernel Hilbert space, the so-called native space
$\mathcal{N}_{\Phi}(\Omega)$. But we will make use of the native space
$\mathcal{N}_{\Phi}(\mathbb{R}^{d})$ which defined in $\mathbb{R}^{d}$ and
described by Fourier transform. $\mathcal{N}_{\Phi}(\mathbb{R}^{d})$ consists
of all functions $f\in L^{2}(\mathbb{R}^{d})$ with

$\|f\|_{\Phi}^{2}=\int_{\mathbb{R}^{d}}\frac{|\widehat{f}(\boldsymbol{\omega})|^{2}}{\widehat{\Phi}(\boldsymbol{\omega})}d\boldsymbol{\omega}<\infty.$
(2.1)

Hence, if the Fourier transform of $\Phi:\mathbb{R}^{d}\rightarrow\mathbb{R}$
has the algebraic decay condition

$c_{1}(1+\|\boldsymbol{\omega}\|_{2}^{2})^{-\sigma}\leq\widehat{\Phi}(\boldsymbol{\omega})\leq
c_{2}(1+\|\boldsymbol{\omega}\|_{2}^{2})^{-\sigma}$ (2.2)

for two fixed constants $0<c_{1}<c_{2}$, then the native space
$\mathcal{N}_{\Phi}(\mathbb{R}^{d})$ will be norm equivalent to the Sobolev
space $W^{\sigma,2}(\mathbb{R}^{d})$. Thus, with a radial function $\Phi$
satisfied (2.2) and a scaling parameter $\delta\in(0,1]$ we can construct a
scaled radial function

$\Phi_{\delta}:=\delta^{-d}\Phi((\textbf{x}-\textbf{y})/\delta),\quad\textbf{x},\textbf{y}\in\mathbb{R}^{d}.$
(2.3)

Thus we can consider the finite dimensional kernel-based trial space generated
by $\Phi_{\delta}$. Let
$Y^{I}=\\{\textbf{y}_{1},\textbf{y}_{2},\ldots,\textbf{y}_{N_{I}}\\}\subseteq\Omega$
and
$Y^{B}=\\{\textbf{y}_{N_{I}+1},\textbf{y}_{N_{I}+2},\ldots,\textbf{y}_{N}\\}\subseteq\partial\Omega$
be some data sites which are arranged on the domain and its boundary
respectively. Then the whole of centers are $Y=Y^{I}\cup Y^{B}$, which can
used to construct a trial space and determine its degrees of freedom. We
denote $B=B(\textbf{0},1)$ as the unit ball in $\mathbb{R}^{d-1}$ defined by
using a number of $C^{k,\gamma}-$diffeomorphisms $\psi_{j}:B\rightarrow
V_{j},\quad 1\leq j\leq K,$ and $V_{j}\subseteq\mathbb{R}^{d}$ are open sets
such that $\partial\Omega=\bigcup_{j=1}^{K}V_{j}$. Some measures are defined
as follows:

$h_{I}:=\sup_{\textbf{y}\in\Omega}\min_{\textbf{y}_{j}\in
Y^{I}}\|\textbf{y}-\textbf{y}_{j}\|_{2},\
q_{I}:=\frac{1}{2}\min_{\textbf{y}_{i},\textbf{y}_{k}\in Y^{I},i\neq
k}\|\textbf{y}_{i}-\textbf{y}_{k}\|_{2},\quad\textrm{for interior centers}\
Y^{I},$ $h_{B}:=\max_{1\leq j\leq K}h_{Bj}=\max_{1\leq j\leq
K}\left\\{\sup_{\textbf{y}\in
B=B(\textbf{0},1)}\min_{\textbf{y}_{j}\in\psi_{j}^{-1}\left(Y^{B}\bigcap
V_{j}\right)}\|\textbf{y}-\textbf{y}_{j}\|_{2}\right\\},\quad\textrm{for
boundary centers}\ Y^{B},$ $q_{B}:=\min_{1\leq j\leq K}q_{Bj}=\min_{1\leq
j\leq
K}\left\\{\frac{1}{2}\min_{\textbf{y}_{i},\textbf{y}_{k}\in\psi_{j}^{-1}\left(Y^{B}\bigcap
V_{j}\right),i\neq
k}\|\textbf{y}_{i}-\textbf{y}_{k}\|_{2}\right\\},\quad\textrm{for boundary
centers}\ Y^{B},$ $h_{Y}:=\max\\{h_{I},h_{B}\\}<1,\quad
q_{Y}:=\min\\{q_{I},q_{B}\\}.$

Using the scaled radial function $\Phi_{\delta}$ as basis functions, then the
radial basis functions trial space is

$W_{\delta}=\textrm{span}\\{\Phi_{\delta}(\cdot-\textbf{y})|\textbf{y}\in
Y\\}.$ (2.4)

In the later theoretical analysis, we need two important auxiliary results:
inverse inequality and sampling inequality. The following inverse inequality
has been proved for RBF approximation in the reference [13] on the bounded
domains.

######  Lemma 2.1.

(inverse inequality) Let $\Omega\subseteq\mathbb{R}^{d}$ be a bounded domain
satisfying an interior cone condtion. Let $Y$ be quasi-uniform in the sense
that $h_{Y}\leq cq_{Y}$. For all $\delta-$dependent functions of the form
$f=\sum_{j=1}^{N}c_{j}\Phi_{\delta}(\cdot-\textbf{x}_{j}),\ \textbf{x}_{j}\in
Y,$ there exists a constant $C$ such that

$\|f\|_{W^{\sigma,2}(\Omega)}\leq
C\delta^{\sigma}h_{Y}^{-\sigma+d/2}\|f\|_{L^{2}(\Omega)}.$ (2.5)

The well-known sampling inequality has been proved in [14, 15].

######  Lemma 2.2.

(sampling inequality) Suppose $\Omega\subseteq\mathbb{R}^{d}$ is a bounded
domain with an interior cone condition. Let $q\in[1,\infty]$ and constants
$0\leq\mu<\mu+d/2<\lfloor m\rfloor$. Then there are positive constant $C$,
$h_{0}$ such that

$\|f\|_{W^{\mu,q}(\Omega)}\leq
C\left(h_{Y}^{m-\mu-d(1/2-1/q)_{+}}\|f\|_{W^{m,2}(\Omega)}+h_{Y}^{-\mu}\|\Pi_{Y}f\|_{\infty,Y}\right)$
(2.6)

holds for every discrete set $Y\subset\Omega$ with fill distance at most
$h_{Y}\leq h_{0}$ and every $f\in W^{m,2}(\Omega)$. Here $\Pi_{Y}$ is the
function evaluation operator at the data sites $Y$.

##  3 Meshfree discretization

###  3.1 Well-posedness

There are several theories which describe the regularity and the well-
posedness of the Monge-Amp$\grave{\textrm{e}}$re equation [16, 17, 18], for
example the existence theorem from [18]:

######  Theorem 3.1.

(existence) Let $\Omega$ be a uniformly convex domain with
$C^{k+2,\gamma}-$boundary, $k\geq 2,\gamma\in(0,1)$. Let $f\in
C^{k,\gamma}(\bar{\Omega})$ be strictly positive. Then for any $g\in
C^{k+2,\gamma}(\partial\Omega)$, there exist a unique solution $u^{*}\in
C^{k+2,\gamma}(\bar{\Omega})$ to the Dirichlet problem (1.1)-(1.2).

In the later discussion, we always assume that the conditions of Theorem 3.1
are satisfied. And we assume that the solution of the equations (1.3)-(1.4)
has the higher regularity $u^{*}\in W^{\sigma,2}(\Omega)$ with $\sigma$ such
that $\lfloor\sigma\rfloor>k+2+d/2$. Of course, $u^{*}\in
C^{k+2,\gamma}(\overline{\Omega})$ according to Sobolev inequality from
section 5.6.3 of [19]. In addition, $F$ is Fr$\acute{\textrm{e}}$chet
differentiable in $K_{R}(u^{*})=\\{u\in
W^{\sigma,2}(\Omega):\|u-u^{*}\|_{W^{\sigma,2}(\Omega)}\leq R\\}\subset D(F)$,
and

$F^{\prime}(u^{*})(v)=u^{*}_{yy}v_{xx}+u^{*}_{xx}v_{yy}-2u^{*}_{xy}v_{xy}.$
(3.1)

Obviously, the convex property of $u^{*}$ implies that the linear operator
$F^{\prime}(u^{*})$ is uniformly elliptic. Thus, we can obtain a prior bounds
of the fully nonlinear problem (1.3)-(1.4) by utilizing the prior estimate of
the linear elliptic problem

$\displaystyle F^{\prime}(u)v(x,y)$ $\displaystyle=$ $\displaystyle
f(x,y),\quad\textrm{in}\ \Omega\subset\mathbb{R}^{2},$ (3.2) $\displaystyle
v(x,y)$ $\displaystyle=$ $\displaystyle g(x,y),\quad\textrm{on}\
\partial\Omega,$ (3.3)

here $u\in K_{R}(u^{*})$.

######  Theorem 3.2.

(a priori bounds of (1.3)-(1.4)) Suppose $\Omega\subseteq\mathbb{R}^{d},d=2$
is a uniformly convex domain with $C^{k+2,\gamma}-$boundary. Then the
Frech$\acute{\textrm{e}}$t derivative $F^{\prime}(u)$ at $u$ be bounded and
Lipschitz continuous, namely

$\|F^{\prime}(u)-F^{\prime}(v)\|:=\|F^{\prime}(u^{*})u-F^{\prime}(u^{*})v\|_{L^{2}(\Omega)}\leq
C^{\prime\prime}\|u-v\|_{W^{\sigma,2}(\Omega)},\quad\textrm{for all}\ u,v\in
K_{R}(u^{*}).$ (3.4)

Assume the radius $R$ to be small enough to satisfy

$C_{l}C^{\prime\prime}R\leq\frac{1}{2},$ (3.5)

then for any $u,v\in K_{R}(u^{*})$, there exists two constants $C,C_{b}$ such
that

$\|u-v\|_{L^{2}(\Omega)}\leq C\\{\|Fu-
Fv\|_{L^{2}(\Omega)}+\|u-v\|_{W^{1/2,2}(\partial\Omega)}\\},$ (3.6) $\|Fu-
Fv\|_{W^{\sigma-2,2}(\Omega)}\leq C_{b}\|u-v\|_{W^{\sigma,2}(\Omega)}.$ (3.7)

###### Proof.

The inequality (3.4) is obvious because $\|F^{\prime}(u)-F^{\prime}(v)\|$ can
be bounded by

$\displaystyle\|F^{\prime}(u^{*})u-F^{\prime}(u^{*})v\|_{L^{2}(\Omega)}$
$\displaystyle=\|u^{*}_{xx}(u-v)_{yy}+u^{*}_{yy}(u-v)_{xx}-2u^{*}_{xy}(u-v)_{xy}\|_{L^{2}(\Omega)}$
$\displaystyle\leq|u^{*}|_{W^{2,2}(\Omega)}|u-v|_{W^{2,2}(\Omega)}$
$\displaystyle\leq C^{{}^{\prime\prime}}\|u-v\|_{W^{\sigma,2}(\Omega)}.$

With the help of the priori inequality of linear elliptic problems (for
example Corollary 8.7 of [20],), we have

$\|u-v\|_{L^{2}(\Omega)}\leq
C_{l}\\{\|F^{\prime}(u)(u-v)\|_{L^{2}(\Omega)}+\|u-v\|_{W^{1/2,2}(\partial\Omega)}\\}.$

Then by using Taylor formula and Lipschitz continuity of $F^{\prime}(u)$, we
have

$\displaystyle\|u-v\|_{L^{2}(\Omega)}$ $\displaystyle\leq
C_{l}\\{\|F^{\prime}(u)(u-v)-Fu+Fv\|_{L^{2}(\Omega)}+\|Fu-
Fv\|_{L^{2}(\Omega)}+\|u-v\|_{W^{1/2,2}(\partial\Omega)}\\}$
$\displaystyle\leq
C_{l}\\{\sup_{w\in\overline{uv}}\|F^{\prime}(w)-F^{\prime}(v)\|\|u-v\|_{L^{2}(\Omega)}+\|Fu-
Fv\|_{L^{2}(\Omega)}+\|u-v\|_{W^{1/2,2}(\partial\Omega)}\\}$
$\displaystyle\leq
C_{l}\\{C^{\prime\prime}\|u-v\|_{W^{\sigma,2}(\Omega)}\|u-v\|_{L^{2}(\Omega)}+\|Fu-
Fv\|_{L^{2}(\Omega)}+\|u-v\|_{W^{1/2,2}(\partial\Omega)}\\}$
$\displaystyle\leq C_{l}\\{C^{\prime\prime}R\|u-v\|_{L^{2}(\Omega)}+\|Fu-
Fv\|_{L^{2}(\Omega)}+\|u-v\|_{W^{1/2,2}(\partial\Omega)}\\}.$

This yields the result (3.6). In addition,

$\displaystyle\|Fu-Fv\|_{W^{\sigma-2,2}(\Omega)}$ $\displaystyle\leq\|Fv-
Fu-F^{\prime}(u)(v-u)\|_{W^{\sigma-2,2}(\Omega)}+\|F^{\prime}(u)(v-u)\|_{W^{\sigma-2,2}(\Omega)}$
$\displaystyle\leq
C^{\prime\prime}R\|u-v\|_{W^{\sigma,2}(\Omega)}+(C^{\prime\prime}R+\|F^{\prime}(u^{*})\|)\|u-v\|_{W^{\sigma,2}(\Omega)}$
$\displaystyle\leq(2C^{\prime\prime}R+\|F^{\prime}(u^{*})\|)\|u-v\|_{W^{\sigma,2}(\Omega)}.$

Setting $C_{b}=2C^{\prime\prime}R+\|F^{\prime}(u^{*})\|$ yields (3.7). ∎

###  3.2 Collocation

We are now consider the meshfree collocation for the Monge-
Amp$\grave{\textrm{e}}$re equation (1.3)-(1.4). Denote $\textbf{x}=(x,y)$ and
$\textbf{x}_{i}=(x_{i},y_{i})$, we use a set $X=X^{I}\cup X^{B}$ with
$X^{I}=\\{\textbf{x}_{1},\textbf{x}_{2},\ldots,\textbf{x}_{M_{I}}\\}\subseteq\Omega$
and
$X^{B}=\\{\textbf{x}_{M_{I}+1},\textbf{x}_{M_{I}+2},\ldots,\textbf{x}_{M}\\}\subseteq\partial\Omega$
as collocation points, and use

$s_{I}:=\sup_{\textbf{x}\in\Omega}\min_{\textbf{x}_{j}\in
X^{I}}\|\textbf{x}-\textbf{x}_{j}\|_{2},$ $s_{B}:=\max_{1\leq j\leq
K}s_{Bj}=\max_{1\leq j\leq K}\left\\{\sup_{\textbf{x}\in
B=B(\textbf{0},1)}\min_{\textbf{x}_{j}\in\psi_{j}^{-1}\left(X^{B}\bigcap
V_{j}\right)}\|\textbf{x}-\textbf{x}_{j}\|_{2}\right\\}$

to represent the fill distance of the data sites $X^{I}$ and $X^{B}$
respectively. Obviously, they have the parallel definitions with $h_{I}$ and
$h_{B}$ on trial side. Hence choose $s_{X}=\max\\{s_{I},s_{B}\\}<1$. Then we
can construct a approximate function with the form

$s(\textbf{x})=\sum_{j=1}^{N}c_{j}\Phi_{\delta}(\textbf{x}-\textbf{y}_{j}),\quad\textbf{y}_{j}\in
Y=Y^{I}\cup Y^{B}\subset\mathbb{R}^{2},$

which satisfies

$\displaystyle Fs(x_{i},y_{i})$
$\displaystyle=f(x_{i},y_{i}),\quad(x_{i},y_{i})\in X^{I},\quad
i=1,\ldots,M_{I},$ (3.8) $\displaystyle s(x_{i},y_{i})$
$\displaystyle=g(x_{i},y_{i}),\quad(x_{i},y_{i})\in X^{B},\quad
i=M_{I+1},\ldots,M.$ (3.9)

(3.8)-(3.9) is a meshfree discretization of the equation (1.3)-(1.4), but with
a nonsquare nonlinear systems. Let $\Pi_{X^{I}}$ and $\Pi_{X^{B}}$ be the
function evaluation operators at the data sites $X^{I}$ and $X^{B}$
respectively, then the nonlinear systems (3.8)-(3.9) can also be rewritten as

$\displaystyle\Pi_{X^{I}}Fs$ $\displaystyle=\Pi_{X^{I}}f=\Pi_{X^{I}}Fu^{*},$
(3.10) $\displaystyle\Pi_{X^{B}}s$
$\displaystyle=\Pi_{X^{B}}g=\Pi_{X^{B}}u^{*}.$ (3.11)

Because the nonlinear systems (3.10)-(3.11) is overdetermined, thus we only
require that one of the nonlinear numerical methods is clever enough to
produce a $s\in W_{\delta}\cap K_{R}(u^{*})$ with the given tolerance

$\max\left\\{\|\Pi_{X^{I}}(Fu^{*}-Fs)\|_{\infty,X^{I}},\|\Pi_{X^{B}}(u^{*}-s)\|_{\infty,X^{B}}\right\\}\leq\emph{tol}.$
(3.12)

This can be done by residual minimization

$s=\arg\min\left\\{\|\Pi_{X^{I}}(Fu^{*}-Fs)\|_{\infty,X^{I}},\|\Pi_{X^{B}}(u^{*}-s)\|_{\infty,X^{B}}:s\in
W_{\delta}\cap K_{R}(u^{*})\right\\}.$ (3.13)

##  4 Convergence analysis

The following theorem is our main theoretical result which proves the
convergence rate of meshfree method for the Monge-Amp$\grave{\textrm{e}}$re
equation (1.3)-(1.4).

######  Theorem 4.1.

Let $\Omega\subseteq\mathbb{R}^{d},d=2$ be a uniformly convex domain has a
$C^{k+2,\gamma}-$boundary with $\gamma\in[0,1)$. In the equation (1.3)-(1.4),
let $f\in C^{k,\gamma}(\bar{\Omega})$ be strictly positive and $g\in
C^{k+2,\gamma}(\partial\Omega)$. We denote the unique solution of (1.3)-(1.4)
as $u^{*}\in W^{\sigma,2}(\Omega)$, $\lfloor\sigma\rfloor>k+2+d/2$. Assume the
radius $R$ of the ball $K_{R}$ to be small enough to satisfy (3.5). Let $Y$ be
quasi-uniform in the sense that $h_{Y}\leq cq_{Y}$. Let the collocation sites
$X$ also be quasi-uniform. Let $s_{I}$ and $s_{B}$ be the fill distance of the
data sites $X^{I}$ and $X^{B}$ respectively, and
$s_{X}=\max\\{s_{I},s_{B}\\}$. Suppose $s\in W_{\delta}$ is the collocation
solution of (3.8)-(3.9) obtained by choosing

$tol=C\cdot
s^{1/2}_{B}\delta^{-\sigma}h_{Y}^{\sigma-2-d/2}\|u^{*}\|_{W^{\sigma,2}(\Omega)}.$
(4.1)

If the testing discretizations is finer than the trial discretization and
satisfies condition

$C\cdot
C_{b}\cdot\delta^{\sigma}s_{X}^{\sigma-2}h_{Y}^{-\sigma+d/2}<\frac{1}{2},$
(4.2)

then there exists a constant $C>0$ such that

$\|u^{*}-s\|_{L^{2}(\Omega)}\leq
C\delta^{-\sigma}h^{\sigma-3}_{Y}\|u^{*}\|_{W^{\sigma,2}(\Omega)}.$

###### Proof.

We want to bound the $L^{2}-$error between the solution $u^{*}$ and its
approximation $s$. We can use the radial basis functions interpolant as a
transition. The interpolant is a map $I_{h}:u^{*}\rightarrow I_{h}u^{*}\in
W_{\delta}$ defined by

$I_{h}u^{*}(\textbf{y}_{k})=\sum_{j=1}^{N}c_{j}\Phi_{\delta}(\textbf{y}_{k}-\textbf{y}_{j})=u^{*}(\textbf{y}_{k}),\quad\textbf{y}_{k}\in
Y\subset\mathbb{R}^{2},k=1,2,\ldots,N,$

for the unknown coefficients $c_{1},c_{2},\ldots,c_{N}$ are determined by
solving the above linear system. Then the $L^{2}-$error between the solution
$u^{*}$ and its approximation $s$ can be split into

$\displaystyle\|u^{*}-s\|_{L^{2}(\Omega)}$
$\displaystyle\leq\|u^{*}-I_{h}u^{*}\|_{L^{2}(\Omega)}+\|I_{h}u^{*}-s\|_{L^{2}(\Omega)}.$

The first term on the right-hand side is the radial basis functions
interpolation error, which can be easily obtained by using the sampling
inequality (Lemma 2.2) and the stability of $I_{h}$ (Lemma 5.1 of [13]),
namely

$\displaystyle\|u^{*}-I_{h}u^{*}\|_{L^{2}(\Omega)}\leq
Ch_{Y}^{\sigma}\|u^{*}-I_{h}u^{*}\|_{W^{\sigma,2}(\Omega)}\leq
C\delta^{-\sigma}h_{Y}^{\sigma}\|u^{*}\|_{W^{\sigma,2}(\Omega)}.$ (4.3)

Hence, our main task in this proof is to bound the second term on the right-
hand side.

Choosing $u=I_{h}u^{*},v=s$ and squaring both sides of (3.6), we have

$\|I_{h}u^{*}-s\|_{L^{2}(\Omega)}^{2}\leq
2C\\{\|F(I_{h}u^{*})-Fs\|_{L^{2}(\Omega)}^{2}+\|I_{h}u^{*}-s\|_{W^{1/2,2}(\partial\Omega)}^{2}\\}.$

Then we can bound the two terms on the right-hand side separately. For the
first term, using Lemma 2.2, the inequality (3.7), Lemma 2.1, and the
stability of radial basis functions interpolation (Lemma 5.1 of [13]) yields

$\displaystyle\|F(I_{h}u^{*})-Fs\|_{L^{2}(\Omega)}^{2}$ $\displaystyle\leq
C\left\\{s_{I}^{2\sigma-4}\|F(I_{h}u^{*})-Fs\|^{2}_{W^{\sigma-2,2}(\Omega)}+\|\Pi_{X^{I}}(F(I_{h}u^{*})-Fs)\|^{2}_{\infty,X^{I}}\right\\}$
$\displaystyle\leq C\cdot C_{b}^{2}\cdot
s_{I}^{2\sigma-4}\|I_{h}u^{*}-s\|^{2}_{W^{\sigma,2}(\Omega)}+C\|\Pi_{X^{I}}(F(I_{h}u^{*})-Fu^{*})\|^{2}_{\infty,X^{I}}+C\cdot
tol^{2}$ $\displaystyle=C\cdot C_{b}^{2}\cdot
s_{I}^{2\sigma-4}\|I_{h}u^{*}-s\|^{2}_{W^{\sigma,2}(\Omega)}+C\|F(I_{h}u^{*})-Fu^{*}\|^{2}_{\infty,X^{I}}+C\cdot
tol^{2}$ $\displaystyle\leq C\cdot
C_{b}^{2}\cdot\delta^{2\sigma}s_{I}^{2\sigma-4}h_{Y}^{-2\sigma+d}\|I_{h}u^{*}-s\|^{2}_{L^{2}(\Omega)}+C\|F(I_{h}u^{*})-Fu^{*}\|^{2}_{\infty,\Omega}+C\cdot
tol^{2}$ $\displaystyle\leq C\cdot
C_{b}^{2}\cdot\delta^{2\sigma}s_{X}^{2\sigma-4}h_{Y}^{-2\sigma+d}\|I_{h}u^{*}-s\|^{2}_{L^{2}(\Omega)}+C\|I_{h}u^{*}-u^{*}\|^{2}_{W^{2,\infty}(\Omega)}+C\cdot
tol^{2}$ $\displaystyle\leq C\cdot
C_{b}^{2}\cdot\delta^{2\sigma}s_{X}^{2\sigma-4}h_{Y}^{-2\sigma+d}\|I_{h}u^{*}-s\|^{2}_{L^{2}(\Omega)}+Ch_{I}^{2(\sigma-2-d/2)}\|I_{h}u^{*}-u^{*}\|^{2}_{W^{\sigma,2}(\Omega)}+C\cdot
tol^{2}$ $\displaystyle\leq C\cdot
C_{b}^{2}\cdot\delta^{2\sigma}s_{X}^{2\sigma-4}h_{Y}^{-2\sigma+d}\|I_{h}u^{*}-s\|^{2}_{L^{2}(\Omega)}+C\delta^{-2\sigma}h_{I}^{2(\sigma-2-d/2)}\|u^{*}\|^{2}_{W^{\sigma,2}(\Omega)}+C\cdot
tol^{2}$ $\displaystyle\leq C\cdot
C_{b}^{2}\cdot\delta^{2\sigma}s_{X}^{2\sigma-4}h_{Y}^{-2\sigma+d}\|I_{h}u^{*}-s\|^{2}_{L^{2}(\Omega)}+C\delta^{-2\sigma}h_{Y}^{2(\sigma-2-d/2)}\|u^{*}\|^{2}_{W^{\sigma,2}(\Omega)}+C\cdot
tol^{2}$ $\displaystyle\leq C\cdot
C_{b}^{2}\cdot\delta^{2\sigma}s_{X}^{2\sigma-4}h_{Y}^{-2\sigma+d}\|I_{h}u^{*}-s\|^{2}_{L^{2}(\Omega)}+C\delta^{-2\sigma}h_{Y}^{2(\sigma-2-d/2)}\|u^{*}\|^{2}_{W^{\sigma,2}(\Omega)}.$

For the boundary term, we denote
$w_{j}=((I_{h}u^{*}-s)\omega_{j})\circ\Psi_{j}\in W^{\sigma-1/2,2}(B)$. From
Lemma 2.2, the trace theorem (Theorem 8.7 of [21]), the Sobolev imbedding
theorems (Theorem 4.12 of [22]), the quasi-uniform property of $X$, and the
quasi-uniform property of $Y$,we can bound

$\displaystyle\quad\|I_{h}u^{*}-s\|^{2}_{W^{1/2,2}(\partial\Omega)}=\sum_{j=1}^{K}\|w_{j}\|^{2}_{W^{1/2,2}(B)}$
$\displaystyle\leq
C\sum_{j=1}^{K}\left(s_{Bj}^{\sigma-1}\|w_{j}\|_{W^{\sigma-1/2,2}(B)}+s_{Bj}^{-1/2}\|\Pi_{\psi_{j}^{-1}\left(X^{B}\bigcap
V_{j}\right)}w_{j}\|_{\infty,\psi_{j}^{-1}\left(X^{B}\bigcap
V_{j}\right)}\right)^{2}$ $\displaystyle\leq
C\sum_{j=1}^{K}\left(s_{Bj}^{2\sigma-2}\|w_{j}\|^{2}_{W^{\sigma-1/2,2}(B)}+s_{Bj}^{-1}\|\Pi_{\psi_{j}^{-1}\left(X^{B}\bigcap
V_{j}\right)}w_{j}\|^{2}_{\infty,\psi_{j}^{-1}\left(X^{B}\bigcap
V_{j}\right)}\right)$ $\displaystyle\leq
Cs_{B}^{2\sigma-2}\|I_{h}u^{*}-s\|^{2}_{W^{\sigma-1/2,2}(\partial\Omega)}+Cs_{B}^{-1}\|\Pi_{X^{B}}(I_{h}u^{*}-s)\|^{2}_{\infty,X^{B}}$
$\displaystyle\leq
Cs_{B}^{2\sigma-2}\|I_{h}u^{*}-s\|^{2}_{W^{\sigma,2}(\Omega)}+Cs_{B}^{-1}\|\Pi_{X^{B}}(I_{h}u^{*}-u^{*})\|^{2}_{\infty,X^{B}}+Cs_{B}^{-1}\cdot
tol^{2}$ $\displaystyle\leq
C\delta^{2\sigma}s_{X}^{2\sigma-2}h_{Y}^{-2\sigma+d}\|I_{h}u^{*}-s\|^{2}_{L^{2}(\Omega)}+Cs_{B}^{-1}\|I_{h}u^{*}-u^{*}\|^{2}_{\infty,X^{B}}+Cs_{B}^{-1}\cdot
tol^{2}$ $\displaystyle\leq
C\delta^{2\sigma}s_{X}^{2\sigma-2}h_{Y}^{-2\sigma+d}\|I_{h}u^{*}-s\|^{2}_{L^{2}(\Omega)}+Cs_{B}^{-1}\max_{1\leq
j\leq K}\|w_{j}\|^{2}_{L^{\infty}(B)}+Cs_{B}^{-1}\cdot tol^{2}$
$\displaystyle\leq
C\delta^{2\sigma}s_{X}^{2\sigma-2}h_{Y}^{-2\sigma+d}\|I_{h}u^{*}-s\|^{2}_{L^{2}(\Omega)}+Cs_{B}^{-1}\max_{1\leq
j\leq
K}h_{Bj}^{2(\sigma-1/2-d/2)}\|w_{j}\|^{2}_{W^{\sigma-1/2,2}(B)}+Cs_{B}^{-1}\cdot
tol^{2}$ $\displaystyle\leq
C\delta^{2\sigma}s_{X}^{2\sigma-2}h_{Y}^{-2\sigma+d}\|I_{h}u^{*}-s\|^{2}_{L^{2}(\Omega)}+Cs_{B}^{-1}h_{B}^{2(\sigma-1/2-d/2)}\sum_{j=1}^{K}\|w_{j}\|^{2}_{W^{\sigma-1/2,2}(B)}+Cs_{B}^{-1}\cdot
tol^{2}$
$\displaystyle=C\delta^{2\sigma}s_{X}^{2\sigma-2}h_{Y}^{-2\sigma+d}\|I_{h}u^{*}-s\|^{2}_{L^{2}(\Omega)}+Cs_{B}^{-1}h_{B}^{2(\sigma-1/2-d/2)}\|I_{h}u^{*}-u^{*}\|^{2}_{W^{\sigma-1/2,2}(\partial\Omega)}+Cs_{B}^{-1}\cdot
tol^{2}$ $\displaystyle\leq
C\delta^{2\sigma}s_{X}^{2\sigma-2}h_{Y}^{-2\sigma+d}\|I_{h}u^{*}-s\|^{2}_{L^{2}(\Omega)}+Cs_{B}^{-1}h_{B}^{2(\sigma-1/2-d/2)}\|I_{h}u^{*}-u^{*}\|^{2}_{W^{\sigma,2}(\Omega)}+Cs_{B}^{-1}\cdot
tol^{2}$ $\displaystyle\leq
C\delta^{2\sigma}s_{X}^{2\sigma-2}h_{Y}^{-2\sigma+d}\|I_{h}u^{*}-s\|^{2}_{L^{2}(\Omega)}+C\delta^{-2\sigma}h_{Y}^{-(\sigma-d/2)/(\sigma-2)}h_{B}^{2(\sigma-1/2-d/2)}\|u^{*}\|^{2}_{W^{\sigma,2}(\Omega)}+Cs_{B}^{-1}\cdot
tol^{2}$ $\displaystyle\leq
C\delta^{2\sigma}s_{X}^{2\sigma-2}h_{Y}^{-2\sigma+d}\|I_{h}u^{*}-s\|^{2}_{L^{2}(\Omega)}+C\delta^{-2\sigma}h_{Y}^{2(\sigma-2-d/2)}\|u^{*}\|^{2}_{W^{\sigma,2}(\Omega)}.$

By using the condition (4.2), combining both above results with the
interpolation error (4.3) and setting $d=2$, we obtain the convergence rate. ∎

##  5 Conclusions

Although the meshfree method has been used for solving the Monge-
Amp$\grave{\textrm{e}}$re equation for a long time, we proved the convergence
of this method until now in this paper. The theoretical convergence was
established by using the regularity theory of the solution, the inverse
inequality, the sampling inequality, and the approximation property of the
radial basis functions trial spaces. However, the convergence rate is
currently not yet optimal and deserve refinement.

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