Search arXiv⌕ Search

arXiv subjects

Ildoo Kim

Publications and source records attributed to Ildoo Kim.

At least 19 recordsLinked to original sources
arXiv2026

Well-Posedness for Cauchy Problems with Singular Time-Measurable Pseudo-Differential Operators in Quasi-decreasing Weighted $\mathrm{L}_2$-Spaces

This study examines Cauchy problems governed by highly singular, time-measurable pseudo-differential operators (singular measurable families of Fourier multipliers). We show that the symbols of these operators can exhibit arbitrary blow-up behavior. In particular, we prove the existence and uniqueness of solutions even when the symbols grow super-exponentially in time and frequency. As a concrete application, we solve evolutionary equations driven by fractional Laplacians of any negative order. Additionally, we establish unique strong solutions under the sole condition that the symbol is locally integrable in frequency, even in the presence of severe blow-up at the initial time.

math.AP↗
arXiv2026

The effect of mean flow speed in soap film channels

The soap film channel has been developed as a model system for two-dimensional hydrodynamics, but its general applicability has been questioned because the mean flow speed is known to alter the experimental outcome considerably, for reasons that have not been examined. In this study, we experimentally investigate how the geometry of a vortex street in a flowing soap film varies with the mean flow speed, using both an inclined and a vertical channel. We quantify the geometry by the Kármán ratio $q$, the ratio of the transverse to the longitudinal spacing between vortices, and find that the variation of $q$ is accounted for by none of the object size, flow speed $u$, or film thickness $δ$ alone. However, the data from both channels collapse onto a single curve when plotted against $u\sqrtδ$, which is proportional to a Mach-like number $M\equiv u/v_m$, where $v_m$ is the Marangoni elastic wave speed. The measured collapse follows $q/q_0=1-M^2$, an $\mathcal{O}(M^2)$ correction characteristic of weakly compressible flow. The breakdown of Reynolds similitude in soap film channels therefore originates from the two-dimensional compressibility of the film.

physics.flu-dyn↗
arXiv2025

A regularity theory for an initial value problem with a time-measurable pseudo-differential operator in a weighted $L_p$-space

In this study, we investigate the existence, uniqueness, and maximal regularity estimates of solutions to homogeneous initial value problems involving time-measurable pseudo-differential operators within the framework of weighted mixed norm Lebesgue spaces. The class of temporal weights in our regularity estimates contains Muckenhoupt's class, and the initial data is in weighted Besov spaces with variable order.

math.AP↗
arXiv2025

An existence and uniqueness result to evolution equations with sign-changing pseudo-differential operators and its applications to logarithmic Laplacian operators and second-order differential operators without ellipticity

We broaden the domain of the Fourier transform to contain all distributions without using the Paley-Wiener theorem and devise a new weak formulation built upon this extension. This formulation is applicable to evolution equations involving pseudo-differential operators, even when the signs of their symbols may vary over time. Notably, our main operator includes the logarithmic Laplacian operator $\log (-Δ)$ and a second-order differential operator whose leading coefficients are not positive semi-definite.

math.AP↗
arXiv2025

Size distribution of decaying foam bubbles

The most studies on the stability of foam bubbles investigated the mechanical stability of thin films between bubbles due to the drainage by gravity. In the current work, we take an alternative approach by assuming the rupture of bubbles as a series of random events and by investigating the time evolution of the size distribution of foam bubbles over a long time up to several hours. For this purpose, we first prepared layers of bubbles on Petri dishes by shaking soap solutions of a few different concentrations, and then we monitored the Petri dishes by using a time-lapse video imaging technique. We analyzed the captured images by custom software to count the bubble size distribution with respect to the initial concentration and elapsed time. From the statistics on our data, we find that the total bubble volume decreases exponentially in time, and the exponent, i.e. the mean lifetime, is a function of the bubble size. The mean lifetimes of larger bubbles are observed to be shorter than those of smaller bubbles, by approximately a factor of 2.

physics.flu-dyn↗
arXiv2024

An existence and uniqueness theory to stochastic partial differential equations involving pseudo-differential operators driven by space-time white noise

In this paper, we aim to develop a new weak formulation that ensures well-posedness for a broad range of stochastic partial differential equations with pseudo-differential operators whose symbols depend only on time and spatial frequencies. The main focus of this paper is to relax the conditions on the symbols of pseudo-differential operators and data while still ensuring that the stochastic partial differential equations remain well-posed in a weak sense. Specifically, we allow symbols to be random and remove all regularity and ellipticity conditions on them. As a result, our main operators include many interesting rough operators that cannot generate any regularity gain or integrability improvement from the equations. In addition, our data do not need to be regular or possess finite stochastic moments.

math.PR↗
arXiv2023

A weighted $L_p$-regularity theory for parabolic partial differential equations with time measurable pseudo-differential operators

We obtain the existence, uniqueness, and regularity estimates of the following Cauchy problem \begin{equation}\label{ab eqn} \begin{cases} \partial_t u(t,x)=ψ(t,-i\nabla)u(t,x)+f(t,x),\quad &(t,x)\in(0,T)\times\mathbb{R}^d,\\ u(0,x)=0,\quad & x\in\mathbb{R}^d \end{cases} \end{equation} in (Muckenhoupt) weighted $L_p$-spaces with time-measurable pseudo-differential operators \begin{equation} \label{ab op} ψ(t,-i\nabla)u(t,x):=\mathcal{F}^{-1}\left[ψ(t,\cdot)\mathcal{F}[u](t,\cdot)\right](x). \end{equation} More precisely, we find sufficient conditions of the symbol $ψ(t,ξ)$ (especially depending on the smoothness of the symbol with respect to $ξ$) to guarantee that equation is well-posed in (Muckenhoupt) weighted $L_p$-spaces. Here the symbol $ψ(t,ξ)$ is merely measurable with respect to $t$, and the sufficient smoothness of $ψ(t,ξ)$ with respect to $ξ$ is characterized by a property of each weight. In particular, we prove the existence of a positive constant $N$ such that for any solution $u$ to the equation, \begin{equation} \label{ab est} \int_0^T \int_{\mathbb{R}^d} |(-Δ)^{γ/2} u(t,x) |^p (t^2 + |x|^2)^{α/2} \mathrm{d}x\mathrm{d}t \leq N\int_0^T \int_{\mathbb{R}^d} |f(t,x)|^p (t^2 + |x|^2)^{α/2} \mathrm{d}x\mathrm{d}t \end{equation} and \begin{equation} \label{ab est 2} \int_0^T \left(\int_{\mathbb{R}^d} |(-Δ)^{γ/2} u(t,x) |^p |x|^{α_2} \mathrm{d}x \right)^{q/p} t^{α_1}\mathrm{d}t \leq N\int_0^T \left(\int_{\mathbb{R}^d} |f(t,x) |^p |x|^{α_2} \mathrm{d}x \right)^{q/p} t^{α_1}\mathrm{d}t, \end{equation} where $p,q\in(1,\infty)$, $-d-1<α< (d+1)(p-1)$, $-1 < α_1 < q-1$, $-d <α_2< d(p-1)$, and $γ$ is the order of the operator $ψ(t,-i\nabla)$.

math.AP↗
arXiv2023

The effect of menisci on vortex streets on soap film flows

In soap film experiments, the insertion of an external object is necessary to produce vorticity. However, this insertion causes local thickness changes, or simply {\it meniscus}, near the object. Because the meniscus formation may alter the flow near the object, the characterization of meniscus is of considerable importance for the accurate interpretation of data. In this study, we insert cylindrical cones made of aluminum, titanium, and glass to measure the size of the menisci by using a long-range microscope. In all material tested, we find that the size of meniscus is less than 0.2 mm, much shorter than the capillary length. In addition, by comparing the formation of vortex streets behind objects of different materials, we conclude that the meniscus acts as an added length to the size of the object itself. This added length effect can be non-negligible if the size of object is comparable to the size of meniscus.

physics.flu-dyn↗
arXiv2023

A maximal $L_p$-regularity theory to initial value problems with time measurable nonlocal operators generated by additive processes

Let $Z=(Z_t)_{t\geq0}$ be an additive process with a bounded triplet $(0,0,Λ_t)_{t\geq0}$. Suppose that for any Schwartz function $φ$ on $\mathbb{R}^d$ whose Fourier transform is in $C_c^{\infty}(B_{c_s} \setminus B_{c_s^{-1}} )$, there exist positive constants $N_0$, $N_1$, and $N_2$ such that \begin{equation*} \int_{\mathbb{R}^d}|\mathbb{E}[φ(x+r^{-1}Z_t)]|dx\leq N_0 e^{- \frac{N_1 t}{s(r)}},\quad \forall (r,t)\in(0,1)\times[0,T], \end{equation*} and $$ \|ψ^μ(r^{-1}D)φ\|_{L_1(\mathbb{R}^d)}\leq \frac{N_2}{s(r)},\quad \forall r\in(0,1). $$ where $s$ is a scaling function (Definition 2.4), $c_s$ is a positive constant related to $s$, $μ$ is a symmetric Lévy measure on $\mathbb{R}^d$, $ψ^μ(r^{-1}D)φ(x)= \mathcal{F}^{-1} \left[ ψ^μ(r^{-1}ξ) \mathcal{F}[φ]\right](x)$ and $$ψ^μ(ξ):=\int_{\mathbb{R}^d}(e^{iy\cdotξ}-1-iy\cdotξ1_{|y|\leq 1})μ(dy).$$ In this paper, we establish the $L_p$-solvability to the initial value problem \begin{equation} \frac{\partial u}{\partial t}(t,x)=\mathcal{A}_Z(t)u(t,x),\quad u(0,\cdot)=u_0,\quad (t,x)\in(0,T)\times\mathbb{R}^d, \end{equation} In other words, there exists a unique solution $u$ to equation satisfying $$ \|u\|_{L_q((0,T);H_p^{μ;γ}(\mathbb{R}^d))}\leq N\|u_0\|_{B_{p,q}^{s;γ-\frac{2}{q}}(\mathbb{R}^d)}, $$ where $N$ is independent of $u$ and $u_0$, and the spaces $B_{p,q}^{s;γ-\frac{2}{q}}(\mathbb{R}^d)$ and $H_p^{μ;γ}(\mathbb{R}^d)$ are scaled Besov spaces (see Definition 2.8) and generalized Bessel potential spaces (see Definition 2.3), respectively.

math.PR↗
arXiv2023

A weighted $L_q(L_p)$-theory for fully degenerate second-order evolution equations with unbounded time-measurable coefficients

We study the fully degenerate second-order evolution equation $u_t=a^{ij}(t)u_{x^ix^j} +b^i(t) u_{x^i} + c(t)u+f, \quad t>0, x\in \mathbb{R}^d$ given with the zero initial data. Here $a^{ij}(t)$, $b^i(t)$, $c(t)$ are merely locally integrable functions, and $(a^{ij}(t))_{d \times d}$ is a nonnegative symmetric matrix with the smallest eigenvalue $δ(t)\geq 0$. We show that there is a positive constant $N$ such that $\int_0^{T} \left(\int_{\mathbb{R}^d} \left(|u|+|u_{xx} |\right)^{p} dx \right)^{q/p} e^{-q\int_0^t c(s)ds} w(α(t)) δ(t) dt \leq N \int_0^{T} \left(\int_{\mathbb{R}^d} \left|f\left(t,x\right)\right|^{p} dx \right)^{q/p} e^{-q\int_0^t c(s)ds} w(α(t)) (δ(t))^{1-q} dt,$ where $p,q \in (1,\infty)$, $α(t)=\int_0^t δ(s)ds$, and $w$ is a Muckenhoupt's weight.

math.AP↗
arXiv2022

The Liver Tumor Segmentation Benchmark (LiTS)

In this work, we report the set-up and results of the Liver Tumor Segmentation Benchmark (LiTS), which was organized in conjunction with the IEEE International Symposium on Biomedical Imaging (ISBI) 2017 and the International Conferences on Medical Image Computing and Computer-Assisted Intervention (MICCAI) 2017 and 2018. The image dataset is diverse and contains primary and secondary tumors with varied sizes and appearances with various lesion-to-background levels (hyper-/hypo-dense), created in collaboration with seven hospitals and research institutions. Seventy-five submitted liver and liver tumor segmentation algorithms were trained on a set of 131 computed tomography (CT) volumes and were tested on 70 unseen test images acquired from different patients. We found that not a single algorithm performed best for both liver and liver tumors in the three events. The best liver segmentation algorithm achieved a Dice score of 0.963, whereas, for tumor segmentation, the best algorithms achieved Dices scores of 0.674 (ISBI 2017), 0.702 (MICCAI 2017), and 0.739 (MICCAI 2018). Retrospectively, we performed additional analysis on liver tumor detection and revealed that not all top-performing segmentation algorithms worked well for tumor detection. The best liver tumor detection method achieved a lesion-wise recall of 0.458 (ISBI 2017), 0.515 (MICCAI 2017), and 0.554 (MICCAI 2018), indicating the need for further research. LiTS remains an active benchmark and resource for research, e.g., contributing the liver-related segmentation tasks in \url{http://medicaldecathlon.com/}. In addition, both data and online evaluation are accessible via \url{www.lits-challenge.com}.

cs.CV↗
arXiv2022

Contrastive Regularization for Semi-Supervised Learning

Consistency regularization on label predictions becomes a fundamental technique in semi-supervised learning, but it still requires a large number of training iterations for high performance. In this study, we analyze that the consistency regularization restricts the propagation of labeling information due to the exclusion of samples with unconfident pseudo-labels in the model updates. Then, we propose contrastive regularization to improve both efficiency and accuracy of the consistency regularization by well-clustered features of unlabeled data. In specific, after strongly augmented samples are assigned to clusters by their pseudo-labels, our contrastive regularization updates the model so that the features with confident pseudo-labels aggregate the features in the same cluster, while pushing away features in different clusters. As a result, the information of confident pseudo-labels can be effectively propagated into more unlabeled samples during training by the well-clustered features. On benchmarks of semi-supervised learning tasks, our contrastive regularization improves the previous consistency-based methods and achieves state-of-the-art results, especially with fewer training iterations. Our method also shows robust performance on open-set semi-supervised learning where unlabeled data includes out-of-distribution samples.

cs.LG↗
arXiv2021

ViLT: Vision-and-Language Transformer Without Convolution or Region Supervision

Vision-and-Language Pre-training (VLP) has improved performance on various joint vision-and-language downstream tasks. Current approaches to VLP heavily rely on image feature extraction processes, most of which involve region supervision (e.g., object detection) and the convolutional architecture (e.g., ResNet). Although disregarded in the literature, we find it problematic in terms of both (1) efficiency/speed, that simply extracting input features requires much more computation than the multimodal interaction steps; and (2) expressive power, as it is upper bounded to the expressive power of the visual embedder and its predefined visual vocabulary. In this paper, we present a minimal VLP model, Vision-and-Language Transformer (ViLT), monolithic in the sense that the processing of visual inputs is drastically simplified to just the same convolution-free manner that we process textual inputs. We show that ViLT is up to tens of times faster than previous VLP models, yet with competitive or better downstream task performance. Our code and pre-trained weights are available at https://github.com/dandelin/vilt.

stat.ML↗
arXiv2021

The Medical Segmentation Decathlon

International challenges have become the de facto standard for comparative assessment of image analysis algorithms given a specific task. Segmentation is so far the most widely investigated medical image processing task, but the various segmentation challenges have typically been organized in isolation, such that algorithm development was driven by the need to tackle a single specific clinical problem. We hypothesized that a method capable of performing well on multiple tasks will generalize well to a previously unseen task and potentially outperform a custom-designed solution. To investigate the hypothesis, we organized the Medical Segmentation Decathlon (MSD) - a biomedical image analysis challenge, in which algorithms compete in a multitude of both tasks and modalities. The underlying data set was designed to explore the axis of difficulties typically encountered when dealing with medical images, such as small data sets, unbalanced labels, multi-site data and small objects. The MSD challenge confirmed that algorithms with a consistent good performance on a set of tasks preserved their good average performance on a different set of previously unseen tasks. Moreover, by monitoring the MSD winner for two years, we found that this algorithm continued generalizing well to a wide range of other clinical problems, further confirming our hypothesis. Three main conclusions can be drawn from this study: (1) state-of-the-art image segmentation algorithms are mature, accurate, and generalize well when retrained on unseen tasks; (2) consistent algorithmic performance across multiple tasks is a strong surrogate of algorithmic generalizability; (3) the training of accurate AI segmentation models is now commoditized to non AI experts.

eess.IV↗
arXiv2021

Spatially Consistent Representation Learning

Self-supervised learning has been widely used to obtain transferrable representations from unlabeled images. Especially, recent contrastive learning methods have shown impressive performances on downstream image classification tasks. While these contrastive methods mainly focus on generating invariant global representations at the image-level under semantic-preserving transformations, they are prone to overlook spatial consistency of local representations and therefore have a limitation in pretraining for localization tasks such as object detection and instance segmentation. Moreover, aggressively cropped views used in existing contrastive methods can minimize representation distances between the semantically different regions of a single image. In this paper, we propose a spatially consistent representation learning algorithm (SCRL) for multi-object and location-specific tasks. In particular, we devise a novel self-supervised objective that tries to produce coherent spatial representations of a randomly cropped local region according to geometric translations and zooming operations. On various downstream localization tasks with benchmark datasets, the proposed SCRL shows significant performance improvements over the image-level supervised pretraining as well as the state-of-the-art self-supervised learning methods. Code is available at https://github.com/kakaobrain/scrl

cs.CV↗
arXiv2021

On the morphology of two-dimensional laminar vortex streets behind triangles

The two-dimensional laminar vortex streets behind a triangle have two morphologically distinct structures depending on the Reynolds number and the aspect ratio of the triangle. These two structures are the conventional structure and the separated rows structure, where the latter is characterized by a thin layer of irrotational fluid between two vortex rows. In this paper, by means of numerical simulation, we find that the separated rows structure occurs when the thickness of boundary layers is less than 25% of their separation distance. We also show from the linear stability analysis that the criterion is related to the coupling of two boundary layers in producing unstable modes.

physics.flu-dyn↗
arXiv2020

An $L_p$-maximal regularity estimate of moments of solutions to second-order stochastic partial differential equations

We obtain uniqueness and existence of a solution $u$ to the following second-order stochastic partial differential equation (SPDE) : \begin{align} \label{abs eqn} du= \left( \bar a^{ij}(ω,t)u_{x^ix^j}+ f \right)dt + g^k dw^k_t, \quad t \in (0,T); \quad u(0,\cdot)=0, \end{align} where $T \in (0,\infty)$, $w^k$ $(k=1,2,\ldots)$ are independent Wiener processes, $(\bar a^{ij}(ω,t))$ is a (predictable) nonnegative symmetric matrix valued stochastic process such that $$ κ|ξ|^2 \leq \bar a^{ij}(ω,t) ξ^i ξ^j \leq K |ξ|^2 \qquad \forall (ω,t,ξ) \in Ω\times (0,T) \times {\mathbf{R}}^d $$ for some $κ, K \in (0,\infty)$, $$ f \in L_p\left( (0,T) \times {\mathbf{R}}^d, dt \times dx ; L_r(Ω, {\mathscr{F}} ,dP) \right), $$ and $$ g, g_x \in L_p\left( (0,T) \times {\mathbf{R}}^d, dt \times dx ; L_r(Ω, {\mathscr{F}} ,dP; l_2) \right) $$ with $2 \leq r \leq p < \infty$ and appropriate measurable conditions.

math.PR↗