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Ildoo Kim

Publications and source records attributed to Ildoo Kim.

At least 37 records · Page 2Linked to original sources
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↗
arXiv2020

Learning Loss for Test-Time Augmentation

Data augmentation has been actively studied for robust neural networks. Most of the recent data augmentation methods focus on augmenting datasets during the training phase. At the testing phase, simple transformations are still widely used for test-time augmentation. This paper proposes a novel instance-level test-time augmentation that efficiently selects suitable transformations for a test input. Our proposed method involves an auxiliary module to predict the loss of each possible transformation given the input. Then, the transformations having lower predicted losses are applied to the input. The network obtains the results by averaging the prediction results of augmented inputs. Experimental results on several image classification benchmarks show that the proposed instance-aware test-time augmentation improves the model's robustness against various corruptions.

cs.CV↗
arXiv2020

AutoCLINT: The Winning Method in AutoCV Challenge 2019

NeurIPS 2019 AutoDL challenge is a series of six automated machine learning competitions. Particularly, AutoCV challenges mainly focused on classification tasks on visual domain. In this paper, we introduce the winning method in the competition, AutoCLINT. The proposed method implements an autonomous training strategy, including efficient code optimization, and applies an automated data augmentation to achieve the fast adaptation of pretrained networks. We implement a light version of Fast AutoAugment to search for data augmentation policies efficiently for the arbitrarily given image domains. We also empirically analyze the components of the proposed method and provide ablation studies focusing on AutoCV datasets.

cs.LG↗
arXiv2020

torchgpipe: On-the-fly Pipeline Parallelism for Training Giant Models

We design and implement a ready-to-use library in PyTorch for performing micro-batch pipeline parallelism with checkpointing proposed by GPipe (Huang et al., 2019). In particular, we develop a set of design components to enable pipeline-parallel gradient computation in PyTorch's define-by-run and eager execution environment. We show that each component is necessary to fully benefit from pipeline parallelism in such environment, and demonstrate the efficiency of the library by applying it to various network architectures including AmoebaNet-D and U-Net. Our library is available at https://github.com/kakaobrain/torchgpipe .

cs.DC↗
arXiv2020

Spatially Attentive Output Layer for Image Classification

Most convolutional neural networks (CNNs) for image classification use a global average pooling (GAP) followed by a fully-connected (FC) layer for output logits. However, this spatial aggregation procedure inherently restricts the utilization of location-specific information at the output layer, although this spatial information can be beneficial for classification. In this paper, we propose a novel spatial output layer on top of the existing convolutional feature maps to explicitly exploit the location-specific output information. In specific, given the spatial feature maps, we replace the previous GAP-FC layer with a spatially attentive output layer (SAOL) by employing a attention mask on spatial logits. The proposed location-specific attention selectively aggregates spatial logits within a target region, which leads to not only the performance improvement but also spatially interpretable outputs. Moreover, the proposed SAOL also permits to fully exploit location-specific self-supervision as well as self-distillation to enhance the generalization ability during training. The proposed SAOL with self-supervision and self-distillation can be easily plugged into existing CNNs. Experimental results on various classification tasks with representative architectures show consistent performance improvements by SAOL at almost the same computational cost.

cs.CV↗
arXiv2019

Approximate analytic solution of the potential flow around a rectangle

In undergraduate classes, the potential flow that goes around a circular cylinder is designed for complemental understanding of mathematical technique to handle the Laplace equation with Neumann boundary conditions and the physical concept of the multipolar expansion. The simplicity of the standard problem is suited for the introductory level, however, it has a drawback. The discussion of higher order multipoles is often missed because the exact analytic solution contains only the dipole term. In this article, we present a modified problem of the potential flow around a rectangle as an advanced problem. Although the exact solution of this case is intractable, the approximate solution can be obtained by the discretization and the optimization using multiple linear regression. The suggested problem is expected to deepen the students' insight on the concept of multipoles and also provides an opportunity to discuss the formalism of the regression analysis, which in many physics curricula is lacking even though it has a significant importance in experimental physics.

physics.flu-dyn↗
arXiv2019

Scalable Neural Architecture Search for 3D Medical Image Segmentation

In this paper, a neural architecture search (NAS) framework is proposed for 3D medical image segmentation, to automatically optimize a neural architecture from a large design space. Our NAS framework searches the structure of each layer including neural connectivities and operation types in both of the encoder and decoder. Since optimizing over a large discrete architecture space is difficult due to high-resolution 3D medical images, a novel stochastic sampling algorithm based on a continuous relaxation is also proposed for scalable gradient based optimization. On the 3D medical image segmentation tasks with a benchmark dataset, an automatically designed architecture by the proposed NAS framework outperforms the human-designed 3D U-Net, and moreover this optimized architecture is well suited to be transferred for different tasks.

cs.LG↗
arXiv2019

Fast AutoAugment

Data augmentation is an essential technique for improving generalization ability of deep learning models. Recently, AutoAugment has been proposed as an algorithm to automatically search for augmentation policies from a dataset and has significantly enhanced performances on many image recognition tasks. However, its search method requires thousands of GPU hours even for a relatively small dataset. In this paper, we propose an algorithm called Fast AutoAugment that finds effective augmentation policies via a more efficient search strategy based on density matching. In comparison to AutoAugment, the proposed algorithm speeds up the search time by orders of magnitude while achieves comparable performances on image recognition tasks with various models and datasets including CIFAR-10, CIFAR-100, SVHN, and ImageNet.

cs.LG↗
arXiv2019

A sharp $L_p$-regularity result for second-order stochastic partial differential equations with unbounded and fully degenerate leading coefficients

We present existence, uniqueness, and sharp regularity results of solution to the stochastic partial differential equation (SPDE) \begin{align} \label{abs eqn} du=(a^{ij}(ω,t)u_{x^ix^j}+f)dt + (σ^{ik}(ω,t)u_{x^i}+g^k)dw^k_t, \quad u(0,x)=u_0, \end{align} where $\{w^k_t:k=1,2,\cdots\}$ is a sequence of independent Brownian motions. The coefficients are merely measurable in $(ω,t)$ and can be unbounded and fully degenerate, that is, coefficients $a^{ij}$, $σ^{ik}$ merely satisfy \begin{align} \label{abs only} \left(α^{ij}(ω,t)\right)_{d\times d}:= \left(a^{ij}(ω,t)-\frac{1}{2}\sum_{k=1}^{\infty} σ^{ik}(ω,t)σ^{jk}(ω,t)\right) \geq 0. \end{align} In this article, we prove that there exists a unique solution $u$ to \eqref{abs eqn}, and \begin{align} \notag \|u_{xx}\|_{\mathbb{H}^γ_p(τ,δ)} &\leq N(d,p) \bigg( \|u_0\|_{\mathbb{B}_p^{γ+2 \left(1-1/ p \right)}} + \| f\|_{\mathbb{H}^γ_p( τ,δ^{1-p} )} \label{abs est} &\qquad \qquad+\|g_x\|^p_{\mathbb{H}^γ_p( τ, |σ|^p δ^{1-p},l_2)}+ \| g_x\|_{\mathbb{H}^γ_p( τ,δ^{1-p/2},l_2)} \bigg), \end{align} where $p\geq 2$, $γ\in \mathbf{R}$, $τ$ is an arbitrary stopping time, $δ(ω, t)$ is the smallest eigenvalue of $α^{ij}(ω, t)$, $\mathbb{H}_p^γ(τ, δ)$ is a weighted stochastic Sobolev space, and $\mathbb{B}_p^{γ+2 \left(1-1/ p \right)}$ is a stochastic Besov space.

math.PR↗
arXiv2018

Tunneling of Micro-sized Droplets Through a Flowing Soap Film

When a micron-sized water droplet impacts on a freely suspended soap film with speed $v_{i}$, there exists a critical impact velocity of penetration $v_{C}$. For the droplet with $v_{i} v_{C}$, it tunnels through. In all cases, the film remains intact despite the fact that the droplet radius ($R_{0}=26\,μm$) is much greater than the film thickness ($0<h\lesssim10\,μm$). The critical velocity $v_{C}$ was measured as a function of $h$, and interestingly $v_{C}$ approaches an asymptotic value $v_{C0}\simeq520\,$ cm/s in the limit $h\rightarrow0$. This indicates that in addition to an inertial effect, a deformation or stretching energy of the film is required for penetration. Quantitatively, we found that this deformation energy corresponds to the creation of $\sim14$ times of the cross-sectional area of the droplet ($14πR_{0}^{2}$) or a critical Weber number ${\rm We_{C} } (\equiv2R_{0}ρ_{w}v_{C0}^{2}/σ)\simeq44$, where $ρ_{w}$ and $σ$ are respectively the density and the surface tension of water. Key results: The interaction between liquid droplet and soap films is studied. When the impact velocity is higher than a critical velocity, the droplet penetrates the soap film without breaking it. The experimental results are rationalized using the mechanical collision model with the film stretching.

physics.flu-dyn↗
arXiv2018

Separated Rows structure of vortex streets behind triangular objects

We discuss two distinct spatial structures of vortex streets. The `conventional mushroom' structure is commonly discuss in many experimental studies, but the exotic `separated rows' structure is characterized by a thin irrotational fluid between two rows of vortices. In a two-dimensional soap film channel, we generated the exotic vortex arrangement by using triangular objects. This setup allows us to vary the thickness of boundary layers and their separation distance independently. We find that the separated rows structure appears only when the boundary layer thickness is less than 40% of the separation distance. We also discuss two physical mechanisms of the breakdown of vortex structures. The conventional mushroom structure decays due to the action of viscosity, and the separated rows structure decays because its arrangement is hydrodynamically unstable.

physics.flu-dyn↗
arXiv2018

On the second order derivative estimates for degenerate parabolic equations

We study the parabolic equation \begin{align} \notag &u_t(t,x)=a^{ij}(t)u_{x^ix^j}(t,x)+f(t,x), \quad (t,x) \in [0,T] \times \mathbf{R}^d \\ &u(0,x)=u_0(x) \label{main eqn} \end{align} with the full degeneracy of the leading coefficients, that is, \begin{align} (a^{ij}(t)) \geq δ(t)I_{d\times d} \geq 0. \end{align} It is well known that if $f$ and $u_0$ are not smooth enough, say $f\in \mathbb{L}_p(T):=L_p([0,T] ; L_p(\mathbf{R}^d))$ and $u_0\in L_p(\mathbf{R}^d)$, then in general the solution is only in $C([0,T];L_p(\mathbf{R}^d))$, and thus derivative estimates are not possible. In this article we prove that $u_{xx}(t,\cdot)\in L_p(\mathbf{R}^d)$ on the set $\{t: δ(t)>0 \}$ and \begin{align*} \int^T_0 \|u_{xx}(t)\|^p_{L_p} δ(t)dt\leq N(d,p) \left(\int^T_0 \|f(t)\|^p_{L_p}δ^{1-p}(t)dt + \|u_0\|^p_{B^{2-2/ p}_p} \right), \end{align*} where $B^{2-2/ p}_p$ is the Besov space of order $2-2/p$. We also prove that $u_{xx}(t,\cdot)\in L_p(\mathbf{R}^d)$ for all $t>0$ and \begin{equation} \label{10.13.3} \int^T_0 \|u_{xx}\|^p_{L_p(\mathbf{R}^d)}\,dt \leq N \|u_0\|^p_{B^{2-2/(βp)}_p}, \end{equation} if $f=0$, $\int^t_0 δ(s)ds>0$ for each $t>0$, and a certain asymptotic behavior of $δ(t)$ holds near $t=0$ (see (1.3)). Here $β>0$ is the constant related to the asymptotic behavior in (1.3). For instance, if $d=1$ and $a^{11}(t)=δ(t)=1+\sin(1/t)$, then the estimate holds with $β=1$, which actually equals the maximal regularity of the heat equation $u_t=Δu$.

math.AP↗
arXiv2017

Surface tension of flowing soap films

The surface tension of flowing soap films is measured with respect to the film thickness and the concentration of soap solution. We perform this measurement by measuring the curvature of the nylon wires that bound the soap film channel and use the measured curvature to parametrize the relation between the surface tension and the tension of the wire. We find the surface tension of our soap films increases when the film is relatively thin or made of soap solution of low concentration, otherwise it approaches an asymptotic value 30 mN/m. A simple adsorption model with only two parameters describes our observations reasonably well. With our measurements, we are also able to measure Gibbs elasticity for our soap film.

physics.flu-dyn↗
arXiv2017

An $L_p$-theory for diffusion equations related to stochastic processes with non-stationary independent increment

Let $X=(X_t)_{t \ge 0}$ be a stochastic process which has an (not necessarily stationary) independent increment on a probability space $(Ω, \mathbb{P})$. In this paper, we study the following Cauchy problem related to the stochastic process $X$: $\label{main eqn} \frac{\partial u}{\partial t}(t,x) = \cA(t)u(t,x) +f(t,x), \quad u(0,\cdot)=0, \quad (t,x) \in (0,T) \times \mathbf{R}^d, \end{align} where $f \in L_p( (0,T) ; L_p(\mathbf{R}^d))=L_p( (0,T) ; L_p)$ and \begin{align*} \cA(t)u(t,x) = \lim_{h \downarrow 0}\frac{\mathbb{E}\left[u(t,x+X_{t+h}-X_t)-u(t,x)\right]}{h}$. We provide a sufficient condition on $X$ to guarantee the unique solvability of equation (\ref{ab main}) in $L_p\left( [0,T] ; H^ϕ_{p}\right)$, where $H^ϕ_{p}$ is a $ϕ$-potential space on $\mathbf{R}^d$ . Furthemore we show that for this solution, \| u\|_{L_p\left( [0,T] ; H^ϕ_{p}\right)} \leq N \|f\|_{L_p\left( [0,T] ; L_p\right)}, where $N$ is independent of $u$ and $f$.

math.AP↗
arXiv2017

A Unified Strouhal-Reynolds Number Relationship for Laminar Vortex Streets Generated by Different Shaped Obstacles

A new Strouhal-Reynolds number relationship, $St=1/(A+B/Re)$, has been recently proposed based on observations of laminar vortex shedding from circular cylinders in a flowing soap film. Since the new $St$-$Re$ relation was derived from a general physical consideration, it raises the possibility that it may be applicable to vortex shedding from bodies other than circular ones. The work presented herein provides experimental evidence that this is the case. Our measurements also show that in the asymptotic limit ($Re\rightarrow\infty$), $St_{\infty}=1/A\simeq0.21$ is constant independent of rod shapes, leaving $B$ the only parameter that is shape dependent.

physics.flu-dyn↗
arXiv2017

An $L_p$-Lipschitz theory for parabolic equations with time measurable pseudo-differential operators

In this article we prove the existence and uniqueness of a (weak) solution $u$ in $L_p\left((0,T) , Λ_{γ+m}\right)$ to the Cauchy problem \begin{align} \notag &\frac{\partial u}{\partial t}(t,x)=ψ(t,i\nabla)u(t,x)+f(t,x),\quad (t,x) \in (0,T) \times \mathbf{R}^d \label{main eqn} & u(0,x)=0, \end{align} where $d \in \mathbb{N}$, $p \in (1,\infty]$, $γ,m \in (0,\infty)$, $Λ_{γ+m}$ is the Lipschitz space on $\mathbf{R}^d$ whose order is $γ+m$, $f \in L_p\left((0,T) , Λ_γ \right)$, and $ψ(t,i\nabla)$ is a time measurable pseudo-differential operator whose symbol is $ψ(t,ξ)$, i.e. $$ ψ(t,i\nabla)u(t,x)=\cF^{-1}\left[ψ(t,ξ)\cF\left[u(t,\cdot)\right](ξ)\right](x), $$ with the assumptions \begin{align*} \Re[ψ(t,ξ)] \leq -ν|ξ|^γ, \end{align*} and \begin{align*} |D_ξ^αψ(t,ξ)|\leqν^{-1}|ξ|^{γ-|α|}. \end{align*} Furthermore, we show \begin{align} \label{e 1028 1} \int_0^T \|u(t,\cdot)\|^p_{Λ_{γ+m}} dt \leq N \int_0^T \|f(t,\cdot)\|^p_{Λ_{m}} dt, \end{align} where $N$ is a positive constant depending only on $d$, $p$, $γ$, $ν$, $m$, and $T$, The unique solvability of equation (\ref{main eqn}) in $L_p$-Hölder space is also considered. More precisely, for any $f \in L_p((0,T);C^{n+α})$, there exists a unique solution $u \in L_p((0,T);C^{γ+n+α}(\mathbf{R}^d))$ to equation (\ref{main eqn}) and for this solution $u$, \begin{align} \label{e 1029 1} \int_0^T \|u(t,\cdot)\|^p_{C^{γ+n+α}}dt \leq N \int_0^T \|f(t,\cdot)\|^p_{C^{n+α}}dt, \end{align} where $n \in \mathbb{Z}_+$, $α\in (0,1)$, and $γ+α\notin \mathbb{Z}_+$.

math.AP↗
arXiv2017

An lp-boundedness of stochastic singular integral operators and its application to spdes

In this article we introduce a stochastic counterpart of the Hörmander condtion on the kernel $K(r,t,x,y)$: there exists a pseudo-metric $ρ$ on $(0,\infty)\times R^d$ and a positive constant $C_0$ such that for $X=(t,x), Y=(s,y), Z=(r,z) \in (0,\infty) \times R^d$, $$ \sup_{X,Y}\int_{0}^\infty \left[ \int_{ρ(X,Z) \geq C_0 ρ(X,Y)} | K(r,t, z,x) - K(r,s, z,y)| ~dz\right]^2 dr <\infty. $$ We prove that the stochastic singular integral of the type $$ \mathbb{T} g(t,x) :=\int_0^{t} \int_{R^d} K(t,s,x,y) g(s,y)dy dW_s $$ is a bounded operator on $\mathbb{L}_p=L_p(Ω\times (0,\infty); L_{p}(R^d))$ for any $p\geq 2$ if it is bounded when $p=2$ and stochastic Hörmander condition holds. Here $Ω$ is a probability space and $W_t$ is a Wiener process on $Ω$. Proving the $L_p$-boundedness of such integral operators is the key step in constructing an $L_p$-theory for linear stochastic partial differential equations (SPDEs in short). As a byproduct of our result on stochastic singular operators we obtain the maximal $L_p$-regularity result for a very wide class of SPDEs.

math.PR↗