We study the Cauchy problem for inhomogeneous evolution equations with time-dependent, potentially degenerate, and unbounded coefficients. A key feature of our work is allowing the principal coefficients to undergo arbitrary blow-up at both the initial and terminal times.

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.

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.

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.

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.

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 $\delta(t)\geq 0$. We show that there is a positive constant $N$ such that $\int_0^{T} \left(\int_{\mathbb{R}^d} \left(|u|+|u_{xx} |\right)^{p} dx \right)^{q/p} e^{-q\int_0^t c(s)ds} w(\alpha(t)) \delta(t) dt \leq N \int_0^{T} \left(\int_{\mathbb{R}^d} \left|f\left(t,x\right)\right|^{p} dx \right)^{q/p} e^{-q\int_0^t c(s)ds} w(\alpha(t)) (\delta(t))^{1-q} dt,$ where $p,q \in (1,\infty)$, $\alpha(t)=\int_0^t \delta(s)ds$, and $w$ is a Muckenhoupt's weight.

We obtain the existence, uniqueness, and regularity estimates of the following Cauchy problem 0.1 {[ ∂ _tu(t,x)=ψ (t,-i∇ )u(t,x)+f(t,x), (t,x)∈ (0,T)×ℝ^d,; u(0,x)=0, x∈ℝ^d, ]. in (Muckenhoupt) weighted L_p -spaces with time-measurable pseudo-differential operators 0.2 ψ (t,-i∇ )u(t,x):=ℱ^-1[ ψ (t,· )ℱ[u](t,· )] (x). 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 (0.1) 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 equation (0.1), 0.3 ∫ _0^T ∫ _ℝ^d |(-Δ )^γ /2 u(t,x) |^p (t^2 + |x|^2)^α /2dxdt ≤ N∫ _0^T ∫ _ℝ^d |f(t,x)|^p (t^2 + |x|^2)^α /2dxdt and 0.4 ∫ _0^T ( ∫ _ℝ^d |(-Δ )^γ /2 u(t,x) |^p |x|^α _2dx ) ^q/p t^α _1dt ≤ N∫ _0^T ( ∫ _ℝ^d |f(t,x) |^p |x|^α _2dx ) ^q/p t^α _1dt, where p,q∈ (1,∞ ) , -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∇ ) .

In this study, we present an efficient and novel unconditionally stable Monte Carlo simulation (MCS) for solving the multi-dimensional Allen–Cahn (AC) equation, which can model the motion by mean curvature flow of a hypersurface. We use an operator splitting method, where the diffusion and nonlinear terms are solved separately. The diffusion term is calculated using MCS for the stochastic differential equation, while the nonlinear term is locally computed for each particle in a virtual grid. Several numerical experiments are presented to demonstrate the performance of the proposed algorithm. The computational results confirm that the proposed algorithm can solve the AC equation more efficiently as the dimension of space increases.

We present existence, uniqueness, and sharp regularity results of solution to the stochastic partial differential equation (SPDE)(0.1)du=(aij(ω,t)uxixj+f)dt+(σik(ω,t)uxi+gk)dwtk,u(0,x)=u0, where {wtk:k=1,2,⋯} 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 aij, σik merely satisfy(0.2)(αij(ω,t))d×d:=(aij(ω,t)−12∑k=1∞σik(ω,t)σjk(ω,t))≥0. In this article, we prove that there exists a unique solution u to (0.1), and(0.3)‖uxx‖Hpγ(τ,δ)≤N(d,p)(‖u0‖Bpγ+2(1−1/p)+‖f‖Hpγ(τ,δ1−p)+‖gx‖Hpγ(τ,|σ|pδ1−p,l2)p+‖gx‖Hpγ(τ,δ1−p/2,l2)), where p≥2, γ∈R, τ is an arbitrary stopping time, δ(ω,t) is the smallest eigenvalue of αij(ω,t), Hpγ(τ,δ) is a weighted stochastic Sobolev space, and Bpγ+2(1−1/p) is a stochastic Besov space.

Let Z=(Z_t)_t≥ 0 be an additive process with a bounded triplet (0,0, _t)_t≥ 0 . Then the infinitesimal generators of Z is given by time dependent nonlocal operators as follows: 𝒜_Z(t)u(t,x) =lim _h ↓ 0𝔼[u(t,x+Z_t+h-Z_t)-u(t,x)]/h =∫ _ℝ^d(u(t,x+y)-u(t,x)-y·∇ _x u(t,x)1_|y|≤ 1) _t(dy). Suppose that for any Schwartz function φ on ℝ^d whose Fourier transform is in C_c^∞(B_c_s∖ B_c_s^-1 ) , there exist positive constants N_0 , N_1 , and N_2 such that ∫ _ℝ^d|𝔼[φ (x+r^-1Z_t)]|dx≤ N_0 e^- N_1 t/s(r), ∀ (r,t)∈ (0,1)× [0,T], and ‖ψ ^μ(r^-1D)φ‖ _L_1(ℝ^d)≤N_2/s(r), ∀ r∈ (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 ℝ^d , ψ ^μ(r^-1D)φ (x)= ℱ^-1[ ψ ^μ(r^-1ξ ) ℱ[φ ]] (x) and ψ ^μ(ξ ):=∫ _ℝ^d(e^iy·ξ-1-iy·ξ 1_|y|≤ 1)μ (dy) . In particular, above assumptions hold for Lévy measures _t having a nice lower bound and μ satisfying a weak-scaling property (Propositions 3.3 , 3.5 , and 3.6 ). We emphasize that there is no regularity condition on Lévy measures _t and they do not have to be symmetric. In this paper, we establish the L_p -solvability to the initial value problem 0.2 ∂ u/∂ t(t,x)=𝒜_Z(t)u(t,x), u(0,· )=u_0, (t,x)∈ (0,T)×ℝ^d, where u_0 is contained in a scaled Besov space B_p,q^s;γ -2/q(ℝ^d) (see Definition 2.8 ) with a scaling function s , exponent p ∈ (1,∞ ) , q∈ [1,∞ ) , and order γ∈ [0,∞ ) . We show that equation ( 0.2 ) is uniquely solvable and the solution u obtains full-regularity gain from the diffusion generated by a stochastic process Z . In other words, there exists a unique solution u to equation ( 0.2 ) in L_q((0,T);H_p^μ ;γ(ℝ^d)) , where H_p^μ ;γ(ℝ^d) is a generalized Bessel potential space (see Definition 2.3 ). Moreover, the solution u satisfies ‖ u‖ _L_q((0,T);H_p^μ ;γ(ℝ^d))≤ N‖ u_0‖ _B_p,q^s;γ -2/q(ℝ^d), where N is independent of u and u_0 . We finally remark that our operators 𝒜_Z(t) include logarithmic operators such as -a(t)log (1- ) (Corollary 3.2 ) and operators whose symbols are non-smooth such as -∑ _j=1^dc_j(t)(- )^α /2_x^j (Corollary 3.9 ).

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.

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 pseudolabels, 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-ofthe-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.

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 http://medicaldecathlon.com/. In addition, both data and online evaluation are accessible via https://competitions.codalab.org/competitions/17094.

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.

We obtain uniqueness and existence of a solution u to the following second-order stochastic partial differential equation: 1 $$\begin{aligned} du= \left( {\bar{a}}^{ij}(\omega ,t)u_{x^ix^j}+ f \right) dt + g^k dw^k_t, \quad t \in (0,T); \quad u(0,\cdot )=0, \end{aligned}$$ where $$T \in (0,\infty )$$ , $$w^k$$ $$(k=1,2,\ldots )$$ are independent Wiener processes, $$({\bar{a}}^{ij}(\omega ,t))$$ is a (predictable) nonnegative symmetric matrix valued stochastic process such that $$\begin{aligned} \kappa |\xi |^2 \le {\bar{a}}^{ij}(\omega ,t) \xi ^i \xi ^j \le K |\xi |^2 \quad \forall \;(\omega ,t,\xi ) \in \Omega \times (0,T) \times {\mathbf {R}}^d \end{aligned}$$ for some $$\kappa , K \in (0,\infty )$$ , $$\begin{aligned} f \in L_p\left( (0,T) \times {\mathbf {R}}^d, dt \times dx ; L_r(\Omega , {\mathscr {F}} ,dP) \right) , \end{aligned}$$ and $$\begin{aligned} g, g_x \in L_p\left( (0,T) \times {\mathbf {R}}^d, dt \times dx ; L_r(\Omega , {\mathscr {F}} ,dP; l_2) \right) \end{aligned}$$ with $$2 \le r \le p < \infty $$ and appropriate measurable conditions. Moreover, for the solution u, we obtain the following maximal regularity moment estimate 2 $$\begin{aligned}&\int _0^T \int _{{\mathbf {R}}^d}\left( \mathbb {E}\left[ |u(t,x)|^r\right] \right) ^{p/r} dx dt + \int _0^T \int _{{\mathbf {R}}^d}\left( \mathbb {E}\left[ |u_{xx}(t,x)|^r\right] \right) ^{p/r} dx dt \nonumber \\&\le N \bigg (\int _0^T \int _{{\mathbf {R}}^d}\left( \mathbb {E}\left[ |f(t,x)|^r\right] \right) ^{p/r} dx dt + \int _0^T \int _{{\mathbf {R}}^d}\left( \mathbb {E}\left[ |g(t,x)|_{l_2}^r\right] \right) ^{p/r} dx dt \nonumber \\&\quad + \int _0^T \int _{{\mathbf {R}}^d}\left( \mathbb {E}\left[ |g_x(t,x)|_{l_2}^r\right] \right) ^{p/r} dx dt \bigg ), \end{aligned}$$ where N is a positive constant depending only on d, p, r, $$\kappa $$ , K, and T. As an application, for the solution u to (1), the rth moment $$m^r(t,x):=\mathbb {E}|u(t,x)|^r$$ is in the parabolic Sobolev space $$W_{p/r}^{1,2}\left( (0,T) \times \mathbf {R}^d\right) $$ .

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.

Lung cancer is the deadliest type of cancer worldwide and late detection is the major factor for the low survival rate of patients. Low dose computed tomography has been suggested as a potential screening tool but manual screening is costly and time-consuming. This has fuelled the development of automatic methods for the detection, segmentation and characterisation of pulmonary nodules. In spite of promising results, the application of automatic methods to clinical routine is not straightforward and only a limited number of studies have addressed the problem in a holistic way. With the goal of advancing the state of the art, the Lung Nodule Database (LNDb) Challenge on automatic lung cancer patient management was organized. The LNDb Challenge addressed lung nodule detection, segmentation and characterization as well as prediction of patient follow-up according to the 2017 Fleischner society pulmonary nodule guidelines. 294 CT scans were thus collected retrospectively at the Centro Hospitalar e Universitrio de So Joo in Porto, Portugal and each CT was annotated by at least one radiologist. Annotations comprised nodule centroids, segmentations and subjective characterization. 58 CTs and the corresponding annotations were withheld as a separate test set. A total of 947 users registered for the challenge and 11 successful submissions for at least one of the sub-challenges were received. For patient follow-up prediction, a maximum quadratic weighted Cohen’s kappa of 0.580 was obtained. In terms of nodule detection, a sensitivity below 0.4 (and 0.7) at 1 false positive per scan was obtained for nodules identified by at least one (and two) radiologist(s). For nodule segmentation, a maximum Jaccard score of 0.567 was obtained, surpassing the interobserver variability. In terms of nodule texture characterization, a maximum quadratic weighted Cohen’s kappa of 0.733 was obtained, with part solid nodules being particularly challenging to classify correctly. Detailed analysis of the proposed methods and the differences in performance allow to identify the major challenges remaining and future directions - data collection, augmentation/generation and evaluation of under-represented classes, the incorporation of scan-level information for better decision-making and the development of tools and challenges with clinical-oriented goals. The LNDb Challenge and associated data remain publicly available so that future methods can be tested and benchmarked, promoting the development of new algorithms in lung cancer medical image analysis and patient follow-up recommendation.

We prove the existence of a mild solution to the three dimensional incompressible stochastic magnetohydrodynamic equations in the whole space with the initial data which belong to the Sobolev spaces.

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.

In this article we introduce a stochastic counterpart of the Hörmander condition and Calderón-Zygmund theorem. Let W t W_t be a Wiener process in a probability space Ω \Omega and let K ( ω , r , t , x , y ) K(\omega ,r,t,x,y) be a random kernel which is allowed to be stochastically singular in a domain O ⊂ R d \mathcal {O} \subset \mathbf {R}^d in the sense that E | ∫ 0 t ∫ | x − y | > ε | K ( ω , s , t , y , x ) | d y d W s | p = ∞ ∀ t , p , ε > 0 , x ∈ O . \begin{equation*} \mathbb {E} \left |\int _0^{t} \int _{|x-y|>\varepsilon }|K(\omega , s, t,y,x)|dy dW_s\right |^p = \infty \quad \forall \, t, p,\varepsilon >0,\, x\in \mathcal {O}. \end{equation*} We prove that the stochastic integral operator of the type T g ( t , x ) ≔ ∫ 0 t ∫ O K ( ω , s , t , y , x ) g ( s , y ) d y d W s \begin{align} \mathbb {T} g(t,x) \coloneq \int _0^{t} \int _{\mathcal {O}} K(\omega ,s,t,y,x) g(s,y)dy dW_s \end{align} is bounded on L p = L p ( Ω × ( 0 , ∞ ) ; L p ( O ) ) \mathbb {L}_p=L_p \left (\Omega \times (0,\infty ); L_{p}(\mathcal {O}) \right ) for all p ∈ [ 2 , ∞ ) p \in [2,\infty ) if it is bounded on L 2 \mathbb {L}_2 and the following (which we call stochastic Hörmander condition) holds: there exists a quasi-metric ρ \rho on ( 0 , ∞ ) × O (0,\infty )\times \mathcal {O} and a positive constant C 0 C_0 such that for X = ( t , x ) , Y = ( s , y ) , Z = ( r , z ) ∈ ( 0 , ∞ ) × O X=(t,x), Y=(s,y), Z=(r,z) \in (0,\infty ) \times \mathcal {O} , sup ω ∈ Ω , X , Y ∫ 0 ∞ [ ∫ ρ ( X , Z ) ≥ C 0 ρ ( X , Y ) | K ( r , t , z , x ) − K ( r , s , z , y ) | d z ] 2 d r > ∞ . \begin{equation*} \sup _{\omega \in \Omega ,X,Y}\int _{0}^\infty \left [ \int _{\rho (X,Z) \geq C_0 \rho (X,Y)} | K(r,t, z,x) - K(r,s, z,y)| ~dz\right ]^2 dr >\infty . \end{equation*} Such a stochastic singular integral naturally appears when one proves the maximal regularity of solutions to stochastic partial differential equations (SPDEs). As applications, we obtain the sharp L p L_p -regularity result for a wide class of SPDEs, which includes SPDEs with time measurable pseudo-differential operators and SPDEs defined on non-smooth angular domains.