High-throughput biological technologies (e.g. ChIPseq, RNA-seq and single-cell RNA-seq) rapidly accelerate the accumulation of genome-wide omics data in
diverse interrelated biological scenarios (e.g. cells,
tissues and conditions). Integration and differential
analysis are two common paradigms for exploring
and analyzing such data. However, current integrative methods usually ignore the differential part, and
typical differential analysis methods either fail to
identify combinatorial patterns of difference or require matched dimensions of the data. Here, we propose a flexible framework CSMF to combine them
into one paradigm to simultaneously reveal Common
and Specific patterns via Matrix Factorization from
data generated under interrelated biological scenarios. We demonstrate the effectiveness of CSMF with
four representative applications including pairwise
ChIP-seq data describing the chromatin modification
map between K562 and Huvec cell lines; pairwise
RNA-seq data representing the expression profiles of
two different cancers; RNA-seq data of three breast
cancer subtypes; and single-cell RNA-seq data of human embryonic stem cell differentiation at six time
points. Extensive analysis yields novel insights into
hidden combinatorial patterns in these multi-modal
data. Results demonstrate that CSMF is a powerful
tool to uncover common and specific patterns with
significant biological implications from data of interrelated biological scenarios.
We study the well-posedness of a unified system of coupled forward-backward stochastic differential equations (FB-SDEs)
with Levy jumps and double completely-S skew reflections. Owing to the reflections, the solution to an embedded Skorohod problem may be not unique, i.e., bifurcations may occur at reflection boundaries, the well-known contraction mapping approach can not be extended directly to solve our problem. Thus, we develop a weak convergence method to prove the well-posedness of an adapted 6-tuple weak solution in the sense of distribution to the unified system. The proof heavily depends on newly established Malliavin calculus for vector-valued Levy processes together with a generalized linear growth and Lipschitz condition that guarantees the well-posedness of the unified system even under a random environment. Nevertheless,
if a more strict boundary condition is imposed, i.e., the spectral radii in certain sense for the reflections are strictly less than the unity, a unique adapted 6-tuple strong solution in the sense of sample pathwise is concerned. In addition, as applications and economical studies of our unified system, we also develop new techniques including deriving a generalized mutual information formula for signal processing over possible non-Gaussian channels with multi-input multi-output (MIMO) antennas and dynamics driven by Levy processes.
We define piecewise continuous almost automorphic (p.c.a.a.) functions in the manners of Bochner, Bohr and Levitan, respectively, to describe almost automorphic motions in impulsive systems, and prove that with certain prefixed possible discontinuities they are equivalent to quasi-uniformly continuous Stepanov almost automorphic ones. Spatially almost automorphic sets on the line, which serve as suitable objects containing discontinuities of p.c.a.a. functions, are characterized in the manners of Bochner, Bohr and Levitan, respectively, and shown to be equivalent. Two Favard's theorems are established to illuminate the importance and convenience of p.c.a.a. functions in the study of almost periodically forced impulsive systems.
In this paper, we discuss the Weyl problem in warped product spaces. We apply the method of continuity and prove the openness
of the Weyl problem. A counterexample is constructed to show that the isometric embedding of the sphere with canonical metric
is not unique up to an isometry if the ambient warped product space is not a space form. Then, we study the rigidity of the
standard sphere if we fix its geometric center in the ambient space. Finally, we discuss a Shi-Tam type of inequality for the
Schwarzschild manifold as an application of our findings.
We consider the reconstruction of the Robin impedance coefficient of a heat conduction system in a two-dimensional spatial domain from the time-average measurement specified on the boundary. By applying the potential representation of a solution, this nonlinear inverse problem is transformed into an ill-posed integral system coupling the density function for potential and the
unknown boundary impedance. The uniqueness as well as the conditional stability of this inverse problem is established from the integral system. Then we propose to find the boundary impedance by solving a non-convex regularizing optimization problem. The well-posedness of this optimization problem together with the convergence property of the minimizer is analyzed. Finally, based on the singularity decomposition of the potential representation of the solution, two iteration schemes with their numerical realizations are proposed to solve this optimization problem
We consider an inverse problem of recovering a time-dependent factor of an unknown source on some subboundary for a diffusion equation with time fractional derivative by nonlocal measurement data. Such fractional-order equations describe anomalous diffusion of some contaminants in heterogeneous media such as soil and model the contamination process from an unknown source located on a part of the boundary of the concerned domain. For this inverse problem, we firstly establish the well-posedness in some Sobolev space. Then we propose two regularizing schemes in order to reconstruct an unknown boundary source stably in terms of the noisy measurement data. The first regularizing scheme is based on an integral equation of the second kind which an unknown boundary source solves, and we prove a convergence rate of regularized solutions with a suitable choice strategy of the regularizing parameter. The second regularizing scheme relies directly on
discretization by the radial basis function for the initial-boundary value problem for fractional diffusion equation, and we carry out numerical tests, which show the validity of our proposed regularizing scheme.
We estimate the heat conducted by a cluster of many small cavities. We show that the dominating heat is a sum, over the number of cavities, of the heat generated by each cavity after interacting with each other. This interaction is described through densities computable as solutions of a closed, and invertible, system of time domain integral equations of second kind. As an application of these expansions, we derive the effective heat conductivity which generates approximately the same heat as the cluster of cavities, distributed in a three-dimensional bounded domain, with explicit error estimates in terms of that cluster. At the analysis level, we use time domain integral equations. Doing that, we have two choices. First, we can favor the space variable by reducing the heat potentials to the ones related to the Laplace operator (avoiding Laplace transform). Second, we can favor the time variable by reducing the representation to the Abel integral operator. As the model under investigation has time-independent parameters, we follow the first approach here.
We consider the problem of reconstructing unknown inclusions inside a thermal conductor from boundary measurements, which arises from active thermography and is formulated as an inverse boundary value problem for the heat equation. In our previous works, we proposed a sampling-type method for reconstructing the boundary of the unknown inclusion and gave its rigorous mathematical justification. In this paper, we continue our previous works and provide a further investigation of the reconstruction method from both the theoretical and numerical points of view. First, we analyze the solvability of the Neumann-to-Dirichlet map gap equation and establish a relation of its solution to the Green function of an interior transmission problem for the inclusion. This naturally provides a way of computing this Green function from the Neumann-to-Dirichlet map. Our new findings reveal the essence of the reconstruction method. A convergence result for noisy measurement data is also proved. Second, based on the heat layer potential argument, we perform a numerical implementation of the reconstruction method for the homogeneous inclusion case. Numerical results are presented to show the efficiency and stability of the proposed method.
Consider the problem of reconstructing unknown Robin inclusions inside a heat conductor from boundary measurements. This problem arises from active thermography and is formulated as an inverse boundary value problem for the heat equation. In our previous works, we proposed a sampling-type method for reconstructing the boundary of the Robin inclusion and gave its rigorous mathematical justification. This method is non-iterative and based on the characterization of the solution to the so-called Neumann-to-Dirichlet map gap equation. In this paper, we give a further investigation of the reconstruction method from both the theoretical and numerical points of view. First, we clarify the solvability of the Neumann-to-Dirichlet map gap equation and establish a relation of its solution to the Green function associated with an initial-boundary value problem for the heat equation inside the Robin inclusion. This naturally provides a way of computing this Green function from the Neumann-to-Dirichlet map and explains what is the input for the linear sampling method. Assuming that the Neumann-to-Dirichlet map gap equation has a unique solution, we also show the convergence of our method for noisy measurements. Second, we give the numerical implementation of the reconstruction method for two-dimensional spatial domains. The measurements for our inverse problem are simulated by solving the forward problem via the boundary integral equation method. Numerical results are presented to illustrate the efficiency and stability of the proposed method. By using a finite sequence of transient input over a time interval, we propose a new sampling method over the time interval by single measurement which is most likely to be practical.
We study detecting a boundary corrosion damage in the inaccessible part of a rectangular shaped electrostatic conductor from a one set of Cauchy data specied on an accessible boundary part of conductor. For this nonlinear ill-posed problem, we prove the uniqueness in a very general framework. Then we establish the conditional stability of Holder type based on some a priori assumptions on the unknown impedance and the electrical current input specied in the accessible part. Finally a regularizing scheme of double regularizing parameters, using the truncation of the series expansion of the solution, is proposed with the convergence analysis on the explicit regularizing solution in terms of a practical average norm for measurement data.
We consider an inverse time-dependent source problem governed by a distributed time-fractional diffusion equation using interior measurement data. Such a problem arises in some ultra-slowdiffusion phenomena in many applied areas. Based on the regularity result of the solution to the direct problem, we establish the solvability of this inverse problem as well as the conditional stability
in suitable function space with a weak norm. By a variational identity connecting the unknown time-dependent source and the interior measurement data, the conjugate gradient method is also introduced to construct the inversion algorithm under the framework of regularizing scheme. We show the validity of the proposed scheme by several numerical examples.
Consider the image restoration from incomplete noisy frequency data with total variation and sparsity regularizing penalty terms. Firstly, we establish an unconstrained optimization model with dierent smooth approximations on the regularizing terms. Then, to weaken the amount of computations for cost functional with total variation term, the alternating iterative scheme is developed to obtain the exact solution through shrinkage thresholding in inner loop, while the nonlinear Euler equation is appropriately linearized at each iteration in exterior loop, yielding a linear system with diagonal coefficient matrix in frequency domain. Finally the linearized iteration is proven to be convergent in generalized sense for suitable regularizing parameters, and the error between the linearized iterative solution and the one gotten from the exact nonlinear Euler equation is rigorously estimated, revealing the essence of the proposed alternative iteration scheme. Numerical tests for different configurations show the validity of the proposed scheme, compared with some existing algorithms.
. Motivated by block partitioned problems arising from group sparsity representation
and generalized noncooperative potential games, this paper presents a basic decomposition method
for a broad class of multiblock nonsmooth optimization problems subject to coupled linear constraints on the variables that may additionally be individually constrained. The objective of such an
optimization problem is given by the sum of two nonseparable functions minus a sum of separable,
pointwise maxima of finitely many convex differentiable functions. One of the former two nonseparable functions is of the class LC1
, i.e., differentiable with a Lipschitz gradient, while the other
summand is multiconvex. The subtraction of the separable, pointwise maxima of convex functions
induces a partial difference-of-convex (DC) structure in the overall objective; yet with all three terms
together, the objective is nonsmooth and non-DC, but is blockwise directionally differentiable. By
taking advantage of the (negative) pointwise maximum structure in the objective, the developed
algorithm and its convergence result are aimed at the computation of a blockwise directional stationary solution, which arguably is the sharpest kind of stationary solutions for this class of nonsmooth
problems. This aim is accomplished by combining the alternating direction method of multipliers
(ADMM) with a semilinearized Gauss–Seidel scheme, resulting in a decomposition of the overall
problem into subproblems each involving the individual blocks. To arrive at a stationary solution of
the desired kind, our algorithm solves multiple convex subprograms at each iteration, one per convex
function in each pointwise maximum. In order to lessen the potential computational burden in each
iteration, a probabilistic version of the algorithm is presented and its almost sure convergence is
We develop Random Batch Methods for interacting particle systems with large number of particles. These methods use small but random batches for particle interactions, thus the computational cost is reduced from $O(N^2)$ per time step to $O(N)$, for a system with $N$ particles with binary interactions. On one hand, these methods are efficient Asymptotic-Preserving schemes for the underlying particle systems, allowing $N$-independent time steps and also capture, in the $N \to \infty$ limit, the solution of the mean field limit which are nonlinear Fokker-Planck equations; on the other hand, the stochastic processes generated by the algorithms can also be regarded as new models for the underlying problems. For one of the methods, we give a particle number independent error estimate under some special interactions. Then, we apply these methods to some representative problems in mathematics, physics, social and data sciences, including the Dyson Brownian motion from random matrix theory, Thomson's problem, distribution of wealth, opinion dynamics and clustering. Numerical results show that the methods can capture both the transient solutions and the global equilibrium in these problems.
A variational formula for the Lutwak affine surface areas j of convex bodies in Rn is
established when 1 ≤ j ≤ n − 1. By using introduced new ellipsoids associated with
projection functions of convex bodies, we prove a sharp isoperimetric inequality for j ,
which opens up a new passage to attack the longstanding Lutwak conjecture in convex
Existence and uniqueness of the solution to the Lp Minkowski
problem for the electrostatic p-capacity are proved when p > 1
and 1 < p < n. These results are nonlinear extensions of the very
recent solution to the Lp Minkowski problem for p-capacity when
p = 1 and 1 < p < n by Colesanti et al. and Akman et al., and
the classical solution to the Minkowski problem for electrostatic
capacity when p = 1 and p = 2 by Jerison.
There is much research on the dynamical complexity on irregular sets andlevel sets of ergodic average from the perspective of density in base space, theHausdorff dimension, Lebesgue positive measure, positive or full topological entropy (andtopological pressure), etc. However, this is not the case from the viewpoint of chaos.There are many results on the relationship of positive topological entropy and variouschaos. However, positive topological entropy does not imply a strong version of chaos,called DC1. Therefore, it is non-trivial to study DC1 on irregular sets and level sets. Inthis paper, we will show that, for dynamical systems with specification properties, thereexist uncountable DC1-scrambled subsets in irregular sets and level sets. Meanwhile, weprove that several recurrent level sets of points with different recurrent frequency haveuncountable DC1-scrambled subsets. The major argument in proving the above results isthat there exists uncountable DC1-scrambled subsets in saturated sets.
For any dynamical system T : X → X of a compact metric
space X with g-almost product property and uniform separation property, under the assumptions that the periodic points
are dense in X and the periodic measures are dense in the
space of invariant measures, we distinguish various periodiclike recurrences and find that they all carry full topological
entropy and so do their gap-sets. In particular, this implies
that any two kind of periodic-like recurrences are essentially
different. Moreover, we coordinate periodic-like recurrences
with (ir)regularity and obtain lots of generalized multifractal analyses for all continuous observable functions. These
results are suitable for all β-shifts (β > 1), topological mixing
subshifts of finite type, topological mixing expanding maps or
topological mixing hyperbolic diffeomorphisms, etc.
Roughly speaking, we combine many different “eyes” (i.e.,
observable functions and periodic-like recurrences) to observe
the dynamical complexity and obtain a Refined Dynamical
Structure for Recurrence Theory and Multi-fractal Analysis.
Li LongDepartment of Mathematics and Center of Geophysics, Harbin Institute of Technology, 150001 Harbin, People’s Republic of ChinaVidard ArthurUniversité Grenoble Alpes, Inria, CNRS, Grenoble INP, LJK, 38000 Grenoble, FranceLe Dimet Francois-Xavier Université Grenoble Alpes, Inria, CNRS, Grenoble INP, LJK, 38000 Grenoble, FranceMa JianweiDepartment of Mathematics and Center of Geophysics, Harbin Institute of Technology, 150001 Harbin, People’s Republic of China
This work combines a level-set approach and the optimal transport-based Wasserstein distance in a data assimilation framework. The primary motivation of this work is to reduce assimilation artifacts resulting from the position and observation error in the tracking and forecast of pollutants present on the surface of oceans or lakes. Both errors lead to spurious effect on the forecast that need to be corrected. In general, the geometric contour of such pollution can be retrieved from observation while more detailed characteristics such as concentration remain unknown. Herein, level sets are tools of choice to model such contours and the dynamical evolution of their topology structures. They are compared with contours extracted from observation using the Wasserstein distance. This allows to better capture position mismatches between both sources compared with the more classical Euclidean distance. Finally, the viability of this approach is demonstrated through academic test cases and its numerical performance is discussed.