Corrector Functions Research Articles

CrossTx: Cross-Cell-Line Transcriptomic Signature Predictions

Predicting the cell response to drugs is central to drug discovery, drug repurposing, and personalized medicine. To this end, large datasets of drug signatures have been curated, most notably the Connectivity Map (CMap). A multitude of in silico approaches have also been formulated, but strategies for predicting drug signatures in unseen cells—cell lines not in the reference datasets—are still lacking. In this work, we developed a simple-yet-efficacious computational strategy, called CrossTx, for predicting the drug transcriptomic signatures of an unseen target cell line using drug transcriptome data of reference cell lines and unlabeled transcriptome data of the target cells. Our strategy involves the combination of Predictor and Corrector steps. The Predictor generates cell-line-agnostic drug signatures using the reference dataset, while the Corrector produces target-cell-specific drug signatures by projecting the signatures from the Predictor onto the transcriptomic latent space of the target cell line. Testing different Predictor–Corrector functions using the CMap revealed the combination of averaging (Mean) as a Predictor and Principal Component Analysis (PCA) followed by Autoencoder (AE) as a Corrector to be the best. Yet, using Mean as a Predictor and PCA as a Corrector achieved comparatively high accuracy with much lower computational requirements when compared to the best combination.

Homogenization of non-local nonlinear p-Laplacian equation with variable index and periodic structure

This paper deals with the homogenization of a one-dimensional nonlinear non-local variable index p(x)-Laplacian operator Lɛ with a periodic structure and convolution kernel. By constructing a scale diffusive model and two corrector functions χ1 and χ2, as scale parameter ɛ → 0+, we first obtain that the limit operator L is a p-Laplacian operator with constant exponent and coefficients such that Lu=Rddx(|u′(x)|p−2u′(x)). Then, for a given function f∈Lq(R)(q=pp−1), we prove the asymptotic behavior of the solution uɛ(x) to the equation (Lɛ − I)uɛ(x) = f(x) such that uε(x)=u(x)+εχ1(xε)u′(x)+ε2χ2(xε)u″(x)+o(1)(ε→0+) in Lp(R), where u(x) is the solution of equation (L − I)u(x) = f(x).

Homogenization of nonlinear nonlocal diffusion equation with periodic and stationary structure

<abstract><p>This paper is devoted to the homogenization of a class of nonlinear nonlocal parabolic equations with time dependent coefficients in a periodic and stationary structure. In the first part, we consider the homogenization problem with a periodic structure. Inspired by the idea of Akagi and Oka for local nonlinear homogenization, by a change of unknown function, we transform the nonlinear nonlocal term in space into a linear nonlocal scaled diffusive term, while the corresponding linear time derivative term becomes a nonlinear one. By constructing some corrector functions, for different time scales $ r $ and the nonlinear parameter $ p $, we obtain that the limit equation is a local nonlinear diffusion equation with coefficients depending on $ r $ and $ p $. In addition, we also consider the homogenization of the nonlocal porous medium equation with non negative initial values and get similar homogenization results. In the second part, we consider the previous problem in a stationary environment and get some similar homogenization results. The novelty of this paper is two folds. First, for the determination equation with a periodic structure, our study complements the results in literature for $ r = 2 $ and $ p = 1 $. Second, we consider the corresponding equation with a stationary structure.</p></abstract>

A Space-Time Multiscale Method for Parabolic Problems

We present a space-time multiscale method for a parabolic model problem with an underlying coefficient that may be highly oscillatory with respect to both the spatial and the temporal variables. The method is based on the framework of the Variational Multiscale Method in the context of a space-time formulation and computes a coarse-scale representation of the differential operator that is enriched by auxiliary space-time corrector functions. Once computed, the coarse-scale representation allows us to efficiently obtain well-approximating discrete solutions for multiple right-hand sides. We prove first-order convergence independently of the oscillation scales in the coefficient and illustrate how the space-time correctors decay exponentially in both space and time, making it possible to localize the corresponding computations. This localization allows us to define a practical and computationally efficient method in terms of complexity and memory, for which we provide a posteriori error estimates and present numerical examples.

Viscosity approximation of the solution to Burgers' equations with shock layers

ABSTRACT Viscous Burgers' equations with a small viscosity are considered and convergence of vanishing viscosity limit problem is investigated. We examine interior layers of a solution to viscous Burgers' equations, , as a viscosity parameter ε tends to zero. The inviscid model, i.e. when , possesses the structure of scalar hyperbolic conservation laws, hence our studies deliver an important idea that arises in the field of shock discontinuities of nonlinear hyperbolic waves. The heart of the paper is to establish asymptotic expansions and utilize inner solutions of sharp transition, which are called a corrector function. With aid of corrector functions and energy estimates, we improve the convergence rate of to as in ( in ) in the regions including shocks under an entropy condition.

Boundary layer analysis for the stochastic nonlinear reaction–diffusion equations

Singularly perturbed stochastic (and deterministic) nonlinear reaction–diffusion equations are considered. We first study the governing problem posed in the channel domain with lateral periodicity and extend the results to general smooth domains. Introducing corrector functions, which correct the boundary values discrepancies, we are able to develop the convergence analysis. For the analysis, we make use of the maximum principle to estimate the corrector functions. The stochastic problems also rely on the deterministic corrector functions, which lead to simpler computations than those of the stochastic version of the correctors.

Enriched Spectral Method for Stiff Convection-Dominated Equations

A novel and simple numerical method for stiff convection-dominated problems is studied in presence of boundary or interior layers. A version of the spectral Chevyshev-collocation method enriched with the so-called corrector functions is investigated. The corrector functions here are designed to capture the stiffness of the layers (see the Appendix), and the proposed method does not rely on the adaptive grid points. The extensive numerical results demonstrate that the enriched spectral methods are very accurate with low computational cost.

Hybrid Multiscale Finite Volume Method for Advection-Diffusion Equations Subject to Heterogeneous Reactive Boundary Conditions

We present a hybrid scheme for the coupling of macro- and microscale continuum models for reactive contaminant transport in fractured and porous media. The transport model considered is the advection-diffusion equation, subject to linear heterogeneous reactive boundary conditions. The multiscale finite volume method (MsFV) is employed to define an approximation to the microscale concentration field defined in terms of macroscopic or global degrees of freedom, together with local interpolator and corrector functions capturing microscopic spatial variability. The macroscopic mass balance relations for the MsFV global degrees of freedom are coupled with the macroscopic model, resulting in a global problem for the simultaneous time-stepping of all macroscopic degrees of freedom throughout the domain. In order to perform the hybrid coupling, the micro- and macroscale models are applied over overlapping subdomains of the simulation domain, with the overlap denoted as the handshake subdomain $\Omega^{hs}$, over which continuity of concentration and transport fluxes between models is enforced. Continuity of concentration is enforced by posing a restriction relation between models over $\Omega^{hs}$. Continuity of fluxes is enforced by prolongating to microscopic resolution the macroscopic model fluxes across the boundary of $\Omega^{hs}$. The microscopic interpolator and corrector functions are solutions to local microscopic advection-diffusion problems decoupled from the global degrees of freedom and from each other by virtue of the MsFV decoupling ansatz. The error introduced by the decoupling ansatz is reduced iteratively by the preconditioned GMRES algorithm, with the hybrid MsFV operator serving as the preconditioner.

A least-squares formulation for the approximation of controls for the Stokes system

This work deals with the approximation of distributed null controls for the Stokes system. The goal is to compute an approximation of controls that drives the solution from a prescribed initial state at $$t=0$$ to zero at $$t=T$$ . The existence of square-integrable controls has been obtained in Fursikov and Imanuvilov (Controllability of evolution equations, pp 1–163, 1996) via Carleman type estimates. We introduce and analyze a least-squares formulation of the controllability problem, and we show that it allows the construction of convergent sequences of functions toward null controls for the Stokes system. The approach consists first in introducing a class of functions satisfying a priori the boundary conditions in space and time—in particular the null controllability condition at time $$T$$ —and then finding among this class one element satisfying the Stokes system. This second step is done by minimizing a quadratic functional, among the admissible corrector functions of the Stokes system. Numerical experiments for the two-dimensional case are performed in the framework of finite element approximations and demonstrate the interest of the approach. The method described here does not make use of duality arguments and, therefore, avoids the introduction of numerical ill-posed problem, as is typical when parabolic-type equation is considered. This work extends (Munch and Pedregal in Eur J Appl Math, 2014) where the case of the heat equation is discussed.

Numerical null controllability of the heat equation through a least squares and variational approach

This work is concerned with the numerical computation of null controls for the heat equation. The goal is to compute an approximation of controls that drives the solution from a prescribed initial state at t=0 to zero at t=T. In spite of the diffusion of the heat equation, recent developments indicate that this issue is difficult and still largely open. Most of the existing literature, concerned with controls of minimal L2-norm, make use of dual convex arguments and introduce backward adjoint system. In practice, the null control problem is then reduced to the minimization of a dual conjugate function with respect to the final condition of the adjoint state. As a consequence of the highly regularizing property of the heat kernel, this final condition – which may be seen as the Lagrange multiplier for the null controllability condition – does not belong to L2, but to a much larger space than can hardly be approximated by finite (discrete) dimensional basis. This phenomenon, unavoidable whatever be the numerical approximation used, strongly deteriorates the efficiency of minimization algorithms. In this work, we do not use duality arguments and in particular do not introduce any backward heat equation. For the boundary case, the approach consists first in introducing a class of functions satisfying a priori the boundary conditions in space and time, in particular the null controllability condition at time T, and then finding among this class one element satisfying the heat equation. This second step is done by minimizing a convex functional among the admissible corrector functions of the heat equation. The inner case is performed in a similar way. We present the (variational) approach, discuss the main features of it and then describe some numerical experiments highlighting the interest of the method. The method holds in any dimension but, for the sake of simplicity, we provide details in the one-space dimensional case.

Stochastic Averaging for a Hamiltonian System with Skew Random Perturbations

We consider a stochastic process that arises due to small random perturbations of a particle moving along the orbits of a double-well Hamiltonian. The perturbing noise is assumed to have skewness along the infinity-shaped homoclinic orbit of the Hamiltonian. As the noise strength goes to zero, we characterize the limiting dynamics in terms of a graph-valued Markov process. The effort centers around the construction of certain perturbed test functions needed to prove convergence. Following Sowers, we accomplish this through the development of suitable corrector functions in a boundary layer near the homoclinic orbit.

A Small-Molecule Modulator Interacts Directly with ΔPhe508-CFTR to Modify Its ATPase Activity and Conformational Stability

The deletion of Phe-508 (DeltaPhe508) constitutes the most prevalent of a number of mutations in the cystic fibrosis transmembrane conductance regulator (CFTR) that cause cystic fibrosis (CF). This mutation leads to CFTR misfolding and retention in the endoplasmic reticulum, as well as impaired channel activity. The biosynthetic defect can be partially overcome by small-molecule "correctors"; once at the cell surface, small-molecule "potentiators" enhance the channel activity of DeltaPhe508-CFTR. Certain compounds, such as VRT-532, exhibit both corrector and potentiator functions. In the current studies, we confirmed that the inherent chloride channel activity of DeltaPhe508-CFTR (after biosynthetic rescue) is potentiated in studies of intact cells and membrane vesicles. It is noteworthy that we showed that the ATPase activity of the purified and reconstituted mutant protein is directly modulated by binding of VRT-532 [4-methyl-2-(5-phenyl-1H-pyrazol-3-yl)-phenol] ATP turnover by reconstituted DeltaPhe508-CFTR is decreased by VRT-532 treatment, an effect that may account for the increase in channel open time induced by this compound. To determine whether the modification of DeltaPhe508-CFTR function caused by direct VRT-532 binding is associated with structural changes, we evaluated the effect of VRT-532 binding on the protease susceptibility of the major mutant. We found that binding of VRT-532 to DeltaPhe508-CFTR led to a minor but significant decrease in the trypsin susceptibility of the full-length mutant protein and a fragment encompassing the second half of the protein. These findings suggest that direct binding of this small molecule induces and/or stabilizes a structure that promotes the channel open state and may underlie its efficacy as a corrector of DeltaPhe508-CFTR.

STOCHASTIC AVERAGING NEAR LONG HETEROCLINIC ORBITS

We refine some of the bounds of [10]. There, we considered the effect of diffusive perturbations on a two-dimensional ODE with a heteroclinic cycle. We constructed corrector functions for asymptotically "glueing" together behavior of periodic orbits in the boundary layer near the heteroclinic cycle. Here, we adapt the analysis of [10] to allow for "long" heteroclinic cycles.

A boundary layer theory for diffusively perturbed transport around a heteroclinic cycle

AbstractWe consider a two‐dimensional transport equation subject to small diffusive perturbations. The transport equation is given by a Hamiltonian flow near a compact and connected heteroclinic cycle. We investigate approximately harmonic functions corresponding to the generator of the perturbed transport equation. In particular, we investigate such functions in the boundary layer near the heteroclinic cycle; the space of these functions gives information about the likelihood of a particle moving a mesoscopic distance into one of the regions where the transport equation corresponds to periodic oscillations (i.e., a “well” of the Hamiltonian). We find that we can construct such approximately harmonic functions (which can be used as “corrector functions” in certain averaging questions) when certain macroscopic “gluing conditions” are satisfied. This provides a different perspective on some previous work of Freidlin and Wentzell on stochastic averaging of Hamiltonian systems. © 2004 Wiley Periodicals, Inc.

A General Method of Improving the Accuracy of Approximate Solutions of Circular Cylindrical Shells

Although Flugge’s differential equations have been known for many years, very few problems have been solved with them due to the high complexity of these equations. Hence a set of simplified differential equations (Donnell’s equations) were developed and have gained wide usage. However, it has been found that the solutions of the simplified equations may be in error by 25% for point load problems, and even more for very long finite shells. This paper introduces a general method of improving the accuracy of the approximate solutions without substantially increasing the amount of labor, by employing only the simplest set of differential equations. Through the introduction of corrector functions and application of an iterative procedure, the improved solutions are expressed in a general form consisting of the solution of Donnell’s equations plus a series of correction terms. It is shown through examples that the improved solutions are very close to those of Flugge’s equations, and that they may be obtained i...