Search for: All records

The nonlinear Poisson-Boltzmann equation (NPBE) is an elliptic partial differential equation used in applications such as protein interactions and biophysical chemistry (among many others). It describes the nonlinear electrostatic potential of charged bodies submerged in an ionic solution. The kinetic presence of the solvent molecules introduces randomness to the shape of a protein, and thus a more accurate model that incorporates these random perturbations of the domain is analyzed to compute the statistics of quantities of interest of the solution. When the parameterization of the random perturbations is high-dimensional, this calculation is intractable as it is subject to the curse of dimensionality. However, if the solution of the NPBE varies analytically with respect to the random parameters, the problem becomes amenable to techniques such as sparse grids and deep neural networks. In this paper, we show analyticity of the solution of the NPBE with respect to analytic perturbations of the domain by using the analytic implicit function theorem and the domain mapping method. Previous works have shown analyticity of solutions to linear elliptic equations with interfaces but not for nonlinear problems. We further show how to derive a priori bounds on the size of the region of analyticity. This method is applied to the Cucurbita Maxima Trypsin Inhibitor I (CMTI-I) molecule to demonstrate that the convergence rates of the quantity of interest are consistent with the analyticity result. Furthermore, the approach developed here is general enough to be applied to other nonlinear problems in uncertainty quantification. 

The nonlinear Poisson-Boltzmann equation (nPBE) is a fundamental partial differential equation (PDE) in electrostatics, widely used in computational biology and chemistry to model potential fields in solvents or plasmas. In this paper, we consider the problem of quantifying the statistical uncertainty of the stochastic nPBE solution under random variations in its coefficients. We establish the existence and uniqueness of solutions of the complexified nPBE using a contraction mapping argument, as conventional convex optimization techniques for the real-valued nPBE do not naturally extend to the complex setting. Using the existence and uniqueness result, we demonstrate that the solutions admit analytic extensions over a well-defined region in the complex hyperplane The analyticity makes the computation for statistics of real-valued quantities of interest amenable to numerical techniques such as sparse grids. Sparse grids are applied to uniformly approximate analytic functions with algebraic to sub-exponential error with respect to the number of knots, thus allowing for efficient approximations of high-dimensional integrals. Our numerical experiments confirm the predicted error behavior. 

Providing rich and useful information regarding spectrum activities and propagation channels, radiomaps characterize the detailed distribution of power spectral density (PSD) and are important tools for network planning in modern wireless systems. Generally, radiomaps are constructed from radio strength measurements by deployed sensors and user devices. However, not all areas are accessible for radio measurements due to physical constraints and security considerations, leading to non-uniformly spaced measurements and blanks on a radiomap. In this work, we explore distribution of radio spectrum strengths in view of surrounding environments, and propose two radiomap inpainting approaches for the reconstruction of radiomaps that cover missing areas. Specifically, we first define a propagation based priority before integrating exemplar-based inpainting with radio propagation model for fine-resolution small-size missing area reconstruction on a radiomap. We next introduce a novel radio depth map and propose a two-step template-perturbation approach for large-size restricted region inpainting. Our experimental results demonstrate the power of the proposed propagation priority and radio depth map in capturing PSD distribution, as well as their efficacy in radiomap reconstruction. 

Abstract We prove that the solutions to the discrete nonlinear Schrödinger equation with non-local algebraically decaying coupling converge strongly in $$L^2({\mathbb {R}}^2)$$ L 2 ( R 2 ) to those of the continuum fractional nonlinear Schrödinger equation, as the discretization parameter tends to zero. The proof relies on sharp dispersive estimates that yield the Strichartz estimates that are uniform in the discretization parameter. An explicit computation of the leading term of the oscillatory integral asymptotics is used to show that the best constants of a family of dispersive estimates blow up as the non-locality parameter $$\alpha \in (1,2)$$ α ∈ ( 1 , 2 ) approaches the boundaries. 

The advent of Persistent Memory (PM) devices enables systems to actively persist information at low costs, including program state traditionally in volatile memory. However, this trend poses a reliability challenge in which multiple classes of soft faults that go away after restart in traditional systems turn into hard (recurring) faults in PM systems. In this paper, we first characterize this rising problem with an empirical study of 28 real-world bugs. We analyze how they cause hard faults in PM systems. We then propose Arthas, a tool to effectively recover PM systems from hard faults. Arthas checkpoints PM states via fine-grained versioning and uses program slicing of fault instructions to revert problematic PM states to good versions. We evaluate Arthas on 12 real-world hard faults from five large PM systems. Arthas successfully recovers the systems for all cases while discarding 10× less data on average compared to state-of-the-art checkpoint-rollback solutions.