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Efficiency of a micro-macro acceleration method for scale-separated stochastic differential equationsFeb 21 2019We discuss through multiple numerical examples the accuracy and efficiency of a micro-macro acceleration method for stiff stochastic differential equations (SDEs) with a time-scale separation between the fast microscopic dynamics and the evolution of ... More
A Joint Deep Learning Approach for Automated Liver and Tumor SegmentationFeb 21 2019Hepatocellular carcinoma (HCC) is the most common type of primary liver cancer in adults, and the most common cause of death of people suffering from cirrhosis. The segmentation of liver lesions in CT images allows assessment of tumor load, treatment ... More
An adaptive stochastic Galerkin tensor train discretization for randomly perturbed domainsFeb 20 2019A linear PDE problem for randomly perturbed domains is considered in an adaptive Galerkin framework. The perturbation of the domain's boundary is described by a vector valued random field depending on a countable number of random variables in an affine ... More
Interpolation of scattered data in $\mathbb{R}^3$ using minimum $L_p$-norm networks, $1<p<\infty$Feb 19 2019We consider the extremal problem of interpolation of scattered data in $\mathbb{R}^3$ by smooth curve networks with minimal $L_p$-norm of the second derivative for $1<p<\infty$. The problem for $p=2$ was set and solved by Nielson (1983). Andersson et ... More
Gain function approximation in the Feedback Particle FilterFeb 19 2019This paper is concerned with numerical algorithms for the problem of gain function approximation in the feedback particle filter. The exact gain function is the solution of a Poisson equation involving a probability-weighted Laplacian. The numerical problem ... More
A Sequential Homotopy Method for Mathematical Programming ProblemsFeb 19 2019We propose a sequential homotopy method for the solution of mathematical programming problems formulated in abstract Hilbert spaces under the Guignard constraint qualification. The method is equivalent to performing projected backward Euler timestepping ... More
Manifold interpolation and model reductionFeb 18 2019One approach to parametric and adaptive model reduction is via the interpolation of orthogonal bases, subspaces or positive definite system matrices. In all these cases, the sampled inputs stem from matrix sets that feature a geometric structure and thus ... More
Local Fourier Slice PhotographyFeb 16 2019Light field cameras provide intriguing possibilities, such as post-capture refocus or the ability to look behind an object. This comes, however, at the price of significant storage requirements. Compression techniques can be used to reduce these but refocusing ... More
Fractional Operators Applied to Geophysical ElectromagneticsFeb 13 2019A growing body of applied mathematics literature in recent years has focussed on the application of fractional calculus to problems of anomalous transport. In these analyses, the anomalous transport (of charge, tracers, fluid, etc.) is presumed attributable ... More
Topology Optimization under Uncertainty using a Stochastic Gradient-based ApproachFeb 11 2019Topology optimization under uncertainty (TOuU) often defines objectives and constraints by statistical moments of geometric and physical quantities of interest. Most traditional TOuU methods use gradient-based optimization algorithms and rely on accurate ... More
Equivalent Polyadic Decompositions of Matrix Multiplication TensorsFeb 11 2019Invariance transformations of polyadic decompositions of matrix multiplication tensors define an equivalence relation on the set of such decompositions. In this paper, we present an algorithm to efficiently decide whether two polyadic decompositions of ... More
Acceleration via Symplectic Discretization of High-Resolution Differential EquationsFeb 11 2019We study first-order optimization methods obtained by discretizing ordinary differential equations (ODEs) corresponding to Nesterov's accelerated gradient methods (NAGs) and Polyak's heavy-ball method. We consider three discretization schemes: an explicit ... More
Fully Adaptive Multilevel Stochastic Collocation Method for Randomized Elliptic PDEsFeb 09 2019In this paper, we propose and analyse a new adaptive multilevel stochastic collocation method for randomized elliptic PDEs. A hierarchical sequence of adaptive mesh refinements for the spatial approximation is combined with adaptive anisotropic sparse ... More
Numerically Stable Recurrence Relations for the Communication Hiding Pipelined Conjugate Gradient MethodFeb 08 2019Pipelined Krylov subspace methods (also referred to as communication-hiding methods) have been proposed in the literature as a scalable alternative to classic Krylov subspace algorithms for iteratively computing the solution to a large linear system in ... More
A robust and non-singular formulation of the boundary integral method for the potential problemFeb 08 2019A non-singular formulation of the boundary integral method (BIM) is presented for the Laplace equation whereby the well-known singularities that arise from the fundamental solution are eliminated analytically. A key advantage of this approach is that ... More
Lattices from tight frames and vertex transitive graphsFeb 07 2019We show that real tight frames that generate lattices must be rational. In the case of irreducible group frames, we show that the corresponding lattice is always strongly eutactic. We use this observation to describe a construction of strongly eutactic ... More
On interface conditions for flows in coupled free-porous mediaFeb 07 2019Many processes in nature (e.g., physical and biogeochemical processes in hyporheic zones, and arterial mass transport) occur near the interface of free-porous media. A firm understanding of these processes needs an accurate prescription of flow dynamics ... More
Dual-Reference Design for Holographic Coherent Diffraction ImagingFeb 07 2019A new reference design is introduced for Holographic Coherent Diffraction Imaging. This consists of two reference portions - being "block" and "pinhole" shaped regions - adjacent to the imaging specimen. Expected error analysis on data following a Poisson ... More
On maximum volume submatrices and cross approximation for symmetric semidefinite and diagonally dominant matricesFeb 06 2019The problem of finding a $k \times k$ submatrix of maximum volume of a matrix $A$ is of interest in a variety of applications. For example, it yields a quasi-best low-rank approximation constructed from the rows and columns of $A$. We show that such a ... More
A Fast Volume Integral Equation Solver with Linear Basis Functions for the Accurate Computation of Electromagnetic Fields in MRIFeb 06 2019Objective: This paper proposes a stable volume integral equation (VIE) solver based on polarization/magnetization currents, for the accurate and efficient computation of the electromagnetic scattering from highly inhomogeneous and high contrast objects. ... More
A Multilevel Monte Carlo Algorithm for Parabolic Advection-Diffusion Problems with Discontinuous CoefficientsFeb 06 2019The Richards' equation is a model for flow of water in unsaturated soils. The coefficients of this (nonlinear) partial differential equation describe the permeability of the medium. Insufficient or uncertain measurements are commonly modeled by random ... More
Rigorous numerics of blow-up solutions for ODEs with exponential nonlinearityFeb 05 2019Our concerns here are blow-up solutions for ODEs with exponential nonlinearity from the viewpoint of dynamical systems and their numerical validations. As an example, the finite difference discretization of $u_t = u_{xx} + e^{u^m}$ with the homogeneous ... More
A non-iterative method for robustly computing the intersections between a line and a curve or surfaceFeb 05 2019The need to compute the intersections between a line and a high-order curve or surface arises in a large number of finite element applications. Such intersection problems are easy to formulate but hard to solve robustly. We introduce a non-iterative method ... More
Randomized Riemannian Preconditioning for Quadratically Constrained ProblemsFeb 05 2019Optimization problem with quadratic equality constraints are prevalent in machine learning. Indeed, two important examples are Canonical Correlation Analysis (CCA) and Linear Discriminant Analysis (LDA). Unfortunately, methods for solving such problems ... More
Distributed-memory parallelization of the aggregated unfitted finite element methodFeb 04 2019The aggregated unfitted finite element method (AgFEM) is a methodology recently introduced in order to address conditioning and stability problems associated with embedded, unfitted, or extended finite element methods. The method is based on removal of ... More
Is There an Analog of Nesterov Acceleration for MCMC?Feb 04 2019We formulate gradient-based Markov chain Monte Carlo (MCMC) sampling as optimization on the space of probability measures, with Kullback-Leibler (KL) divergence as the objective function. We show that an underdamped form of the Langevin algorithm perform ... More
Stochastic first-order methods: non-asymptotic and computer-aided analyses via potential functionsFeb 03 2019We provide a novel computer-assisted technique for systematically analyzing first-order methods for optimization. In contrast with previous works, the approach is particularly suited for handling sublinear convergence rates and stochastic oracles. The ... More
MorteX method for contact along real and embedded surfaces: coupling X-FEM with the Mortar methodFeb 03 2019A method to treat frictional contact problems along embedded surfaces in the finite element framework is developed. Arbitrarily shaped embedded surfaces, cutting through finite element meshes, are handled by the X-FEM. The frictional contact problem is ... More
Stabilized MorteX method for mesh tying along embedded interfacesFeb 03 2019We present a unified framework to tie overlapping meshes in solid mechanics applications. This framework is a combination of the X-FEM method and the mortar method, which uses Lagrange multipliers to fulfill the tying constraints. As known, mixed formulations ... More
Convergence study and optimal weight functions of an explicit particle method for the incompressible Navier--Stokes equationsFeb 03 2019Feb 05 2019To increase the reliability of simulations by particle methods for incompressible viscous flow problems, convergence studies and improvements of accuracy are considered for a fully explicit particle method for incompressible Navier--Stokes equations. ... More
Alternating Directions Implicit Integration in a General Linear Method FrameworkFeb 02 2019Alternating Directions Implicit (ADI) integration is an operator splitting approach to solve parabolic and elliptic partial differential equations in multiple dimensions based on solving sequentially a set of related one-dimensional equations. Classical ... More
Parametric FEM for Shape Optimization applied to Golgi StackFeb 02 2019The thesis is about an application of the shape optimization to the morphological evolution of Golgi stack. Golgi stack consists of multiple layers of cisternae. It is an organelle in the biological cells. Inspired by the Helfrich Model \cite{Helfrich}, ... More
A block preconditioner for non-isothermal flow in porous mediaJan 31 2019In petroleum reservoir simulation, the industry standard preconditioner, the constrained pressure residual method (CPR), is a two-stage process which involves solving a restricted pressure system with Algebraic Multigrid (AMG). Initially designed for ... More
A nested Schur complement solver with mesh-independent convergence for the time domain photonics modelingJan 31 2019A nested Schur complement solver is proposed for iterative solution of linear systems arising in exponential and implicit time integration of the Maxwell equations with perfectly matched layer (PML) nonreflecting boundary conditions. These linear systems ... More
Approximation to uniform distribution in SO(3)Jan 30 2019Using the theory of determinantal point processes we give upper bounds for the Green and Riesz energies for the rotation group SO(3), with Riesz parameter up to 3. The Green function is computed explicitly, and a lower bound for the Green energy is established, ... More
An approximate Itô-SDE based simulated annealing algorithm for multivariate design optimization problemsJan 30 2019This research concerns design optimization problems involving numerous design parameters and large computational models. These problems generally consist in non-convex constrained optimization problems in large and sometimes complex search spaces. The ... More
Practicable Simulation-Free Model Order Reduction by Nonlinear Moment MatchingJan 30 2019In this paper, a practicable simulation-free model order reduction method by nonlinear moment matching is developed. Based on the steady-state interpretation of linear moment matching, we comprehensively explain the extension of this reduction concept ... More
Statistical inference of probabilistic origin-destination demand using day-to-day traffic dataJan 29 2019Recent transportation network studies on uncertainty and reliability call for modeling the probabilistic O-D demand and probabilistic network flow. Making the best use of day-to-day traffic data collected over many years, this paper develops a novel theoretical ... More
Two-sided bounds for cost functionals of time-periodic parabolic optimal control problemsJan 28 2019In this paper, a new technique is applied on deriving computable, guaranteed lower bounds of functional type (minorants) for two different cost functionals subject to a parabolic time-periodic boundary value problem. Together with previous results on ... More
A Faster Solution to Smale's 17th Problem I: Real Binomial SystemsJan 28 2019Suppose $F:=(f_1,\ldots,f_n)$ is a system of random $n$-variate polynomials with $f_i$ having degree $\leq\!d_i$ and the coefficient of $x^{a_1}_1\cdots x^{a_n}_n$ in $f_i$ being an independent complex Gaussian of mean $0$ and variance $\frac{d_i!}{a_1!\cdots ... More
Sundial: Using Sunlight to Reconstruct Global TimestampsJan 28 2019This paper investigates postmortem timestamp reconstruction in environmental monitoring networks. In the absence of a time-synchronization protocol, these networks use multiple pairs of (local, global) timestamps to retroactively estimate the motes' clock ... More
CURE: Curvature Regularization For Missing Data RecoveryJan 28 2019Missing data recovery is an important and yet challenging problem in imaging and data science. Successful models often adopt certain carefully chosen regularization. Recently, the low dimension manifold model (LDMM) was introduced by S.Osher et al. and ... More
Numerical analysis comparing ODE approach and level set method for evolving spirals by crystalline eikonal-curvature flowJan 26 2019In this paper, the evolution of a polygonal spiral curve by the crystalline curvature flow with a pinned center is considered with two view points, discrete model consist of an ODE system of facet lengths and a level set method. We investigate the difference ... More
Plantinga-Vegter algorithm takes average polynomial timeJan 26 2019We exhibit a condition-based analysis of the adaptive subdivision algorithm due to Plantinga and Vegter. The first complexity analysis of the PV Algorithm is due to Burr, Gao and Tsigaridas who proved a $O\big(2^{\tau d^{4}\log d}\big)$ worst-case cost ... More
Using adjoint CFD to quantify the impact of manufacturing variations on a heavy duty turbine vaneJan 25 2019We consider the evaluation of manufacturing variations to the aerodynamic performace of turbine vanes using the adjoint method. The empirical data is based on 102 white light scans from casted parts. We compare expensive calculations by the finite disfference ... More
The conjugate gradient algorithm on well-conditioned Wishart matrices is almost deteriministicJan 25 2019Feb 01 2019We prove that the number of iterations required to solve a random positive definite linear system with the conjugate gradient algorithm is almost deterministic for large matrices. We treat the case of Wishart matrices $W = XX^*$ where $X$ is $n \times ... More
Pricing options and computing implied volatilities using neural networksJan 25 2019This paper proposes a data-driven approach, by means of an Artificial Neural Network (ANN), to value financial options and to calculate implied volatilities with the aim of accelerating the corresponding numerical methods. With ANNs being universal function ... More
Scalable solvers for complex electromagnetics problemsJan 25 2019In this work, we present scalable balancing domain decomposition by constraints methods for linear systems arising from arbitrary order edge finite element discretizations of multi-material and heterogeneous 3D problems. In order to enforce the continuity ... More
Strong convergence rate of Euler-Maruyama method for stochastic differential equations with Hölder continuous drift coefficient driven by symmetric $α$-stable processJan 25 2019Euler-Maruyama method is studied to approximate stochastic differential equations driven by the symmetric $\alpha$-stable additive noise with the $\beta$ H\"older continuous drift coefficient. When $\alpha \in (1,2)$ and $\beta \in (0,\alpha/2)$, for ... More
Distributed Matrix-Vector Multiplication: A Convolutional Coding ApproachJan 25 2019Distributed computing systems are well-known to suffer from the problem of slow or failed nodes; these are referred to as stragglers. Straggler mitigation (for distributed matrix computations) has recently been investigated from the standpoint of erasure ... More
Parallelization and scalability analysis of inverse factorization using the Chunks and Tasks programming modelJan 23 2019Jan 24 2019We present three methods for distributed memory parallel inverse factorization of block-sparse Hermitian positive definite matrices. The three methods are a recursive variant of the AINV inverse Cholesky algorithm, iterative refinement, and localized ... More
Coupling the reduced-order model and the generative model for an importance sampling estimatorJan 23 2019In this work, we develop an importance sampling estimator by coupling the reduced-order model and the generative model in a problem setting of uncertainty quantification. The target is to estimate the probability that the quantity of interest (QoI) in ... More
On orthogonal projections for dimension reduction and applications in variational loss functions for learning problemsJan 22 2019The use of orthogonal projections on high-dimensional input and target data in learning frameworks is studied. First, we investigate the relations between two standard objectives in dimension reduction, maximizing variance and preservation of pairwise ... More
Convergence and stability of a micro-macro acceleration method:linear slow-fast stochastic differential equations with additive noiseJan 22 2019We analyse the convergence and stability of a micro-macro acceleration algorithm for Monte Carlo simulations of stiff stochastic differential equations with a time-scale separation between the fast evolution of the individual stochastic realizations and ... More
B-spline-like bases for $C^2$ cubics on the Powell-Sabin 12-splitJan 21 2019For spaces of constant, linear, and quadratic splines of maximal smoothness on the Powell-Sabin 12-split of a triangle, the so-called S-bases were recently introduced. These are simplex spline bases with B-spline-like properties on the 12-split of a single ... More
A Deterministic Approach to Avoid Saddle PointsJan 21 2019Loss functions with a large number of saddle points are one of the main obstacles to training many modern machine learning models. Gradient descent (GD) is a fundamental algorithm for machine learning and converges to a saddle point for certain initial ... More
Iterative Refinement for $\ell_p$-norm RegressionJan 21 2019We give improved algorithms for the $\ell_{p}$-regression problem, $\min_{x} \|x\|_{p}$ such that $A x=b,$ for all $p \in (1,2) \cup (2,\infty).$ Our algorithms obtain a high accuracy solution in $\tilde{O}_{p}(m^{\frac{|p-2|}{2p + |p-2|}}) \le \tilde{O}_{p}(m^{\frac{1}{3}})$ ... More
Deterministic constructions of high-dimensional sets with small dispersionJan 20 2019The dispersion of a point set $P\subset[0,1]^d$ is the volume of the largest box with sides parallel to the coordinate axes, which does not intersect $P$. Here, we show a construction of low-dispersion point sets, which can be deduced from solutions of ... More
Holographic Phase Retrieval and Optimal Reference DesignJan 19 2019A general mathematical framework and recovery algorithm is presented for the holographic phase retrieval problem. In this problem, which arises in holographic coherent diffraction imaging, a "reference" portion of the signal to be recovered via phase ... More
Unreasonable effectiveness of Monte CarloJan 18 2019This is a comment on the article "Probabilistic Integration: A Role in Statistical Computation?" by F.-X. Briol, C. J. Oates, M. Girolami, M. A. Osborne and D. Sejdinovic to appear in Statistical Science. There is a role for statistical computation in ... More
Algorithms for high-dimensional non-linear filtering and smoothing problemsJan 18 2019Several numerical tools designed to overcome the challenges of smoothing in a high dimensional nonlinear setting are investigated for a class of particle smoothers. The considered family of smoothers is induced by the class of Linear Ensemble Transform ... More
Strongly Asymptotically Optimal Schemes for the Strong Approximation of Non-Lipschitzian Stochastic Differential Equations with respect to the Supremum ErrorJan 18 2019Our subject of study is strong approximation of systems of stochastic differential equations (SDEs) with respect to the supremum error criterion, and we seek approximations that perform strongly asymptotically optimal. In this context, we focus on two ... More
Weak convergence rates for temporal numerical approximations of stochastic wave equations with multiplicative noiseJan 16 2019In numerical analysis for stochastic partial differential equations one distinguishes between weak and strong convergence rates. Often the weak convergence rate is twice the strong convergence rate. However, there is no standard way to prove this: to ... More
A Newton method for harmonic mappings in the planeJan 16 2019We present an iterative root finding method for harmonic mappings in the complex plane, which is a generalization of Newton's method for analytic functions. The complex formulation of the method allows an analysis in a complex variables spirit. For zeros ... More
Algorithms for $\ell_p$-based semi-supervised learning on graphsJan 15 2019We develop fast algorithms for solving the variational and game-theoretic $p$-Laplace equations on weighted graphs for $p>2$. The graph $p$-Laplacian for $p>2$ has been proposed recently as a replacement for the standard ($p=2$) graph Laplacian in semi-supervised ... More
Discrete Spectra of Convolutions on Disks using Sturm-Liouville TheoryJan 15 2019This paper presents a systematic study for analytic aspects of discrete spectra methods for convolution of functions supported on disks, according to the Sturm-Liouville theory. We then investigate different aspects of the presented theory in the cases ... More
A Hybrid High-Order method for finite elastoplastic deformations within a logarithmic strain frameworkJan 14 2019We devise and evaluate numerically a Hybrid High-Order (HHO) method for finite plasticity within a logarithmic strain framework. The HHO method uses as discrete unknowns piecewise polynomials of order $k\ge1$ on the mesh skeleton, together with cell-based ... More
On the Convergence of the Laplace Approximation and Noise-Level-Robustness of Laplace-based Monte Carlo Methods for Bayesian Inverse ProblemsJan 13 2019The Bayesian approach to inverse problems provides a rigorous framework for the incorporation and quantification of uncertainties in measurements, parameters and models. We are interested in designing numerical methods which are robust w.r.t. the size ... More
Stability estimates for phase retrieval from discrete Gabor measurementsJan 12 2019Phase retrieval refers to the problem of recovering some signal (which is often modelled as an element of a Hilbert space) from phaseless measurements. It has been shown that, in the deterministic setting, phase retrieval from frame coefficients is always ... More
A discretization of Caputo derivatives with application to time fractional SDEs and gradient flowsJan 10 2019We consider a discretization of Caputo derivatives resulted from deconvolving a scheme for the corresponding Volterra integral. Some important properties of this discretization are proved by its linkage to Volterra integrals with completely monotone kernels. ... More
Statistical closure modeling for reduced-order models of stationary systems by the ROMES methodJan 09 2019This work proposes a technique for constructing a statistical closure model for reduced-order models (ROMs) applied to stationary systems modeled as parameterized systems of algebraic equations. The proposed technique extends the reduced-order-model error ... More
A sparse FFT approach for ODE with random coefficientsJan 06 2019The paper presents a general strategy to solve ordinary differential equations (ODE), where some coefficient depend on the spatial variable and on additional random variables. The approach is based on the application of a recently developed dimension-incremental ... More
Quantum spectral methods for differential equationsJan 04 2019Recently developed quantum algorithms address computational challenges in numerical analysis by performing linear algebra in Hilbert space. Such algorithms can produce a quantum state proportional to the solution of a $d$-dimensional system of linear ... More
A mesh-free method for interface problems using the deep learning approachJan 03 2019In this paper, we propose a mesh-free method to solve interface problems using the deep learning approach. Two interface problems are considered. The first one is an elliptic PDE with a discontinuous and high-contrast coefficient. While the second one ... More
Nearly optimal lattice simulation by product formulasJan 03 2019Product formulas provide a straightforward yet surprisingly efficient approach to quantum simulation. We show that this algorithm can simulate an $n$-qubit Hamiltonian with nearest-neighbor interactions evolving for time $t$ using only $(nt)^{1+o(1)}$ ... More
High order numerical schemes for solving fractional powers of elliptic operatorsJan 01 2019In many recent applications when new materials and technologies are developed it is important to describe and simulate new nonlinear and nonlocal diffusion transport processes. A general class of such models deals with nonlocal fractional power elliptic ... More
Allocation strategies for high fidelity models in the multifidelity regimeDec 30 2018We propose a novel approach to allocating resources for expensive simulations of high fidelity models when used in a multifidelity framework. Allocation decisions that distribute computational resources across several simulation models become extremely ... More
New conformal map for the Sinc approximation for exponentially decaying functions over the semi-infinite intervalDec 30 2018Jan 08 2019The Sinc approximation has shown high efficiency for numerical methods in many fields. Conformal maps play an important role in the success, i.e., appropriate conformal map must be employed to elicit high performance of the Sinc approximation. Appropriate ... More
Vilin: Unconstrained Numerical Optimization ApplicationDec 28 2018We introduce an application for executing and testing different unconstrained optimization algorithms. The application contains a library of various test functions with pre-defined starting points. A several known classes of methods as well as different ... More
Isogeometric Mortar Coupling for Electromagnetic ProblemsDec 27 2018This paper discusses and analyses two domain decomposition approaches for electromagnetic problems that allow the combination of domains discretised by either N\'ed\'elec-type polynomial finite elements or spline-based isogeometric analysis. The first ... More
Optimal approximation of stochastic integrals in analytic noise modelDec 27 2018We study approximate stochastic It\^o integration of processes belonging to a class of progressively measurable stochastic processes that are H\"older continuous in the $r$th mean. Inspired by increasingly popularity of computations with low precision ... More
Random batch methods (RBM) for interacting particle systemsDec 26 2018We 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 ... More
ART: adaptive residual--time restarting for Krylov subspace matrix exponential evaluationsDec 25 2018In this paper a new restarting method for Krylov subspace matrix exponential evaluations is proposed. Since our restarting technique essentially employs the residual, some convergence results for the residual are given. We also discuss how the restart ... More
Adaptive time-stepping for Stochastic Partial Differential Equations with non-Lipschitz driftDec 21 2018We introduce an explicit, adaptive time-stepping scheme for simulation of SPDEs with non-Lipschitz drift coefficients. Strong convergence is proven for the full space-time discretisation with multiplicative noise by considering the space and time discretisation ... More
Efficient and scalable data structures and algorithms for goal-oriented adaptivity of space-time FEM codesDec 20 2018The cost- and memory-efficient numerical simulation of coupled volume-based multi-physics problems like flow, transport, wave propagation and others remains a challenging task with finite element method (FEM) approaches. Goal-oriented space and time adaptive ... More
Stochastic Galerkin reduced basis methods for parametrized linear elliptic PDEsDec 20 2018We consider the estimation of parameter-dependent statistics of functional outputs of elliptic boundary value problems with parametrized random and deterministic inputs. For a given value of the deterministic parameter, a stochastic Galerkin finite element ... More
On bipartization of networksDec 20 2018Relations between discrete quantities such as people, genes, or streets can be described by networks, which consist of nodes that are connected by edges. Network analysis aims to identify important nodes in a network and to uncover structural properties ... More
Model reduction of dynamical systems on nonlinear manifolds using deep convolutional autoencodersDec 20 2018Nearly all model-reduction techniques project the governing equations onto a linear subspace of the original state space. Such subspaces are typically computed using methods such as balanced truncation, rational interpolation, the reduced-basis method, ... More
An arbitrary order time-stepping algorithm for tracking particles in inhomogeneous magnetic fieldsDec 19 2018The Lorentz equations describe the motion of electrically charged particles in electric and magnetic fields and are used widely in plasma physics. The most popular numerical algorithm for solving them is the Boris method, a variant of the St\"ormer-Verlet ... More
Lattice Identification and Separation: Theory and AlgorithmDec 19 2018Motivated by lattice mixture identification and grain boundary detection, we present a framework for lattice pattern representation and comparison, and propose an efficient algorithm for lattice separation. We define new scale and shape descriptors, which ... More
Breaking Reversibility Accelerates Langevin Dynamics for Global Non-Convex OptimizationDec 19 2018Dec 26 2018Langevin dynamics (LD) has been proven to be a powerful technique for optimizing a non-convex objective as an efficient algorithm to find local minima while eventually visiting a global minimum on longer time-scales. LD is based on the first-order Langevin ... More
Semi-Riemannian Manifold OptimizationDec 18 2018We introduce in this paper a manifold optimization framework that utilizes semi-Riemannian structures on the underlying smooth manifolds. Unlike in Riemannian geometry, where each tangent space is equipped with a positive definite inner product, a semi-Riemannian ... More
Pathwise space approximations of semi-linear parabolic SPDEs with multiplicative noiseDec 18 2018We provide convergence rates for space approximations of semi-linear stochastic differential equations with multiplicative noise in a Hilbert space. The space approximations we consider are spectral Galerkin and finite elements, and the type of convergence ... More
Method of Green's potentials for elliptic PDEs in domains with random boundariesDec 18 2018Problems with topological uncertainties appear in many fields ranging from nano-device engineering to the design of bridges. In many of such problems, a part of the domains boundaries is subjected to random perturbations making inefficient conventional ... More
Strong convergence rate of a full discretization for stochastic Cahn--Hilliard equation driven by space-time white noiseDec 15 2018Jan 02 2019In this article, we consider the stochastic Cahn--Hilliard equation driven by space-time white noise. We discretize this equation by using a spatial spectral Galerkin method and a temporal accelerated implicit Euler method. The optimal regularity properties ... More
Algorithmic Theory of ODEs and Sampling from Well-conditioned Logconcave DensitiesDec 15 2018Sampling logconcave functions arising in statistics and machine learning has been a subject of intensive study. Recent developments include analyses for Langevin dynamics and Hamiltonian Monte Carlo (HMC). While both approaches have dimension-independent ... More
Impulse response of bilinear systems based on Volterra series representationDec 13 2018Jan 17 2019This paper focuses on the systems theory of bilinear dynamical systems using the Volterra series representation. The main contributions are threefold. First, we gain an input-output representation in the frequency domain, where the Laplace transform of ... More
Winding number and Cutting number of Harmonic cycleDec 12 2018Dec 13 2018A harmonic cycle $\lambda$, also called a discrete harmonic form, is a solution of the Laplace's equation with the combinatorial Laplace operator obtained from the boundary operators of a chain complex. By the combinatorial Hodge theory, harmonic spaces ... More
Multi-scale variance reduction methods based on multiple control variates for kinetic equations with uncertaintiesDec 12 2018The development of efficient numerical methods for kinetic equations with stochastic parameters is a challenge due to the high dimensionality of the problem. Recently we introduced a multiscale control variate strategy which is capable to accelerate considerably ... More
On the regularisation of the noise for the Euler-Maruyama scheme with irregular driftDec 11 2018The strong rate of convergence of the Euler-Maruyama scheme for nondegenerate SDEs with irregular drift coefficients is considered. In the case of $\alpha$-H\"older drift in recent literature the rate $\alpha/2$ was proved in many related situations. ... More