Article
Mathematics, Applied
Tucker Hartland, Georg Stadler, Mauro Perego, Kim Liegeois, Noemi Petra
Summary: Obtaining lightweight and accurate approximations of discretized objective functional Hessians is crucial for solving large-scale inverse problems efficiently. This work demonstrates that the hierarchical off-diagonal low-rank (HODLR) matrix approximations can effectively approximate the Hessians arising from inverse problems governed by partial differential equations (PDEs). The HODLR matrix format is utilized for efficient sampling of the posterior distribution and compression of covariance, providing computational advantages over traditional methods.
Article
Mathematics, Applied
Hector Orera, Juan Manuel Pena
Summary: It is proven that any B-pi(R)-matrix has a positive determinant, and norm bounds for the inverses of B-pi(R)-matrices, as well as error bounds for linear complementarity problems associated with B-pi(R)-matrices, are provided for pi > 0. The new bounds in the last case are simpler than previous ones and can be used without prior knowledge of whether a B-pi(R)-matrix exists. Numerical examples show that these new bounds can be significantly more accurate than previous ones.
COMPUTATIONAL & APPLIED MATHEMATICS
(2021)
Article
Engineering, Multidisciplinary
Jan Povala, Ieva Kazlauskaite, Eky Febrianto, Fehmi Cirak, Mark Girolami
Summary: Inverse problems involving partial differential equations (PDEs) are commonly used in science and engineering. While Markov Chain Monte Carlo (MCMC) has been the go-to method for sampling from posterior probability measures, it is computationally infeasible for large-scale problems. Variational Bayes (VB) has emerged as a more computationally tractable alternative, approximating posterior distributions with simpler trial distributions. This work presents a flexible and efficient approach to solving inverse problems using VB methods.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2022)
Article
Computer Science, Artificial Intelligence
Elie Bretin, Pierre Millien, Laurent Seppecher
Summary: In this article, stability estimates for the finite element discretization of a class of inverse parameter problems are provided. The problem involves recovering an unknown parameter mu in a domain Omega of R-d, where the functions S and f are known. A Lipschitz-type stability estimate in a hyperplane of L-2(Omega) is proved, as uniqueness is not guaranteed. The discrete inf-sup constant or LBB constant is adapted to a large class of first order differential operators to obtain this stability.
SIAM JOURNAL ON IMAGING SCIENCES
(2023)
Article
Mathematics
Petr Martyshko, Igor Ladovskii, Denis Byzov
Summary: This paper introduces a gravity data inversion method based on parallel algorithms, ensuring stability by selecting a density model and solution search set. It proposes a new upward and downward continuation algorithm for separating shallow and deep sources, restoring density distribution in a 3D grid. The iterative parallel algorithms enable density value restoration and optimization of gravity field calculation, speeding up the inversion process.
Article
Mathematics
Mengmeng Zhou, Jianlong Chen, Dingguo Wang
Summary: The paper presents the core inverses of linear combinations of two core invertible matrices, as well as the dual core inverses of linear combinations of two dual core invertible matrices. It also provides sufficient conditions for guaranteeing that the difference of two core invertible matrices is core invertible.
LINEAR & MULTILINEAR ALGEBRA
(2021)
Article
Mathematics, Applied
Abeynaya Gnanasekaran, Eric Darve
Summary: In this work, a fast hierarchical solver is developed for solving large, sparse least squares problems. The solver utilizes a multifrontal QR approach and low-rank approximation to reduce computational complexity and memory usage, resulting in an approximate factorization of the matrix stored as a sequence of sparse orthogonal and upper-triangular factors.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
Article
Mathematics
Ekaterina Gribanova
Summary: This research focuses on goal setting in performance management, presenting an algorithm for solving problems related to goal setting and conducting computational experiments to validate the solutions.
Article
Mathematics
Yao Sun, Lijuan He, Bo Chen
Summary: This paper explores the application of machine learning and neural networks in solving inverse elastic scattering problems. It approximates displacements through linear combinations of fundamental tensors and uses a two-layer neural network method to reconstruct the shape of unknown elastic bodies. The convergence of the method is proven, and the feasibility and effectiveness are validated through numerical examples.
ELECTRONIC RESEARCH ARCHIVE
(2023)
Article
Mathematics, Applied
Xianping Wu
Summary: This paper presents error bounds for linear complementarity problems with an H+-matrix based on the absolute value equation for minimizing two vectors. Some computable bounds are provided by specifying the particular diagonal parameter matrix D. The proposed bounds can improve existing ones when D is selected properly.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2022)
Article
Automation & Control Systems
Mohammed Rayyan Sheriff, Debasish Chatterjee
Summary: This article delves into ill-posed Linear Inverse Problems (LIP) and introduces a slightly generalized version of the error constrained linear inverse problem, presenting a novel and equivalent convex-concave min-max reformulation through exposition on its convex geometry. The saddle points of the min-max problem are characterized in terms of a solution to LIP, and applying simple algorithms based on saddle point seeking ascend-descent type provides solutions to the LIP. Additionally, the reformulation of LIP as the min-max problem is crucial for developing methods to solve the dictionary learning problem with almost sure recovery constraints.
JOURNAL OF MACHINE LEARNING RESEARCH
(2022)
Article
Mathematics, Interdisciplinary Applications
Rana Muhammad Zulqarnain, Imran Siddique, Rifaqat Ali, Fahd Jarad, Abdul Samad, Thabet Abdeljawad
Summary: The concept of the neutrosophic hypersoft set (NHSS) is a parameterized family that deals with subattributes of parameters and is an extension of the neutrosophic soft set. The follow-up study develops the theory of neutrosophic hypersoft matrix (NHSM) and introduces logical operators for NHSMs. The proposed decision-making methodology is validated through numerical illustration and comparative analysis with existing studies.
Article
Computer Science, Interdisciplinary Applications
Yoshiki Fukada
Summary: This study introduces an efficient approximation method for the Moore-Penrose pseudo-inverse, which significantly reduces computational cost by adding a small-amplitude diagonal matrix and utilizing a projection matrix to remove components of zero eigenvectors. The method is applied to stiffness matrices in support-free elasticity problems, showing excellent accuracy and efficiency. Conducting robust topology optimization on fine-mesh problems leads to structures with biological features.
COMPUTERS & STRUCTURES
(2021)
Article
Engineering, Geological
Ivan Depina, Saket Jain, Sigurdur Mar Valsson, Hrvoje Gotovac
Summary: This paper investigates the application of Physics-Informed Neural Networks (PINNs) to inverse problems in unsaturated groundwater flow. PINNs solve inverse problems by reformulating the loss function of a deep neural network to simultaneously satisfy measured values and unknown values at collocation points distributed across the problem domain. Results demonstrate that PINNs are capable of efficiently solving the inverse problem with relatively accurate approximation of the solution to the Richards equation and estimates of the van Genuchten model parameters.
GEORISK-ASSESSMENT AND MANAGEMENT OF RISK FOR ENGINEERED SYSTEMS AND GEOHAZARDS
(2022)
Article
Mathematics, Applied
Yingxia Zhao, Lanlan Liu, Feng Wang
Summary: An upper bound of the infinity norm for the inverse of S DD1 matrix is provided, along with a new error bound and lower bound for the smallest singular value. Numerical examples demonstrate the validity of the results.
Article
Environmental Sciences
Lijing Wang, Peter K. Kitanidis, Jef Caers
Summary: Bayesian inversion is commonly used to quantify uncertainty of hydrological variables. This paper proposes a hierarchical Bayesian framework to quantify uncertainty of both global and spatial variables. The authors present a machine learning-based inversion method and a local dimension reduction method to efficiently estimate posterior probabilities and update spatial fields. Using three case studies, they demonstrate the importance of quantifying uncertainty of global variables for predictions and the acceleration effect of the local PCA approach.
WATER RESOURCES RESEARCH
(2022)
Article
Thermodynamics
Pragneshkumar Rajubhai Rana, Krithika Narayanaswamy, Sivaram Ambikasaran
Summary: Ignition delay time is a significant combustion property. This study proposes a data-driven approach to obtain the ignition delay time for new fuels. The proposed algorithm uses regression-based clustering and incorporates fuel structure to establish models. The accuracy of the algorithm is demonstrated using straight-chain alkane data.
COMBUSTION THEORY AND MODELLING
(2022)
Article
Mathematics, Applied
Arvind K. Saibaba, Rachel Minster, Misha E. Kilmer
Summary: This paper introduces a method for multivariate function approximation using Chebyshev polynomials and tensor compression techniques, and provides detailed analysis and experiments. The results show that the proposed method has significant advantages in terms of computational and storage efficiency.
ADVANCES IN COMPUTATIONAL MATHEMATICS
(2022)
Article
Water Resources
Mojtaba Forghani, Yizhou Qian, Jonghyun Lee, Matthew Farthing, Tyler Hesser, Peter K. Kitanidis, Eric F. Darve
Summary: This article presents a reduced-order model (ROM) based approach that utilizes a variational autoencoder (VAE) to compress bathymetry and flow velocity information, allowing for fast solving of bathymetry inverse problems. By constructing ROMs on a nonlinear manifold and employing a Hierarchical Bayesian setting, variational inference and efficient uncertainty quantification can be achieved using a small number of ROM runs.
ADVANCES IN WATER RESOURCES
(2022)
Article
Computer Science, Theory & Methods
Suman Majumder, Yawen Guan, Brian J. Reich, Arvind K. Saibaba
Summary: In this study, we propose a novel approximate inference methodology for analyzing massive spatial datasets using a Gaussian process model. Our method effectively estimates spatial covariance parameters and makes accurate predictions with uncertainty quantification for point-referenced spatial data. It is applicable to various types of observations and covariance functions. Extensive simulation studies and a real data application demonstrate that our method significantly reduces computation time while maintaining scalability.
STATISTICS AND COMPUTING
(2022)
Article
Environmental Sciences
Xueyuan Kang, Amalia Kokkinaki, Xiaoqing Shi, Hongkyu Yoon, Jonghyun Lee, Peter K. Kitanidis, Jichun Wu
Summary: This study presents a framework that combines a deep-learning-based inversion method with a process-based upscaled model to estimate source zone architecture (SZA) metrics and mass discharge from sparse data. By improving the estimation method, the upscaled model accurately reproduces the concentrations and uncertainties of multistage effluents, providing valuable input for decision making in remediation applications.
WATER RESOURCES RESEARCH
(2022)
Article
Mathematics, Applied
Misha E. Kilmer, Arvind K. Saibaba
Summary: This paper presents a computational framework for approximating structured matrices by mapping them to tensors and using tensor compression algorithms. The resulting matrix approximations are memory efficient, easy to compute with, and preserve the error that is due to tensor compression. The framework is general and can be applied to various applications.
SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS
(2022)
Article
Mathematics, Applied
Hussam Al Daas, Grey Ballard, Paul Cazeaux, Eric Hallman, Agnieszka Miedlar, Mirjeta Pasha, Tim W. Reid, Arvind K. Saibaba
Summary: The tensor-train format is widely used for approximating high-dimensional tensors in the solution of parametrized partial differential equations. Rounding, a fundamental operation used to maintain memory and computational efficiency, truncates the internal ranks of a tensor in tensor-train format. In this study, we propose randomized algorithms for rounding that provide significant reduction in computation compared to deterministic algorithms, especially when rounding a sum of tensor-train tensors.
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2023)
Article
Mechanics
Mohammad Farazmand, Arvind K. Saibaba
Summary: Reconstructing high-resolution flow fields from sparse measurements is a major challenge in fluid dynamics. We introduce a tensor-based method that retains and exploits the multidimensionality of the flow, resulting in more accurate and less variable flow reconstructions compared to vectorized methods. Our method also significantly reduces the storage cost while maintaining comparable computational cost.
JOURNAL OF FLUID MECHANICS
(2023)
Article
Mathematics, Applied
Eric Hallman, Ilse C. F. Ipsen, Arvind K. Saibaba
Summary: This paper presents Monte Carlo estimators for computing the diagonal elements of real symmetric matrices. The probabilistic bounds and experimental results show that the accuracy of the estimators increases with the diagonal dominance of the matrix.
SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS
(2023)
Article
Environmental Sciences
Simon Meunier, Peter K. Kitanidis, Amaury Cordier, Alan M. MacDonald
Summary: This study develops a numerical model to simulate the abstraction capacities of photovoltaic water pumping systems across Africa using openly available data. The model includes realistic geological constraints on pumping depth and sub-hourly irradiance time series. The simulation results show that for much of Africa, groundwater pumping using photovoltaic energy is limited by aquifer conditions rather than irradiance. These findings can help identify regions with high potential for photovoltaic pumping and guide large-scale investments.
COMMUNICATIONS EARTH & ENVIRONMENT
(2023)
Article
Mathematics, Applied
Harbir Antil, Arvind K. Saibaba
Summary: This paper focuses on developing reduced order models for repeated simulation of parameterized elliptic PDEs using the Kato integral formula and discretization methods. The approach improves computational and memory efficiency, and numerical experiments demonstrate its benefits for various model problems.
FINITE ELEMENTS IN ANALYSIS AND DESIGN
(2023)
Article
Physics, Mathematical
Vaishnavi Gujjula, Sivaram Ambikasaran
Summary: This article introduces a fast iterative solver for 2D scattering problems, utilizing the Green's function to represent the scattered field and discretizing the Lippmann-Schwinger equation with appropriate quadrature technique. The iterative solver accelerated by DAFMM and the new NCA method show significant advantages in numerical experiments.
COMMUNICATIONS IN COMPUTATIONAL PHYSICS
(2022)
Article
Geosciences, Multidisciplinary
Taewon Cho, Julianne Chung, Scot M. Miller, Arvind K. Saibaba
Summary: Atmospheric inverse modeling is the process of estimating greenhouse gas fluxes or air pollution emissions at the Earth's surface using observations of these gases in the atmosphere. This article discusses computationally efficient methods for large-scale atmospheric inverse modeling and addresses challenges in computation and practicality. The study develops generalized hybrid projection methods that are efficient, robust, automatic, and flexible. The benefits of these methods are demonstrated with a case study from NASA's OCO-2 satellite.
GEOSCIENTIFIC MODEL DEVELOPMENT
(2022)
Article
Mathematics, Applied
E. T. H. A. N. DUDLEY, A. R. V. I. N. D. K. SAIBABA, A. L. E. N. ALEXANDERIAN
Summary: This paper presents numerical methods for computing the Schatten p-norm of positive semi-definite matrices. It proposes a matrix-free method using a Monte Carlo estimator and extends the convergence and error analysis to the computation of non-integer and large values of p. The performance of the proposed estimators is demonstrated on test matrices and in an application to optimal experimental design for a model inverse problem.
ELECTRONIC TRANSACTIONS ON NUMERICAL ANALYSIS
(2022)
