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Current 3D representations like meshes, voxels, point clouds, and NeRF-based neural implicit fields exhibit significant limitations: they are often task-specific, lacking universal applicability across reconstruction, generation, editing,…
We consider a strongly heterogeneous medium saturated by an incompressible viscous fluid as it appears in geomechanical modeling. This poroelasticity problem suffers from rapidly oscillating material parameters, which calls for a thorough…
We present the spline-interpolation approximate solution of the Dirichlet problem for the Laplace equation in the bodies of revolution, cones and cylinders. Our method is based on reduction of the 3D problem to the sequence of 2D Dirichlet…
This paper introduces the hierarchical interpolative factorization for elliptic partial differential equations (HIF-DE) in two (2D) and three dimensions (3D). This factorization takes the form of an approximate generalized LU/LDL…
Various 3D reconstruction methods have enabled civil engineers to detect damage on a road surface. To achieve the millimetre accuracy required for road condition assessment, a disparity map with subpixel resolution needs to be used.…
Traditional low-rank approximation is a powerful tool to compress the huge data matrices that arise in simulations of partial differential equations (PDE), but suffers from high computational cost and requires several passes over the PDE…
We propose an optimization proxy in terms of iterative implicit gradient methods for solving constrained optimization problems with nonconvex loss functions. This framework can be applied to a broad range of machine learning settings,…
Dual decomposition is a powerful technique for deriving decomposition schemes for convex optimization problems with separable structure. Although the Augmented Lagrangian is computationally more stable than the ordinary Lagrangian, the…
We present an effective harmonic density interpolation method for the numerical evaluation of singular and nearly singular Laplace boundary integral operators and layer potentials in two and three spatial dimensions. The method relies on…
Directed energy deposition (DED), a metal additive manufacturing process, is highly susceptible to process-induced defects such as geometric deviations, lack of fusion, and poor surface finish. This work presents a build-height-synchronized…
The discretization of certain integral equations, e.g., the first-kind Fredholm equation of Laplace's equation, leads to symmetric positive-definite linear systems, where the coefficient matrix is dense and often ill-conditioned. We…
Layered manufacturing inherently suffers from staircase defects along surfaces that are gently slopped with respect to the build direction. Reducing the slice thickness improves the situation but never resolves it completely as flat layers…
We propose an analysis-by-synthesis method for fast multi-view 3D reconstruction of opaque objects with arbitrary materials and illumination. State-of-the-art methods use both neural surface representations and neural rendering. While…
We present a simple, PDE-based proof of the result [M. Johnson, 2001] that the error estimates of [J. Duchon, 1978] for thin plate spline interpolation can be improved by $h^{1/2}$. We illustrate that ${\mathcal H}$-matrix techniques can…
Boundary integral equations are an efficient and accurate tool for the numerical solution of elliptic boundary value problems. The solution is expressed as a layer potential; however, the error in its evaluation grows large near the…
We consider a finite element method for elliptic equation with heterogeneous and possibly high-contrast coefficients based on primal hybrid formulation. A space decomposition as in FETI and BDCC allows a sequential computations of the…
Recent advances in localized implicit functions have enabled neural implicit representation to be scalable to large scenes. However, the regular subdivision of 3D space employed by these approaches fails to take into account the sparsity of…
Low-rank approximations are essential in modern data science. The interpolative decomposition provides one such approximation. Its distinguishing feature is that it reuses columns from the original matrix. This enables it to preserve matrix…
Compression is a crucial solution for data reduction in modern scientific applications due to the exponential growth of data from simulations, experiments, and observations. Compression with progressive retrieval capability allows users to…
This paper uses the Modified Projection Method to examine the errors in solving the boundary integral equation from Laplace equation. The analysis uses weighted norms, and parallel algorithms help solve the independent linear systems. By…