Related papers: A perturbed differential resultant based impliciti…
We develop an interpolation-based modeling framework for parameter-dependent partial differential equations arising in control, inverse problems, and uncertainty quantification. The solution is discretized in the physical domain using…
Differentiable programming is revolutionizing computational science by enabling automatic differentiation (AD) of numerical simulations. While first-order gradients are well-established, second-order derivatives (Hessians) for implicit…
Implicit regularization refers to the tendency of local search algorithms to converge to low-dimensional solutions, even when such structures are not explicitly enforced. Despite its ubiquity, the mechanism underlying this behavior remains…
This paper aims to investigate the numerical approximation of semilinear non-autonomous stochastic partial differential equations (SPDEs) driven by multiplicative or additive noise. Such equations are more realistic than autonomous SPDEs…
This article deals with the computation of guaranteed lower bounds of the error in the framework of finite element (FE) and domain decomposition (DD) methods. In addition to a fully parallel computation, the proposed lower bounds separate…
Assume that $n$ is a positive integer, $p_{j}$ ($j=1,2, \cdots, 6)$ are polynomials, $p$ is an irreducible polynomial, and $f$ is an entire function on $\mathbb{C}^{n}.$ Let $ L(f)=\sum_{j=1}^s q_{t_j}f_{z_{t_j}}$ and…
With the immense computing power at our disposal, the numerical solution of partial differential equations (PDEs) is becoming a day-to-day task for modern computational scientists. However, the complexity of real-life problems is such that…
Optimization problems constrained by partial differential equations (PDEs) naturally arise in scientific computing, as those constraints often model physical systems or the simulation thereof. In an implicitly constrained approach, the…
In this paper, we focus on computing the kernel of a map of polynomial rings $\varphi$. This core problem in symbolic computation is known as implicitization. While there are extremely effective Gr\"obner basis methods used to solve this…
When modeling scientific and industrial problems, geometries are typically modeled by explicit boundary representations obtained from computer-aided design software. Unfitted (also known as embedded or immersed) finite element methods offer…
Physics-informed deep learning often faces optimization challenges due to the complexity of solving partial differential equations (PDEs), which involve exploring large solution spaces, require numerous iterations, and can lead to unstable…
Machine learning pipelines often rely on optimization procedures to make discrete decisions (e.g., sorting, picking closest neighbors, or shortest paths). Although these discrete decisions are easily computed, they break the…
In this work, we introduce implicit Finite Operator Learning (iFOL) for the continuous and parametric solution of partial differential equations (PDEs) on arbitrary geometries. We propose a physics-informed encoder-decoder network to…
This paper presents a weakly intrusive strategy for computing a low-rank approximation of the solution of a system of nonlinear parameter-dependent equations. The proposed strategy relies on a Newton-like iterative solver which only…
This article presents an identification methodology to capture general relationships, with application to piecewise nonlinear approximations of model predictive control for constrained (non)linear systems. The mathematical formulation…
The (matricial) solution set of a Linear Matrix Inequality (LMI) is a convex basic non-commutative semi-algebraic set. The main theorem of this paper is a converse, a result which has implications for both semidefinite programming and…
The fully connected conditional random field (CRF) with Gaussian pairwise potentials has proven popular and effective for multi-class semantic segmentation. While the energy of a dense CRF can be minimized accurately using a linear…
When optimizing over-parameterized models, such as deep neural networks, a large set of parameters can achieve zero training error. In such cases, the choice of the optimization algorithm and its respective hyper-parameters introduces…
This paper deals with the algorithmic aspects of solving feasibility problems of semidefinite programming (SDP), aka linear matrix inequalities (LMI). Since in some SDP instances all feasible solutions have irrational entries, numerical…
We consider the problem of complex root classification, i.e., finding the conditions on the coefficients of a univariate polynomial for all possible multiplicity structures on its complex roots. It is well known that such conditions can be…