Efficient Shallow Ritz Method For 1D Diffusion-Reaction Problems
- URL: http://arxiv.org/abs/2407.01496v4
- Date: Sat, 26 Jul 2025 12:06:51 GMT
- Title: Efficient Shallow Ritz Method For 1D Diffusion-Reaction Problems
- Authors: Zhiqiang Cai, Anastassia Doktorova, Robert D. Falgout, César Herrera,
- Abstract summary: This paper studies the shallow Ritz method for solving one-dimensional diffusion-reaction problems.<n>We present a damped block Newton (dBN) method to achieve nearly optimal order of approximation.
- Score: 0.0
- License: http://creativecommons.org/licenses/by-nc-sa/4.0/
- Abstract: This paper studies the shallow Ritz method for solving one-dimensional diffusion-reaction problems. The method is capable of improving the order of approximation for non-smooth problems. By following a similar approach to the one presented in [9], we present a damped block Newton (dBN) method to achieve nearly optimal order of approximation. The dBN method optimizes the Ritz functional by alternating between the linear and non-linear parameters of the shallow ReLU neural network (NN). For diffusion-reaction problems, new difficulties arise: (1) for the linear parameters, the mass matrix is dense and even more ill-conditioned than the stiffness matrix, and (2) for the non-linear parameters, the Hessian matrix is dense and may be singular. This paper addresses these challenges, resulting in a dBN method with computational cost of ${\cal O}(n)$. The ideas presented for diffusion-reaction problems can also be applied to least-squares approximation problems. For both applications, starting with the non-linear parameters as a uniform partition, numerical experiments show that the dBN method moves the mesh points to nearly optimal locations.
Related papers
- A Natural Primal-Dual Hybrid Gradient Method for Adversarial Neural Network Training on Solving Partial Differential Equations [9.588717577573684]
We propose a scalable preconditioned primal hybrid gradient algorithm for solving partial differential equations (PDEs)<n>We compare the performance of the proposed method with several commonly used deep learning algorithms.<n>The numerical results suggest that the proposed method performs efficiently and robustly and converges more stably.
arXiv Detail & Related papers (2024-11-09T20:39:10Z) - Total Uncertainty Quantification in Inverse PDE Solutions Obtained with Reduced-Order Deep Learning Surrogate Models [50.90868087591973]
We propose an approximate Bayesian method for quantifying the total uncertainty in inverse PDE solutions obtained with machine learning surrogate models.
We test the proposed framework by comparing it with the iterative ensemble smoother and deep ensembling methods for a non-linear diffusion equation.
arXiv Detail & Related papers (2024-08-20T19:06:02Z) - A Structure-Guided Gauss-Newton Method for Shallow ReLU Neural Network [18.06366638807982]
We propose a structure-guided Gauss-Newton (SgGN) method for solving least squares problems using a shallow ReLU neural network.
The method effectively takes advantage of both the least squares structure and the neural network structure of the objective function.
arXiv Detail & Related papers (2024-04-07T20:24:44Z) - Purely quantum algorithms for calculating determinant and inverse of matrix and solving linear algebraic systems [43.53835128052666]
We propose quantum algorithms for the computation of the determinant and inverse of an $(N-1) times (N-1)$ matrix.
This approach is a straightforward modification of the existing algorithm for determining the determinant of an $N times N$ matrix.
We provide suitable circuit designs for all three algorithms, each estimated to require $O(N log N)$ in terms of space.
arXiv Detail & Related papers (2024-01-29T23:23:27Z) - Optimizing Solution-Samplers for Combinatorial Problems: The Landscape
of Policy-Gradient Methods [52.0617030129699]
We introduce a novel theoretical framework for analyzing the effectiveness of DeepMatching Networks and Reinforcement Learning methods.
Our main contribution holds for a broad class of problems including Max-and Min-Cut, Max-$k$-Bipartite-Bi, Maximum-Weight-Bipartite-Bi, and Traveling Salesman Problem.
As a byproduct of our analysis we introduce a novel regularization process over vanilla descent and provide theoretical and experimental evidence that it helps address vanishing-gradient issues and escape bad stationary points.
arXiv Detail & Related papers (2023-10-08T23:39:38Z) - An Extreme Learning Machine-Based Method for Computational PDEs in
Higher Dimensions [1.2981626828414923]
We present two effective methods for solving high-dimensional partial differential equations (PDE) based on randomized neural networks.
We present ample numerical simulations for a number of high-dimensional linear/nonlinear stationary/dynamic PDEs to demonstrate their performance.
arXiv Detail & Related papers (2023-09-13T15:59:02Z) - Vectorization of the density matrix and quantum simulation of the von
Neumann equation of time-dependent Hamiltonians [65.268245109828]
We develop a general framework to linearize the von-Neumann equation rendering it in a suitable form for quantum simulations.
We show that one of these linearizations of the von-Neumann equation corresponds to the standard case in which the state vector becomes the column stacked elements of the density matrix.
A quantum algorithm to simulate the dynamics of the density matrix is proposed.
arXiv Detail & Related papers (2023-06-14T23:08:51Z) - A Majorization-Minimization Gauss-Newton Method for 1-Bit Matrix Completion [15.128477070895055]
We propose a novel method for 1-bit matrix completion called Majorization-Minimization Gauss-Newton (MMGN)
Our method is based on the majorization-minimization principle, which converts the original optimization problem into a sequence of standard low-rank matrix completion problems.
arXiv Detail & Related papers (2023-04-27T03:16:52Z) - Decentralized Riemannian natural gradient methods with Kronecker-product
approximations [11.263837420265594]
We present an efficient decentralized natural gradient descent (DRNGD) method for solving decentralized manifold optimization problems.
By performing the communications over the Kronecker factors, a high-quality approximation of the RFIM can be obtained in a low cost.
arXiv Detail & Related papers (2023-03-16T19:36:31Z) - Log-based Sparse Nonnegative Matrix Factorization for Data
Representation [55.72494900138061]
Nonnegative matrix factorization (NMF) has been widely studied in recent years due to its effectiveness in representing nonnegative data with parts-based representations.
We propose a new NMF method with log-norm imposed on the factor matrices to enhance the sparseness.
A novel column-wisely sparse norm, named $ell_2,log$-(pseudo) norm, is proposed to enhance the robustness of the proposed method.
arXiv Detail & Related papers (2022-04-22T11:38:10Z) - Numerical Approximation of Partial Differential Equations by a Variable
Projection Method with Artificial Neural Networks [0.0]
We present a method for solving linear and nonlinear PDEs based on the variable projection (VarPro) framework and artificial neural networks (ANNs)
For linear PDEs, enforcing the boundary/initial value problem on the collocation points leads to a separable nonlinear least squares problem about the network coefficients.
We reformulate this problem by the VarPro approach to eliminate the linear output-layer coefficients, leading to a reduced problem about the hidden-layer coefficients only.
arXiv Detail & Related papers (2022-01-24T22:31:38Z) - Quantum Newton's method for solving system of nonlinear algebraic
equations [0.25782420501870296]
We propose quantum Newton's method (QNM) for solving $N$-dimensional system of nonlinear equations based on Newton's method.
We use a specific quantum data structure and $l_infty$ tomography with sample error $epsilon_s$ to implement the classical-quantum data conversion process.
arXiv Detail & Related papers (2021-09-17T11:20:26Z) - Unfolding Projection-free SDP Relaxation of Binary Graph Classifier via
GDPA Linearization [59.87663954467815]
Algorithm unfolding creates an interpretable and parsimonious neural network architecture by implementing each iteration of a model-based algorithm as a neural layer.
In this paper, leveraging a recent linear algebraic theorem called Gershgorin disc perfect alignment (GDPA), we unroll a projection-free algorithm for semi-definite programming relaxation (SDR) of a binary graph.
Experimental results show that our unrolled network outperformed pure model-based graph classifiers, and achieved comparable performance to pure data-driven networks but using far fewer parameters.
arXiv Detail & Related papers (2021-09-10T07:01:15Z) - Newton-LESS: Sparsification without Trade-offs for the Sketched Newton
Update [88.73437209862891]
In second-order optimization, a potential bottleneck can be computing the Hessian matrix of the optimized function at every iteration.
We show that the Gaussian sketching matrix can be drastically sparsified, significantly reducing the computational cost of sketching.
We prove that Newton-LESS enjoys nearly the same problem-independent local convergence rate as Gaussian embeddings.
arXiv Detail & Related papers (2021-07-15T17:33:05Z) - QUBO formulations for numerical quantum computing [0.0]
Harrow-Hassidim-Lloyd algorithm is a monumental quantum algorithm for solving linear systems on gate model quantum computers.
We will find unconstrained binary optimization (QUBO) models for a vector x that satisfies Ax=b.
We validate those QUBO models on the D-Wave system and discuss the results.
arXiv Detail & Related papers (2021-06-21T02:49:59Z) - Analysis of One-Hidden-Layer Neural Networks via the Resolvent Method [0.0]
Motivated by random neural networks, we consider the random matrix $M = Y Yast$ with $Y = f(WX)$.
We prove that the Stieltjes transform of the limiting spectral distribution satisfies a quartic self-consistent equation up to some error terms.
In addition, we extend the previous results to the case of additive bias $Y=f(WX+B)$ with $B$ being an independent rank-one Gaussian random matrix.
arXiv Detail & Related papers (2021-05-11T15:17:39Z) - Solving and Learning Nonlinear PDEs with Gaussian Processes [11.09729362243947]
We introduce a simple, rigorous, and unified framework for solving nonlinear partial differential equations.
The proposed approach provides a natural generalization of collocation kernel methods to nonlinear PDEs and IPs.
For IPs, while the traditional approach has been to iterate between the identifications of parameters in the PDE and the numerical approximation of its solution, our algorithm tackles both simultaneously.
arXiv Detail & Related papers (2021-03-24T03:16:08Z) - Hybrid Trilinear and Bilinear Programming for Aligning Partially
Overlapping Point Sets [85.71360365315128]
In many applications, we need algorithms which can align partially overlapping point sets are invariant to the corresponding corresponding RPM algorithm.
We first show that the objective is a cubic bound function. We then utilize the convex envelopes of trilinear and bilinear monomial transformations to derive its lower bound.
We next develop a branch-and-bound (BnB) algorithm which only branches over the transformation variables and runs efficiently.
arXiv Detail & Related papers (2021-01-19T04:24:23Z) - Linear-Sample Learning of Low-Rank Distributions [56.59844655107251]
We show that learning $ktimes k$, rank-$r$, matrices to normalized $L_1$ distance requires $Omega(frackrepsilon2)$ samples.
We propose an algorithm that uses $cal O(frackrepsilon2log2fracepsilon)$ samples, a number linear in the high dimension, and nearly linear in the matrices, typically low, rank proofs.
arXiv Detail & Related papers (2020-09-30T19:10:32Z) - Conditional gradient methods for stochastically constrained convex
minimization [54.53786593679331]
We propose two novel conditional gradient-based methods for solving structured convex optimization problems.
The most important feature of our framework is that only a subset of the constraints is processed at each iteration.
Our algorithms rely on variance reduction and smoothing used in conjunction with conditional gradient steps, and are accompanied by rigorous convergence guarantees.
arXiv Detail & Related papers (2020-07-07T21:26:35Z) - Effective Dimension Adaptive Sketching Methods for Faster Regularized
Least-Squares Optimization [56.05635751529922]
We propose a new randomized algorithm for solving L2-regularized least-squares problems based on sketching.
We consider two of the most popular random embeddings, namely, Gaussian embeddings and the Subsampled Randomized Hadamard Transform (SRHT)
arXiv Detail & Related papers (2020-06-10T15:00:09Z) - Multi-Objective Matrix Normalization for Fine-grained Visual Recognition [153.49014114484424]
Bilinear pooling achieves great success in fine-grained visual recognition (FGVC)
Recent methods have shown that the matrix power normalization can stabilize the second-order information in bilinear features.
We propose an efficient Multi-Objective Matrix Normalization (MOMN) method that can simultaneously normalize a bilinear representation.
arXiv Detail & Related papers (2020-03-30T08:40:35Z) - Optimal Randomized First-Order Methods for Least-Squares Problems [56.05635751529922]
This class of algorithms encompasses several randomized methods among the fastest solvers for least-squares problems.
We focus on two classical embeddings, namely, Gaussian projections and subsampled Hadamard transforms.
Our resulting algorithm yields the best complexity known for solving least-squares problems with no condition number dependence.
arXiv Detail & Related papers (2020-02-21T17:45:32Z)
This list is automatically generated from the titles and abstracts of the papers in this site.
This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.