The geometric high-order regularization methods such as mean curvature and Gaussian curvature, have been intensively studied during the last decades due to their abilities in preserving geometric properties including image edges, corners, and contrast. However, the dilemma between restoration quality and computational efficiency is an essential roadblock for high-order methods. In this paper, we propose fast multi-grid algorithms for minimizing both mean curvature and Gaussian curvature energy functionals without sacrificing accuracy for efficiency. Unlike the existing approaches based on operator splitting and the Augmented Lagrangian method (ALM), no artificial parameters are introduced in our formulation, which guarantees the robustness of the proposed algorithm. Meanwhile, we adopt the domain decomposition method to promote parallel computing and use the fine-to-coarse structure to accelerate convergence. Numerical experiments are presented on image denoising, CT, and MRI reconstruction problems to demonstrate the superiority of our method in preserving geometric structures and fine details. The proposed method is also shown effective in dealing with large-scale image processing problems by recovering an image of size $1024\times 1024$ within $40$s, while the ALM method requires around $200$s.
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In this paper we address the numerical solution of the quadratic optimal transport problem in its dynamical form, the so-called Benamou-Brenier formulation. When solved using interior point methods, the main computational bottleneck is the solution of large saddle point linear systems arising from the associated Newton-Raphson scheme. The main purpose of this paper is to design efficient preconditioners to solve these linear systems via iterative methods. Among the proposed preconditioners, we introduce one based on the partial commutation of the operators that compose the dual Schur complement of these saddle point linear systems, which we refer as $\boldsymbol{B}\boldsymbol{B}$-preconditioner. A series of numerical tests show that the $\boldsymbol{B}\boldsymbol{B}$-preconditioner is the most efficient among those presented, with a CPU-time scaling only slightly more than linearly with respect to the number of unknowns used to discretize the problem.
A modification of Newton's method for solving systems of $n$ nonlinear equations is presented. The new matrix-free method relies on a given decomposition of the invertible Jacobian of the residual into invertible sparse local Jacobians according to the chain rule of differentiation. It is motivated in the context of local Jacobians with bandwidth $2m+1$ for $m\ll n$. A reduction of the computational cost by $\mathcal{O}(\frac{n}{m})$ can be observed. Supporting run time measurements are presented for the tridiagonal case showing a reduction of the computational cost by $\mathcal{O}(n).$ Generalization yields the combinatorial Matrix-Free Newton Step problem. We prove is NP-completeness and we present several algorithmic components for building methods for its approximate solution. Inspired by adjoint Algorithmic Differentiation the new method shares several challenges for the latter including the DAG Reversal problem. Further challenges are due to combinatorial problems in sparse linear algebra such as Bandwidth or Directed Elimination Ordering.
An Efficient Multi-solution Solver for the Inverse Kinematics of 3-Section Constant-Curvature Robots
Piecewise constant curvature is a popular kinematics framework for continuum robots. Computing the model parameters from the desired end pose, known as the inverse kinematics problem, is fundamental in manipulation, tracking and planning tasks. In this paper, we propose an efficient multi-solution solver to address the inverse kinematics problem of 3-section constant-curvature robots by bridging both the theoretical reduction and numerical correction. We derive analytical conditions to simplify the original problem into a one-dimensional problem. Further, the equivalence of the two problems is formalised. In addition, we introduce an approximation with bounded error so that the one dimension becomes traversable while the remaining parameters analytically solvable. With the theoretical results, the global search and numerical correction are employed to implement the solver. The experiments validate the better efficiency and higher success rate of our solver than the numerical methods when one solution is required, and demonstrate the ability of obtaining multiple solutions with optimal path planning in a space with obstacles.
Recent studies show that BM25-driven dynamic index skipping can greatly accelerate MaxScore-based document retrieval based on the learned sparse representation derived by DeepImpact. This paper investigates the effectiveness of such a traversal guidance strategy during top k retrieval when using other models such as SPLADE and uniCOIL, and finds that unconstrained BM25-driven skipping could have a visible relevance degradation when the BM25 model is not well aligned with a learned weight model or when retrieval depth k is small. This paper generalizes the previous work and optimizes the BM25 guided index traversal with a two-level pruning control scheme and model alignment for fast retrieval using a sparse representation. Although there can be a cost of increased latency, the proposed scheme is much faster than the original MaxScore method without BM25 guidance while retaining the relevance effectiveness. This paper analyzes the competitiveness of this two-level pruning scheme, and evaluates its tradeoff in ranking relevance and time efficiency when searching several test datasets.
Neural networks are the state-of-the-art for many approximation tasks in high-dimensional spaces, as supported by an abundance of experimental evidence. However, we still need a solid theoretical understanding of what they can approximate and, more importantly, at what cost and accuracy. One network architecture of practical use, especially for approximation tasks involving images, is convolutional (residual) networks. However, due to the locality of the linear operators involved in these networks, their analysis is more complicated than for generic fully connected neural networks. This paper focuses on sequence approximation tasks, where a matrix or a higher-order tensor represents each observation. We show that when approximating sequences arising from space-time discretisations of PDEs we may use relatively small networks. We constructively derive these results by exploiting connections between discrete convolution and finite difference operators. Throughout, we design our network architecture to, while having guarantees, be similar to those typically adopted in practice for sequence approximation tasks. Our theoretical results are supported by numerical experiments which simulate linear advection, the heat equation, and the Fisher equation. The implementation used is available at the repository associated to the paper.
The Immersed Boundary (IB) method of Peskin (J. Comput. Phys., 1977) is useful for problems involving fluid-structure interactions or complex geometries. By making use of a regular Cartesian grid that is independent of the geometry, the IB framework yields a robust numerical scheme that can efficiently handle immersed deformable structures. Additionally, the IB method has been adapted to problems with prescribed motion and other PDEs with given boundary data. IB methods for these problems traditionally involve penalty forces which only approximately satisfy boundary conditions, or they are formulated as constraint problems. In the latter approach, one must find the unknown forces by solving an equation that corresponds to a poorly conditioned first-kind integral equation. This operation can require a large number of iterations of a Krylov method, and since a time-dependent problem requires this solve at each time step, this method can be prohibitively inefficient without preconditioning. In this work, we introduce a new, well-conditioned IB formulation for boundary value problems, which we call the Immersed Boundary Double Layer (IBDL) method. We present the method as it applies to Poisson and Helmholtz problems to demonstrate its efficiency over the original constraint method. In this double layer formulation, the equation for the unknown boundary distribution corresponds to a well-conditioned second-kind integral equation that can be solved efficiently with a small number of iterations of a Krylov method. Furthermore, the iteration count is independent of both the mesh size and immersed boundary point spacing. The method converges away from the boundary, and when combined with a local interpolation, it converges in the entire PDE domain. Additionally, while the original constraint method applies only to Dirichlet problems, the IBDL formulation can also be used for Neumann conditions.
Approximate Bayesian Computation (ABC) is a widely applicable and popular approach to estimating unknown parameters of mechanistic models. As ABC analyses are computationally expensive, parallelization on high-performance infrastructure is often necessary. However, the existing parallelization strategies leave resources unused at times and thus do not optimally leverage them yet. We present look-ahead scheduling, a wall-time minimizing parallelization strategy for ABC Sequential Monte Carlo algorithms, which utilizes all available resources at practically all times by proactive sampling for prospective tasks. Our strategy can be integrated in e.g. adaptive distance function and summary statistic selection schemes, which is essential in practice. Evaluation of the strategy on different problems and numbers of parallel cores reveals speed-ups of typically 10-20% and up to 50% compared to the best established approach. Thus, the proposed strategy allows to substantially improve the cost and run-time efficiency of ABC methods on high-performance infrastructure.
The well-known discrete Fourier transform (DFT) can easily be generalized to arbitrary nodes in the spatial domain. The fast procedure for this generalization is referred to as nonequispaced fast Fourier transform (NFFT). Various applications such as MRI, solution of PDEs, etc., are interested in the inverse problem, i.e., computing Fourier coefficients from given nonequispaced data. In this paper we survey different kinds of approaches to tackle this problem. In contrast to iterative procedures, where multiple iteration steps are needed for computing a solution, we focus especially on so-called direct inversion methods. We review density compensation techniques and introduce a new scheme that leads to an exact reconstruction for trigonometric polynomials. In addition, we consider a matrix optimization approach using Frobenius norm minimization to obtain an inverse NFFT.
We consider sequential state and parameter learning in state-space models with intractable state transition and observation processes. By exploiting low-rank tensor-train (TT) decompositions, we propose new sequential learning methods for joint parameter and state estimation under the Bayesian framework. Our key innovation is the introduction of scalable function approximation tools such as TT for recursively learning the sequentially updated posterior distributions. The function approximation perspective of our methods offers tractable error analysis and potentially alleviates the particle degeneracy faced by many particle-based methods. In addition to the new insights into algorithmic design, our methods complement conventional particle-based methods. Our TT-based approximations naturally define conditional Knothe--Rosenblatt (KR) rearrangements that lead to filtering, smoothing and path estimation accompanying our sequential learning algorithms, which open the door to removing potential approximation bias. We also explore several preconditioning techniques based on either linear or nonlinear KR rearrangements to enhance the approximation power of TT for practical problems. We demonstrate the efficacy and efficiency of our proposed methods on several state-space models, in which our methods achieve state-of-the-art estimation accuracy and computational performance.
As soon as abstract mathematical computations were adapted to computation on digital computers, the problem of efficient representation, manipulation, and communication of the numerical values in those computations arose. Strongly related to the problem of numerical representation is the problem of quantization: in what manner should a set of continuous real-valued numbers be distributed over a fixed discrete set of numbers to minimize the number of bits required and also to maximize the accuracy of the attendant computations? This perennial problem of quantization is particularly relevant whenever memory and/or computational resources are severely restricted, and it has come to the forefront in recent years due to the remarkable performance of Neural Network models in computer vision, natural language processing, and related areas. Moving from floating-point representations to low-precision fixed integer values represented in four bits or less holds the potential to reduce the memory footprint and latency by a factor of 16x; and, in fact, reductions of 4x to 8x are often realized in practice in these applications. Thus, it is not surprising that quantization has emerged recently as an important and very active sub-area of research in the efficient implementation of computations associated with Neural Networks. In this article, we survey approaches to the problem of quantizing the numerical values in deep Neural Network computations, covering the advantages/disadvantages of current methods. With this survey and its organization, we hope to have presented a useful snapshot of the current research in quantization for Neural Networks and to have given an intelligent organization to ease the evaluation of future research in this area.
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