arXiv subjects

Wei Qu

Publications and source records attributed to Wei Qu.

17 recordsLinked to original sources

arXiv2026

Observability and controllability for the Schr\"{o}dinger equation on polyhedra

We consider the observability and controllability for the Schr\"{o}dinger equation on polyhedra. We show that for the Schr\"{o}dinger equation on convex polyhedra in $\mathbb{R}^d\,(d\geq2)$, when the initial data possess $H^s\,(s>d/2)$ regularity, observability and controllability hold, with the control region being an arbitrary nonempty open neighborhood of the singular set of the boundary, which is the same as the control region for eigenfunctions. Our proof of observability is a concrete adaptation of the method in the Burq--Zworski black box control work, where the observability inequality is proved via a resolvent estimate. Hence, before proving observability, we present an observability resolvent estimate on polyhedra. The idea of the proof of this resolvent estimate comes from the case of eigenfunction concentration on polyhedra considered by Ceki\'c--Georgiev--Mukherjee, which uses semiclassical measures and relies closely on the dynamical properties of the billiard flow in polyhedra. Finally, by a standard HUM argument, we show that observability implies controllability.

math.OC↗

arXiv2026

High-Multiplicity Flexible Job Shops: From Exact Recurrent Fluid Attainment to Structure-Guided Finite-Horizon Scheduling

High-multiplicity flexible job shops involve many copies of a small set of job types that must be scheduled on alternative machines. Fluid relaxations provide scalable workload lower bounds, but their fractional machine allocations do not define feasible schedules for individual jobs. We show that, after a suitable finite scaling, an optimal fluid allocation can be realized exactly by a feasible repeating discrete schedule. The construction scales the fluid allocation to integer operation counts, places the resulting operations in nonoverlapping machine intervals, repeats this arrangement, and links operations across repetitions into individual jobs without moving any interval, thereby enforcing job precedence while preserving machine feasibility. For growing finite instances with the same job-type composition, even with a fixed number of extra jobs, the gap between the optimal makespan and the fluid lower bound remains bounded by a constant; hence the relative gap vanishes as the instance grows. Guided by this repeated structure, we develop type-based cyclic template replay (TCTR), which searches job-type templates using an optimization model whose size does not grow with the number of job copies and replays the selected template on the full instance. On 676 multiplicity-expanded public flexible-job-shop instances, TCTR is feasible in every case and achieves a 1.54% mean gap to the fluid lower bound, compared with 5.32% for a job-indexed adaptive large-neighborhood search and 6.18% for a hybrid genetic algorithm.

math.OC↗

arXiv2026

Selecting the optimal Parameters Results in Double Interpolation: Double AFD

Let $f$ belong to the Hardy space $H^2(\mathbb{D})$ of the unit disc, and $e_a$ the normalized Szeg\"o (reproducing) kernel of $H^2(\mathbb{D}).$ It is well known that, due to the reproducing kernel property, for any distinct $n$ points $a_1,\cdots,a_n$ in $\mathbb{D}$ the orthogonal projection of $f$ into ${\rm span}\{e_{a_1},\cdots,e_{a_n}\},$ denoted as $P_{{\rm span}\{e_{a_1},\cdots,e_{a_n}\}}(f),$ interpolates $f$ at the points $a_k$'s. The present study further proves that if the $a_k$'s are optimally selected according to certain energy matching pursuit principle, then $P_{{\rm span}\{e_{a_1},\cdots,e_{a_n}\}}(f)$ double interpolates $f$ at the points $a_k$'s, or order $m=2$ interpolation, that is, \[ P_{{\rm span}\{e_{a_1},\cdots,e_{a_n}\}}(f)(a_k)=f(a_k), \quad {\rm and}\quad P_{{\rm span}\{e_{a_1},\cdots,e_{a_n}\}}'(f)(a_k)=f'(a_k),\quad k=1,\cdots,n.\] With the accordingly newly defined double Takenaka-Malmquist system, the norm convergence for $n\to \infty,$ the $n$-best approximation for $n$ being fixed, and the related boundary function interpolation are studied. The such generated new sparse representation, named as double AFD, is shown to outperform the classical AFD. Pointwise interpolations for orders $m>2,$ meaning to simultaneously interpolates all functions $f,f',\cdots,f^{(m-1)}$ at a set of $a_k$'s are, additionally, discussed. For the Hardy space of the upper-half complex plane there exists a counterpart theory.

math.CV↗

arXiv2025

SeedProteo: Accurate De Novo All-Atom Design of Protein Binders

We present SeedProteo, a diffusion-based model for de novo all-atom protein design. We demonstrate how to repurpose a cutting-edge folding architecture into a powerful generative design framework by effectively integrating self-conditioning features. Extensive benchmarks highlight the model's capabilities across two distinct tasks: in unconditional generation, SeedProteo exhibits superior length generalization and structural diversity, maintaining robustness for long sequences and complex topologies; in binder design, it achieves state-of-the-art performance among open-source methods, attaining the highest in-silico design success rates, structural diversity and novelty. Finally, we validate SeedProteo through wet-lab assays on two therapeutic targets, achieving hit rates of 70%-80% and picomolar-level binding affinities, establishing leading results. To facilitate community adoption, we provide public access to SeedProteo via a webserver (https://seedfold.io/proteinDesign).

q-bio.BM↗

arXiv2025

SeedFold: Scaling Biomolecular Structure Prediction

Highly accurate biomolecular structure prediction is a key component of developing biomolecular foundation models, and one of the most critical aspects of building foundation models is identifying the recipes for scaling the model. In this work, we present SeedFold, a folding model that successfully scales up the model capacity. Our contributions are threefold: first, we identify an effective width-scaling strategy for the Pairformer to increase representation capacity; second, we introduce a novel linear triangular attention that reduces computational complexity to enable efficient scaling; finally, we construct a large-scale distillation dataset to substantially enlarge the training set. Experiments on FoldBench show that SeedFold outperforms AlphaFold3 on most protein-related tasks.

q-bio.BM↗

arXiv2025

Optimal preconditioning techniques for finite volume approximation of three-dimensional conservative space-fractional diffusion equations

A Crank-Nicolson finite volume approximation for three-dimensional conservative space-fractional diffusion equation results in large and dense three-level Toeplitz discrete linear systems. Preconditioned Krylov subspace methods with sine transform-based preconditioners are developed to solve these systems, including the preconditioned conjugate gradient (PCG) method for the symmetric case and the preconditioned generalized minimal residual (PGMRES) method for the non-symmetric case. Moreover, we provide detailed analysis of the convergence of these Krylov subspace methods. Specifically, for the symmetric case, we prove the spectra of the preconditioned matrices are uniformly bounded in the open interval (1/2, 3/2), which results in a linear convergence rate of the PCG method. For the non-symmetric case, we demonstrate that the PGMRES method also achieves a linear convergence rate independent of discretization stepsizes from the residual point of view. These results imply that the iteration counts of the PCG and PGMRES methods are uniformly bounded and independent of the matrix sizes. Numerical experiments in both symmetric and non-symmetric cases in two- and three-dimensions are conducted to confirm the optimal performance of the proposed preconditioned Krylov subspace methods.

math.NA↗

arXiv2025

An optimal preconditioner for high-order scheme arising from multi-dimensional Riesz space fractional diffusion equations with variable coefficients

In this paper, we propose an efficient method for solving multi-dimensional Riesz space fractional diffusion equations with variable coefficients. The Crank-Nicolson (CN) method is used for temporal discretization, while the fourth-order fractional centered difference (4FCD) method is employed for spatial discretization. Using a novel technique, we show that the CN-4FCD scheme for the multi-dimensional case is unconditionally stable and convergent, achieving second-order accuracy in time and fourth-order accuracy in space with respect to the discrete L2-norm. Moreover, leveraging the symmetric multi-level Toeplitz-like structure of the coefficient matrix in the discrete linear systems, we enhance the computational efficiency of the proposed scheme with a sine transform-based preconditioner, ensuring a mesh-size-independent convergence rate for the conjugate gradient method. Finally, two numerical examples validate the theoretical analysis and demonstrate the superior performance of the proposed preconditioner compared to existing methods.

math.NA↗

arXiv2024

Matrix Pre-orthogonal Matching Pursuit and Pseudo-Inverse

We introduce a new fundamental algorithm called Matrix-POAFD to solve the matrix least square problem. The method is based on the matching pursuit principle. The method directly extracts, among the given features as column vectors of the measurement matrix, in the order of their importance, the decisive features for the observing vector. With competitive computational efficiency to the existing sophisticated least square solutions the proposed method, due to its explicit and iterative algorithm process, has the advantage of trading off minimum norms with tolerable error scales. The method inherits recently developed studies in functional space contexts. The second main contribution, also in the algorithm aspect, is to present a two-step iterative computation method for pseudo-inverse. We show that consecutively performing two least square solutions, of which one is to $X$ and the other to $X^*,$ results in the minimum norm least square solution. The two-step algorithm can also be combined into one solving a single least square problem but with respect to $XX^\ast.$ The result is extended to the functional formulation as well. To better explain the idea, as well as for the self-containing purpose, we give short surveys with proofs of key results on closely relevant subjects, including solutions with reproducing kernel Hilbert space setting, AFD type sparse representation in terms of matching pursuit, the general ${\mathcal H}$-$H_K$ formulation and pseudo-inverse of bounded linear operator in Hilbert spaces.

cs.IT↗

arXiv2024

Minimal-norm solution to the Fredholm integral equations of the first kind via the H-HK formulation

The Fredholm integral equations of the first kind is a typical ill-posed problem, so that it is usually difficult to obtain its analytical minimal-norm solution. This paper gives a closed-form minimal-norm solution for the degenerate kernel equations based on the H-HK formulation. Furthermore, it has been shown that the structure of solutions to degenerate kernel equations and matrix equations are consistent. Subsequently, the obtained results are extended to non-degenerate integral equations. Finally, the validity and applicability of the proposed method are demonstrated by some examples.

math.NA↗

arXiv2024

A novel fourth-order scheme for two-dimensional Riesz space fractional nonlinear reaction-diffusion equations and its optimal preconditioned solver

A novel fourth-order finite difference formula coupling the Crank-Nicolson explicit linearized method is proposed to solve Riesz space fractional nonlinear reaction-diffusion equations in two dimensions. Theoretically, under the Lipschitz assumption on the nonlinear term, the proposed high-order scheme is proved to be unconditionally stable and convergent in the discrete $L_2$-norm. Moreover, a $\tau$-matrix based preconditioner is developed to speed up the convergence of the conjugate gradient method with an optimal convergence rate (a convergence rate independent of mesh sizes) for solving the symmetric discrete linear system. Theoretical analysis shows that the spectra of the preconditioned matrices are uniformly bounded in the open interval $(3/8,2)$. To the best of our knowledge, this is the first attempt to develop a preconditioned iterative solver with a mesh-independent convergence rate for the linearized high-order scheme. Numerical examples are given to validate the accuracy of the scheme and the effectiveness of the proposed preconditioned solver.

math.NA↗

arXiv2023

VQGraph: Rethinking Graph Representation Space for Bridging GNNs and MLPs

GNN-to-MLP distillation aims to utilize knowledge distillation (KD) to learn computationally-efficient multi-layer perceptron (student MLP) on graph data by mimicking the output representations of teacher GNN. Existing methods mainly make the MLP to mimic the GNN predictions over a few class labels. However, the class space may not be expressive enough for covering numerous diverse local graph structures, thus limiting the performance of knowledge transfer from GNN to MLP. To address this issue, we propose to learn a new powerful graph representation space by directly labeling nodes' diverse local structures for GNN-to-MLP distillation. Specifically, we propose a variant of VQ-VAE to learn a structure-aware tokenizer on graph data that can encode each node's local substructure as a discrete code. The discrete codes constitute a codebook as a new graph representation space that is able to identify different local graph structures of nodes with the corresponding code indices. Then, based on the learned codebook, we propose a new distillation target, namely soft code assignments, to directly transfer the structural knowledge of each node from GNN to MLP. The resulting framework VQGraph achieves new state-of-the-art performance on GNN-to-MLP distillation in both transductive and inductive settings across seven graph datasets. We show that VQGraph with better performance infers faster than GNNs by 828x, and also achieves accuracy improvement over GNNs and stand-alone MLPs by 3.90% and 28.05% on average, respectively. Code: https://github.com/YangLing0818/VQGraph.

cs.LG↗

arXiv2022

The sparse representation related with fractional heat equations

This study introduces pre-orthogonal adaptive Fourier decomposition (POAFD) to obtain approximations and numerical solutions to the fractional Laplacian initial value problem and the extension problem of Caffarelli and Silvestre (generalized Poisson equation). The method, as the first step, expands the initial data function into a sparse series of the fundamental solutions with fast convergence, and, as the second step, makes use the semigroup or the reproducing kernel property of each of the expanding entries. Experiments show effectiveness and efficiency of the proposed series solutions.

math.NA↗

arXiv2022

AFD Types Sparse Representations vs. the Karhunen-Loeve Expansion for Decomposing Stochastic Processes

This article introduces adaptive Fourier decomposition (AFD) type methods, emphasizing on those that can be applied to stochastic processes and random fields, mainly including stochastic adaptive Fourier decomposition and stochastic pre-orthogonal adaptive Fourier decomposition. We establish their algorithms based on the covariant function and prove that they enjoy the same convergence rate as the Karhunen-Lo\`eve (KL) decomposition. The AFD type methods are compared with the KL decomposition. In contrast with the latter, the AFD type methods do not need to compute eigenvalues and eigenfunctions of the kernel-integral operator induced by the covariance function, and thus considerably reduce the computation complexity and computer consumes. Various kinds of dictionaries offer AFD flexibility to solve problems of a great variety, including different types of deterministic and stochastic equations. The conducted experiments show, besides the numerical convenience and fast convergence, that the AFD type decompositions outperform the KL type in describing local details, in spite of the proven global optimality of the latter.

math.ST↗

arXiv2020

Phase Retrieval in Hardy Space

This paper concerns the study of reconstructing a function $f$ in the Hardy space of the unit disc $\D$ from intensity measurements $|f(z)|,\ z\in \D.$ It's known as the problem of phase retrieval. We transform it into solving the corresponding outer and inner function through the Nevanlinna factorization Theorem. The outer function will be established based on the mechanical quadrature method, while we use two different ways to find out the zero points of Blashcke product, thereby computing the inner function under the assumption that the singular inner function part is trivial. Then the concrete algorithms and illustrative experiments follow. Finally, we give a sparse representation of $f$ by introducing the unwinding adaptive Fourier decomposition.

math.CV↗

arXiv2020

Sparse Approximation to the Dirac-{\delta} Distribution

The Dirac-{\delta} distribution may be realized through sequences of convlutions, the latter being also regarded as approximation to the identity. The present study proposes the so called pre-orthogonal adaptive Fourier decomposition (POAFD) method to realize fast approximation to the identity. The type of sparse representation method has potential applications in signal and image analysis, as well as in system identification.

math.CA↗

arXiv2020

A sufficient condition for n-Best Kernel Approximation in Reproducing Kernel Hilbert Spaces

We show that if a reproducing kernel Hilbert space $H_K,$ consisting of functions defined on ${\bf E},$ enjoys Double Boundary Vanishing Condition (DBVC) and Linear Independent Condition (LIC), then for any preset natural number $n,$ and any function $f\in H_K,$ there exists a set of $n$ parameterized multiple kernels ${\tilde{K}}_{w_1},\cdots,{\tilde{K}}_{w_n}, w_k\in {\bf E}, k=1,\cdots,n,$ and real (or complex) constants $c_1,\cdots,c_n,$ giving rise to a solution of the optimization problem \[ \|f-\sum_{k=1}^n c_k{\tilde{K}}_{w_k}\|=\inf \{\|f-\sum_{k=1}^n d_k{\tilde{K}}_{v_k}\|\ |\ v_k\in {\bf E}, d_k\in {\bf R}\ ({\rm or}\ {\bf C}), k=1,\cdots,n\}.\] By applying the theorem of this paper we show that the Hardy space and the Bergman space, as well as all the weighted Bergman spaces in the unit disc all possess $n$-best approximations. In the Hardy space case this gives a new proof of a classical result. Based on the obtained results we further prove existence of $n$-best spherical Poisson kernel approximation to functions of finite energy on the real-spheres.

math.CV↗

arXiv2018

Rational Approximation in the Bergman Spaces

It is known that adaptive Fourier decomposition (AFD) offers efficient rational approxima- tions to functions in the classical Hardy H2 spaces with significant applications. This study aims at rational approximation in Bergman, and more widely, in weighted Bergman spaces, the functions of which have more singularity than those in the Hardy spaces. Due to lack of an effective inner function theory, direct adaptation of the Hardy-space AFD is not performable. We, however, show that a pre-orthogonal method, being equivalent to AFD in the classical cases, is available for all weighted Bergman spaces. The theory in the Bergman spaces has equal force as AFD in the Hardy spaces. The methodology of approximation is via constructing the rational orthogonal systems of the Bergman type spaces, called Bergman space rational orthog- onal (BRO) system, that have the same role as the Takennaka-Malmquist (TM) system in the Hardy spaces. Subsequently, we prove a certain type direct sum decomposition of the Bergman spaces that reveals the orthogonal complement relation between the span of the BRO system and the zero-based invariant spaces. We provide a sequence of examples with different and ex- plicit singularities at the boundary along with a study on the inclusion relations of the weighted Bergman spaces. We finally present illustrative examples for effectiveness of the approximation.

math.FA↗