SearcharxivSearch

arXiv subjects

Akash Anand

Publications and source records attributed to Akash Anand.

At least 19 recordsLinked to original sources

World Models Under Asynchronous Sensor Observations

Learned world models typically assume that observations arrive synchronously, an abstraction inherited from simulators that return a complete state vector at each environment step. Physical sensing instead operates at heterogeneous rates, leaving most observation channels stale at any given instant. Interpolating stale channels introduces measurements that were never observed, while downsampling to the slowest sensor discards valid measurements. A natural alternative is to zero-order-hold the most recent reading and provide the known sampling schedule to the model through two features, staleness and time-to-refresh. We test this prediction using transformer world models across three regimes of increasing causal coupling: open-loop rollouts in continuous-control locomotion, closed-loop model-predictive planning in which each learned model serves as the planner dynamics, and a linear latched-actuator system in which refresh events apply a zero-order-held command to the plant. Our findings show that the effectiveness of time-to-refresh depends on the causal role of the sampling schedule, specifically when refresh events affect the system rather than merely report its state. These results establish when sampling schedules provide useful information for predictive world models operating under asynchronous physical observations.

cs.AI

A Generalized FC-Gram Approximation Framework with Analysis and Applications

The FC-Gram algorithm constructs high-order trigonometric approximations of nonperiodic functions by periodically extending them to a larger interval, with the quality of the blending continuation of Gram polynomials over the extension interval directly governing the approximation accuracy. We introduce GenFC, a generalized FC-Gram framework in which the continuation of each Gram polynomial is shaped by a cutoff function satisfying prescribed boundary flatness conditions. We establish a convergence theorem showing that for any such family the GenFC approximation error satisfies $O(n^{-\min(r+\beta,\,d)})$ in the supremum norm on the original interval, where $f \in C^r([0,1])$ has an integrable $(r+1)$th derivative, $d$ is the number of Gram polynomials, and $\beta \in [0,1]$ is the Fourier decay exponent of $f^{(r+1)}$. The modified FC-Gram algorithm, recently introduced by the authors, is recovered as a special case, and several explicit families satisfying these conditions are constructed in the paper. Numerical experiments across smooth, limited-regularity, and rapidly oscillating test cases confirm the theoretical predictions. The framework is further applied to high-order solvers for linear ODEs and parabolic PDEs via backward differentiation formulae (BDF) time-stepping, demonstrating high-order accuracy throughout.

math.NA

Which Reconstruction Model Should a Robot Use? Routing Image-to-3D Models for Cost-Aware Robotic Manipulation

Robotic manipulation tasks require 3D mesh reconstructions of varying quality: dexterous manipulation demands fine-grained surface detail, while collision-free planning tolerates coarser representations. Multiple reconstruction methods offer different cost-quality tradeoffs, from Image-to-3D models - whose output quality depends heavily on the input viewpoint - to view-invariant methods such as structured light scanning. Querying all models is computationally prohibitive, motivating per-input model selection. We propose SCOUT, a novel routing framework that decouples reconstruction scores into two components: (1) the relative performance of viewpoint-dependent models, captured by a learned probability distribution, and (2) the overall image difficulty, captured by a scalar partition function estimate. As the learned network operates only over the viewpoint-dependent models, view-invariant pipelines can be added, removed, or reconfigured without retraining. SCOUT also supports arbitrary cost constraints at inference time, accommodating the multi-dimensional cost constraints common in robotics. We evaluate on the Google Scanned Objects, BigBIRD, and YCB datasets under multiple mesh quality metrics, demonstrating consistent improvements over routing baselines adapted from the LLM literature across various cost constraints. We further validate the framework through robotic grasping and dexterous manipulation experiments. We release the code and additional results on our website.

cs.RO

A modified FC-Gram approximation algorithm with provable error bounds

The FC-Gram trigonometric polynomial approximation of a non-periodic function that interpolates the function on equispaced grids was introduced in 2010 by Bruno and Lyon [J. Comput. Phys, 229(6):2009-2033, 2010]. Since then, the approximation algorithm and its further refinements have been used extensively in numerical solutions of various PDE-based problems, and it has had impressive success in handling challenging configurations. While much computational evidence exists in the literature confirming the rapid convergence of FC-Gram approximations, a theoretical convergence analysis has remained open. In this paper, we study a modified FC-Gram algorithm where the implicit least-squares-based periodic extensions of the Gram polynomials are replaced with an explicit extension utilizing two-point Hermite polynomials. This modification brings in two significant advantages - (i) as the extensions are known explicitly, the need to use computationally expensive precomputed extension data is eliminated, which, in turn, facilitates seamlessly changing the extension length, and (ii) allows for establishing provable error bounds for the modified approximations. We show that the numerical convergence rates are consistent with those predicted by the theory through a variety of computational experiments.

math.NA

Computation of highly oscillatory integrals using a Fourier extension approximation

The numerical evaluation of integrals of the form \begin{align*} \int_a^b f(x) e^{ikg(x)}\,dx \end{align*} is an important problem in scientific computing with significant applications in many branches of applied mathematics, science and engineering. The numerical approximation of such integrals using classical quadratures can be prohibitively expensive at high oscillation frequency ($k \gg 1$) as the number of quadrature points needed for achieving a reasonable accuracy must grow proportionally to $k$. To address this significant computational challenge, starting with Filon in 1930, several specialized quadratures have been developed to compute such oscillatory integrals efficiently. A crucial element in such Filon-type quadrature is the accurate evaluation of certain moments which poses a significant challenge when non-linear phase functions $g$ are involved. In this paper, we propose an equispaced-grid Filon-type quadrature for computing such highly oscillatory integrals that utilizes a Fourier extension of the slowly varying envelope $f$. This strategy is primarily aimed at significantly simplifying the moment calculations, even when the phase function has stationary points. Moreover, the proposed approach can also handle certain integrable singularities in the integrand. We analyze the scheme to theoretically establish high-order convergence rates. We also include a wide variety of numerical experiments, including oscillatory integrals with algebraic and logarithmic singularities, to demonstrate the performance of the quadrature.

math.NA

Joint complete monotonicity of rational functions in two variables and toral $m$-isometric pairs

We discuss the problem of classifying polynomials $p : \mathbb R^2_+ \rightarrow (0, \infty)$ for which $\frac{1}{p}=\{\frac{1}{p(m, n)}\}_{m, n \geq 0}$ is joint completely monotone, where $p$ is a linear polynomial in $y.$ We show that if $p(x, y)=a+b x+c y+d xy$ with $a > 0$ and $b, c, d \geq 0,$ then $\frac{1}{p}$ is joint completely monotone if and only if $a d - b c \leq 0.$ We also present an application to the Cauchy dual subnormality problem for toral $3$-isometric weighted $2$-shifts.

math.FA

Fast accurate approximation of convolutions with weakly singular kernel and its applications

In this article, we present an $O(N \log N)$ rapidly convergent algorithm for the numerical approximation of the convolution integral with radially symmetric weakly singular kernels and compactly supported densities. To achieve the reduced computational complexity, we utilize the Fast Fourier Transform (FFT) on a uniform grid of size $N$ for approximating the convolution. To facilitate this and maintain the accuracy, we primarily rely on a periodic Fourier extension of the density with a suitably large period depending on the support of the density. The rate of convergence of the method increases with increasing smoothness of the periodic extension and, in fact, approximations exhibit super-algebraic convergence when the extension is infinitely differentiable. Furthermore, when the density has jump discontinuities, we utilize a certain Fourier smoothing technique to accelerate the convergence to achieve the quadratic rate in the overall approximation. Finally, we apply the integration scheme for numerical solution of certain partial differential equations. Moreover, we apply the quadrature to obtain a fast and high-order Nystöm solver for the solution of the Lippmann-Schwinger integral equation. We validate the performance of the proposed scheme in terms of accuracy as well as computational efficiency through a variety of numerical experiments.

math.NA

Spectral Galerkin boundary element methods for high-frequency sound-hard scattering problems

This paper is concerned with the design of two different classes of Galerkin boundary element methods for the solution of high-frequency sound-hard scattering problems in the exterior of two-dimensional smooth convex scatterers. Both methods require a small increase in the order of $k^ε$ (for any $ε>0$) in the number of degrees of freedom to guarantee frequency independent precisions with increasing wavenumber $k$. In addition, the accuracy of the numerical solutions are independent of frequency provided sufficiently many terms in the asymptotic expansion are incorporated into the integral equation formulation. Numerical results validate our theoretical findings.

math.NA

Fourier smoothed pre-corrected trapezoidal rule for solution of Lippmann-Schwinger integral equation

For the numerical solution of the Lippmann-Schwinger equation, while the pre-corrected trapezoidal rule converges with high-order for smooth compactly supported densities, it exhibits only the linear convergence in the case of discontinuity in material properties across the interface. In this short article, we propose a Nyström solver based on "Fourier smoothed pre-corrected trapezoidal rule" that converges with second order for such scattering problems while maintaining the computational complexity of $O(N \log N)$. Moreover, the method is not only very simple to implement, it is also applicable to problems with geometrically complex inhomogeneities including those with corners and cusps. We present a variety of numerical experiments including comparative studies with competing approaches reported in [J. Comput. Phys., 200(2) (2004), 670--694] by Bruno and Hyde, and in [J. Fourier Anal. Appl., 11(4) (2005), 471-487 ] by Andersson and Holst to exemplify its performance in terms of speed and accuracy. This Fourier smoothed numerical integration scheme can also be adapted to other problems of interest where the convolution integral with discontinuous density is required to be computed.

math.NA

Knowledge synthesis from 100 million biomedical documents augments the deep expression profiling of coronavirus receptors

The COVID-19 pandemic demands assimilation of all available biomedical knowledge to decode its mechanisms of pathogenicity and transmission. Despite the recent renaissance in unsupervised neural networks for decoding unstructured natural languages, a platform for the real-time synthesis of the exponentially growing biomedical literature and its comprehensive triangulation with deep omic insights is not available. Here, we present the nferX platform for dynamic inference from over 45 quadrillion possible conceptual associations extracted from unstructured biomedical text, and their triangulation with Single Cell RNA-sequencing based insights from over 25 tissues. Using this platform, we identify intersections between the pathologic manifestations of COVID-19 and the comprehensive expression profile of the SARS-CoV-2 receptor ACE2. We find that tongue keratinocytes and olfactory epithelial cells are likely under-appreciated targets of SARS-CoV-2 infection, correlating with reported loss of sense of taste and smell as early indicators of COVID-19 infection, including in otherwise asymptomatic patients. Airway club cells, ciliated cells and type II pneumocytes in the lung, and enterocytes of the gut also express ACE2. This study demonstrates how a holistic data science platform can leverage unprecedented quantities of structured and unstructured publicly available data to accelerate the generation of impactful biological insights and hypotheses.

q-bio.GN

The Cauchy dual subnormality problem for cyclic $2$-isometries

The Cauchy dual subnormality problem asks whether the Cauchy dual operator of a $2$-isometry is subnormal. Recently this problem has been solved in the negative. Here we show that it has a negative solution even in the class of cyclic $2$-isometries.

math.FA

Trefftz Finite Elements on Curvilinear Polygons

We present a Trefftz-type finite element method on meshes consisting of curvilinear polygons. Local basis functions are computed using integral equation techniques that allow for the efficient and accurate evaluation of quantities needed in the formation of local stiffness matrices. To define our local finite element spaces in the presence of curved edges, we must also properly define what it means for a function defined on a curved edge to be "polynomial" of a given degree on that edge. We consider two natural choices, before settling on the one that yields the inclusion of complete polynomial spaces in our local finite element spaces, and discuss how to work with these edge polynomial spaces in practice. An interpolation operator is introduced for the resulting finite elements, and we prove that it provides optimal order convergence for interpolation error under reasonable assumptions. We provide a description of the integral equation approach used for the examples in this paper, which was recently developed precisely with these applications in mind. A few numerical examples illustrate this optimal order convergence of the finite element solution on some families of meshes in which every element has at least one curved edge. We also demonstrate that it is possible to exploit the approximation power of locally singular functions that may exist in our finite element spaces in order to achieve optimal order convergence without the typical adaptive refinement toward singular points.

math.NA

Analytic $m$-isometries without the wandering subspace property

The wandering subspace problem for an analytic norm-increasing $m$-isometry $T$ on a Hilbert space $\mathcal H$ asks whether every $T$-invariant subspace of $\mathcal H$ can be generated by a wandering subspace. An affirmative solution to this problem for $m=1$ is ascribed to Beurling-Lax-Halmos, while that for $m=2$ is due to Richter. In this paper, we capitalize on the idea of weighted shift on one-circuit directed graph to construct a family of analytic cyclic $3$-isometries, which do not admit the wandering subspace property and which are norm-increasing on the orthogonal complement of a one-dimensional space. Further, on this one dimensional space, their norms can be made arbitrarily close to $1$. We also show that if the wandering subspace property fails for an analytic norm-increasing $m$-isometry, then it fails miserably in the sense that the smallest $T$-invariant subspace generated by the wandering subspace is of infinite codimension.

math.FA

A Fourier extension based numerical integration scheme for fast and high-order approximation of convolutions with weakly singular kernels

Computationally efficient numerical methods for high-order approximations of convolution integrals involving weakly singular kernels find many practical applications including those in the development of fast quadrature methods for numerical solution of integral equations. Most fast techniques in this direction utilize uniform grid discretizations of the integral that facilitate the use of FFT for $O(n\log n)$ computations on a grid of size $n$. In general, however, the resulting error converges slowly with increasing $n$ when the integrand does not have a smooth periodic extension. Such extensions, in fact, are often discontinuous and, therefore, their approximations by truncated Fourier series suffer from Gibb's oscillations. In this paper, we present and analyze an $O(n\log n)$ scheme, based on a Fourier extension approach for removing such unwanted oscillations, that not only converges with high-order but is also relatively simple to implement. We include a theoretical error analysis as well as a wide variety of numerical experiments to demonstrate its efficacy.

math.NA

A Fourier continuation framework for high-order approximations

It is well known that approximation of functions on $[0,1]$ whose periodic extension is not continuous fail to converge uniformly due to rapid Gibbs oscillations near the boundary. Among several approaches that have been proposed toward the resolution of Gibbs phenomenon in recent years, a Fourier continuation (FC) based approximation scheme has been suggested by Bruno and collaborators in the context of certain PDE solvers where approximation grids used are equispaced. While the practical efficacy of FC based schemes in obtaining a high-order numerical solution of PDEs is well known, theoretical convergence analyses largely remain unavailable. The primary objective of this paper is to take a step in this direction where we analyze the convergence rates of a Fourier continuation framework for approximations based on discrete functional data coming from equispaced grids. In this context, we explore a certain two-point Hermite interpolation strategy for constructing Fourier continuations that, not only simplifies the implementation of such approximations but also makes possible a rigorous analysis of its numerical properties. In particular, we show that the approximations converge with order $r+1$ for functions coming from a subspace of $C^{r,1}([0,1])$, the space of $r$-times continuously differentiable function whose $r$th derivative is Lipschitz continuous. We also demonstrate that theoretical rates are indeed achieved in practice, through a variety of numerical experiments.

math.NA

Complete systems of unitary invariants for some classes of $2$-isometries

The unitary equivalence of $2$-isometric operators satisfying the so-called kernel condition is characterized. It relies on a model for such operators built on operator valued unilateral weighted shifts and on a characterization of the unitary equivalence of operator valued unilateral weighted shifts in a fairly general context. A complete system of unitary invariants for $2$-isometric weighted shifts on rooted directed trees satisfying the kernel condition is provided. It is formulated purely in the langauge of graph-theory, namely in terms of certain generation branching degrees. The membership of the Cauchy dual operators of $2$-isometries in classes $C_{0 \cdot}$ and $C_{\cdot 0}$ is also studied.

math.FA

Improved convergence of fast integral equation solvers for acoustic scattering by inhomogeneous penetrable media with discontinuous material interface

In recent years, several fast solvers for the solution of the Lippmann-Schwinger integral equation that mathematically models the scattering of time-harmonic acoustic waves by penetrable inhomogeneous obstacles, have been proposed. While many of these fast methodologies exhibit rapid convergence for smoothly varying scattering configurations, the rate for most of them reduce to either linear or quadratic when material properties are allowed to jump across the interface. A notable exception to this is a recently introduced Nystr\"{o}m scheme [J. Comput. Phys., 311 (2016), 258--274] that utilizes a specialized quadrature in the boundary region for a high-order treatment of the material interface. In this text, we present a solution framework that relies on the specialized boundary integrator to enhance the convergence rate of other fast, low order methodologies without adding to their computational complexity of $O(N \log N)$ for an $N$-point discretization. In particular, to demonstrate the efficacy of the proposed framework, we explain its implementation to enhance the order to convergence of two schemes, one introduced by Duan and Rokhlin [J. Comput. Phys., 228(6) (2009), 2152--2174] that is based on a pre-corrected trapezoidal rule while the other by Bruno and Hyde [J. Comput. Phys., 200(2) (2004), 670--694] which relies on a suitable decomposition of the Green's function via Addition theorem. In addition to a detailed description of these methodologies, we also present a comparative performance study of the improved versions of these two and the Nystr\"{o}m solver in [J. Comput. Phys., 311 (2016), 258--274] through a wide range of numerical experiments.

math.NA

A solution to the Cauchy dual subnormality problem for 2-isometries

The Cauchy dual subnormality problem asks whether the Cauchy dual operator $T^{\prime}:=T(T^*T)^{-1}$ of a $2$-isometry $T$ is subnormal. In the present paper we show that the problem has a negative solution. The first counterexample depends heavily on a reconstruction theorem stating that if $T$ is a $2$-isometric weighted shift on a rooted directed tree with nonzero weights that satisfies the perturbed kernel condition, then $T^{\prime}$ is subnormal if and only if $T$ satisfies the (unperturbed) kernel condition. The second counterexample arises from a $2$-isometric adjacency operator of a locally finite rooted directed tree again by thorough investigations of positive solutions of the Cauchy dual subnormality problem in this context. We prove that if $T$ is a $2$-isometry satisfying the kernel condition or a quasi-Brownian isometry, then $T^{\prime}$ is subnormal. We construct a $2$-isometric adjacency operator $T$ of a rooted directed tree such that $T$ does not satisfy the kernel condition, $T$ is not a quasi-Brownian isometry and $T^{\prime}$ is subnormal.

math.FA