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We present an adaptive grid matrix-free operator approximation scheme based on a "product-convolution" interpolation of convolution operators. This scheme is appropriate for operators that are locally translation-invariant, even if these…
We introduce a variant of PCPs, that we refer to as rectangular PCPs, wherein proofs are thought of as square matrices, and the random coins used by the verifier can be partitioned into two disjoint sets, one determining the row of each…
Fr\'echet mean and variance provide a way of obtaining mean and variance for general metric space valued random variables and can be used for statistical analysis of data objects that lie in abstract spaces devoid of algebraic structure and…
In this Letter, we strengthen and extend the connection between simulation and estimation to exploit simulation routines that do not exactly compute the probability of experimental data, known as the likelihood function. Rather, we provide…
We consider continuous Dirac operators defined on $\mathbf{R}^d$, $d\in\{1,2,3\}$, together with various discrete versions of them. Both forward-backward and symmetric finite differences are used as approximations to partial derivatives. We…
Building on the free-probability stochastic control framework introduced in arXiv:2502.17329, we connect optimal control problems for $n \times n$ random matrix ensembles with their infinite-dimensional, free-probability analogues. Under…
A finite dimensional abstract approximation and convergence theory is developed for estimation of the distribution of random parameters in infinite dimensional discrete time linear systems with dynamics described by regularly dissipative…
In this article we introduce powerful tools and techniques from invariant theory to free analysis. This enables us to study free maps with involution. These maps are free noncommutative analogs of real analytic functions of several…
Free exponential families have been previously introduced as a special case of the q-exponential family. We show that free exponential families arise also from a procedure analogous to the definition of exponential families by using the…
A random matrix is likely to be well conditioned, and motivated by this well known property we employ random matrix multipliers to advance some fundamental matrix computations. This includes numerical stabilization of Gaussian elimination…
In this paper we revisit the classical Cauchy problem for Laplace's equation as well as two further related problems in the light of regularisation of this highly ill-conditioned problem by replacing integer derivatives with fractional…
Over the past decades, a great body of theoretical and mathematical work has been devoted to random-matrix descriptions of open quantum systems. In these notes, based on lectures delivered at the Les Houches Summer School "Stochastic…
We exhibit several bounds for operator norms of the sum of $\epsilon$-free semicircular random variables introduced in the paper of Speicher and Wysocza\'{n}ski. In particular, using the first and second largest eigenvalues of the adjacency…
Voiculescu's freeness emerges in computing the asymptotic of spectra of polynomials on $N\times N$ random matrices with eigenspaces in generic positions: they are randomly rotated with a uniform unitary random matrix $U_N$. In this article…
This paper considers the problem of approximating the inverse of the wave-equation Hessian, also called normal operator, in seismology and other types of wave-based imaging. An expansion scheme for the pseudodifferential symbol of the…
The relation between continuous functions and random vectors is revealed in the paper that the main meaning is described as, for any given continuous function, there must be a sequence of probability spaces and a sequence of random vectors…
This paper derives a free analog of the Euler-Maruyama method (fEMM) to numerically approximate solutions of free stochastic differential equations (fSDEs). Simply speaking fSDEs are stochastic differential equations in the context of…
Usual quantum mechanics predicts probabilities for the outcomes of measurements carried out at definite moments of time. However, realistic measurements do not take place in an instant, but are extended over a period of time. The assumption…
A similarity transformation is constructed through which a system of particles interacting with inverse-square two-body and harmonic potentials in one dimension, can be mapped identically, to a set of free harmonic oscillators. This…
The random matrix ensembles are applied to the quantum statistical two-dimensional systems of electrons. The quantum systems are studied using the finite dimensional real, complex and quaternion Hilbert spaces of the eigenfunctions. The…