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We introduce an approach for solving PDEs over manifolds using physics informed neural networks whose architecture aligns with spectral methods. The networks are trained to take in as input samples of an initial condition, a time stamp and point(s) on the manifold and then output the solution's value at the given time and point(s). We provide proofs of our method for the heat equation on the interval and examples of unique network architectures that are adapted to nonlinear equations on the sphere and the torus. We also show that our spectral-inspired neural network architectures outperform the standard physics informed architectures. Our extensive experimental results include generalization studies where the testing dataset of initial conditions is randomly sampled from a significantly larger space than the training set., Comment: 25 pages

One of the main challenges in computed tomography (CT) is how to balance between the amount of radiation the patient is exposed to during scan time and the quality of the reconstructed CT image. We propose a mathematical model for adaptive CT sensing whose goal is to reduce dosage levels while maintaining high image quality at the same time. The adaptive algorithm iterates between selective limited sensing and improved reconstruction, with the goal of applying only the dose level required for sufficient image quality. The theoretical foundation of the algorithm is nonlinear Ridgelet approximation and a discrete form of Ridgelet analysis is used to compute the selective acquisition steps that best capture the image edges. We show experimental results where for the same number of line projections, the adaptive model produces higher image quality, when compared with standard limited angle, nonadaptive sensing algorithms.

This paper considers the problem of estimating the delays of a weighted superposition of pulses, called stream of pulses, in a noisy environment. We show that the delays can be estimated using a tractable convex optimization problem with a localization error proportional to the square root of the noise level. Furthermore, all false detections produced by the algorithm have small amplitudes. Numerical and in-vitro ultrasound experiments corroborate the theoretical results and demonstrate their applicability for the ultrasound imaging signal processing.

In this work we consider the problem of recovering an ensemble of Diracs on the sphere from its projection onto spaces of spherical harmonics. We show that under an appropriate separation condition on the unknown locations of the Diracs, the ensemble can be recovered through Total Variation norm minimization. The proof of the uniqueness of the solution uses the method of `dual' interpolating polynomials and is based on [8], where the theory was developed for trigonometric polynomials. We also show that in the special case of non-negative ensembles, a sparsity condition is sufficient for exact recovery.

In this work we develop highly geometric Hardy spaces, for the full range 0

Anisotropic Besov spaces (B-spaces) are developed based on anisotropic multilevel ellipsoid covers (dilations) of R n . This extends earlier results on anisotropic Besov spaces. Furthermore, sequences of anisotropic bases are constructed and uti- lized for two-level-split decompositions of the B-spaces and nonlinear m-term ap- proximation.

We present a construction of anisotropic multiresolution and anisotropic wavelet frames based on multilevel ellipsoid covers (dilations) of ℝn. The wavelets we construct are C∞ functions, can have any prescribed number of vanishing moments and fast decay with respect to the anisotropic quasi-distance induced by the cover. The dual wavelets are also C∞, with the same number of vanishing moments, but with only mild decay with respect to the quasi-distance. An alternative construction yields a meshless frame whose elements do not have vanishing moments, but do have fast anisotropic decay.

We present a new image coding algorithm, the geometric piecewise polynomials (GPP) method, that draws on recent developments in the theory of adaptive multivariate piecewise polynomials approximation. The algorithm relies on a segmentation stage whose goal is to minimize a functional that is conceptually similar to the Mumford-Shah functional except that it measures the smoothness of the segmentation instead of the length. The initial segmentation is ldquoprunedrdquo and the remaining curve portions are lossy encoded. The image is then further partitioned and approximated by low order polynomials on the subdomains. We show examples where our algorithm outperforms state-of-the-art wavelet coding in the low bit-rate range. The GPP algorithm significantly outperforms wavelet based coding methods on graphic and cartoon images. Also, at the bit rate 0.05 bits per pixel, the GPP algorithm achieves on the test image Cameraman, which has a geometric structure, a PSNR of 21.5 dB, while the JPEG2000 Kakadu software obtains PSNR of 20 dB. For the test image Lena, the GPP algorithm obtains the same PSNR as JPEG2000, but with better visual quality at 0.03 bpp.

This paper describes a new and efficient method for low bit-rate image coding which is based on recent development in the theory of multivariate nonlinear piecewise polynomial approximation. It combines a binary space partition scheme with geometric wavelet (GW) tree approximation so as to efficiently capture curve singularities and provide a sparse representation of the image. The GW method successfully competes with state-of-the-art wavelet methods such as the EZW, SPIHT, and EBCOT algorithms. We report a gain of about 0.4 dB over the SPIHT and EBCOT algorithms at the bit-rate 0.0625 bits-per-pixels (bpp). It also outperforms other recent methods that are based on "sparse geometric representation." For example, we report a gain of 0.27 dB over the Bandelets algorithm at 0.1 bpp. Although the algorithm is computationally intensive, its time complexity can be significantely reduced by collecting a "global" GW n-term approximation to the image from a collection of GW trees, each constructed separately over tiles of the image.

It is well known that it is possible to enhance the approximation properties of a kernel operator by increasing its support size. There is an obvious tradeoff between higher approximation order of a kernel and the complexity of algorithms that employ it. A question is then asked: how do we compare the efficiency of kernels with comparable support size? We follow Blu and Unser and choose as a measure of the efficiency of the kernels the first leading constant in a certain error expansion. We use time domain methods to treat the case of globally supported kernels in L p (R d ), 1≤p≤∞.

The Bramble--Hilbert lemma is a fundamental result on multivariate polynomial approximation. It is frequently applied in the analysis of finite elements methods (FEM) used for numerical solutions of PDEs. However, this classical estimate depends on the geometry of the domain and may "blow up" for simple examples such as a sequence of triangles of equivalent diameter that become thinner and thinner. Thus, in FEM applications one usually requires that the mesh has ``quasi-uniform" geometry. This assumption is perhaps too restrictive when one tries to obtain estimates of nonlinear approximation methods that use piecewise polynomials.Our main result that improves upon this point is the following. Let $\Omega \subset \RR^n$ be a bounded convex domain and let $g \in W^m_p (\Omega)$, $m \in \NN$, $1 \le p \le \infty$, where $W^m_p (\Omega)$ is the Sobolev space. Then there exists a polynomial P of total degree $m-1$ for which $$ |g-P|_{k,p} \le C(n,m)\big({\rm diam}\,\Omega\big)^{m-k} |g|_{m,p},\quad k=0,1,\ldot...