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Abstract: Abstract In this paper, acceleration of gradient methods for convex optimization problems with weak levels of convexity and smoothness is considered. Starting from the universal fast gradient method which was designed to be an optimal method for weakly smooth problems whose gradients are Hölder continuous, its momentum is modified appropriately so that it can also accommodate uniformly convex and weakly smooth problems. Different from the existing works, fast gradient methods proposed in this paper do not use the restarting technique but use momentums that are suitably designed to reflect both the uniform convexity and weak smoothness information of the target energy function. Both theoretical and numerical results that support the superiority of the proposed methods are presented. PubDate: 2022-05-24
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Abstract: Abstract This paper proposes a new type of multilevel method for solving the Steklov eigenvalue problem based on Newton’s method. In this iteration method, solving the Steklov eigenvalue problem is replaced by solving a small-scale eigenvalue problem on the coarsest mesh and a sequence of augmented linear problems on refined meshes, derived by Newton step. We prove that this iteration scheme obtains the optimal convergence rate with linear complexity, which improves the overall efficiency of solving the Steklov eigenvalue problem. Moreover, an adaptive iteration scheme for multi eigenvalues based on this new multilevel method is given. Finally, some numerical experiments are provided to illustrate the efficiency of the proposed multilevel scheme. PubDate: 2022-05-13
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Abstract: Abstract Hierarchical \({\mathscr{H}}^{2}\) -matrices are asymptotically optimal representations for the discretizations of non-local operators such as those arising in integral equations or from kernel functions. Their O(N) complexity in both memory and operator application makes them particularly suited for large-scale problems. As a result, there is a need for software that provides support for distributed operations on these matrices to allow large-scale problems to be represented. In this paper, we present high-performance, distributed-memory GPU-accelerated algorithms and implementations for matrix-vector multiplication and matrix recompression of hierarchical matrices in the \({\mathscr{H}}^{2}\) format. The algorithms are a new module of H2Opus, a performance-oriented package that supports a broad variety of \({\mathscr{H}}^{2}\) matrix operations on CPUs and GPUs. Performance in the distributed GPU setting is achieved by marshaling the tree data of the hierarchical matrix representation to allow batched kernels to be executed on the individual GPUs. MPI is used for inter-process communication. We optimize the communication data volume and hide much of the communication cost with local compute phases of the algorithms. Results show near-ideal scalability up to 1024 NVIDIA V100 GPUs on Summit, with performance exceeding 2.3 Tflop/s/GPU for the matrix-vector multiplication, and 670 Gflop/s/GPU for matrix compression, which involves batched QR and SVD operations. We illustrate the flexibility and efficiency of the library by solving a 2D variable diffusivity integral fractional diffusion problem with an algebraic multigrid-preconditioned Krylov solver and demonstrate scalability up to 16M degrees of freedom problems on 64 GPUs. PubDate: 2022-05-10
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Abstract: Abstract For the Oseen problem, we present a new stabilized virtual element method on polygonal meshes that allows us to employ “equal-order” virtual element pairs to approximate both velocity and pressure. By introducing the local projection type stabilization terms to the virtual element method, the method can not only circumvent the discrete Babuška-Brezzi condition, but also maintain the favorable stability and approximation properties of residual-based stabilization methods. In particular, it does not need to calculate complex high-order derivative terms and avoids the strong coupling terms of velocity and pressure. Error estimates are obtained without depending on the inverse of the viscosity, which means that the method is effective in the convective-dominated regime. Some numerical experiments are performed to verify the method has good behaviors. PubDate: 2022-05-10
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Abstract: Abstract The paper introduces an one-step optimization method for solving a monotone equilibrium problem including a Lipschitz-type condition in a Hilbert space. The method uses variable stepsizes and is constructed by the proximal-like mapping associated with the cost bifunction and incorporated with regularization terms. Comparing with the extragradient-like methods, our new method has an elegant and simple structure with a cheap computation over each iteration. By an appropriate choice of stepsizes and regularization parameters, we establish the strong convergence of the iterative sequence generated by the method to a solution of the considered equilibrium problem. We also show the numerical behavior of our new method and illustrate the computational effectiveness of it over other methods via experiments. PubDate: 2022-05-10
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Abstract: Abstract Boundary element methods produce dense linear systems that can be accelerated via multipole expansions. Solved with Krylov methods, this implies computing the matrix-vector products within each iteration with some error, at an accuracy controlled by the order of the expansion, p. We take advantage of a unique property of Krylov iterations that allows lower accuracy of the matrix-vector products as convergence proceeds, and propose a relaxation strategy based on progressively decreasing p. In extensive numerical tests of the relaxed Krylov iterations, we obtained speed-ups of between 1.5 × and 2.3 × for Laplace problems and between 2.7 × and 3.3 × for Stokes problems. We include an application to Stokes flow around red blood cells, computing with up to 64 cells and problem size up to 131k boundary elements and nearly 400k unknowns. The study was done with an in-house multi-threaded C++ code, on a hexa-core CPU. The code is available on its version-control repository, https://github.com/barbagroup/fmm-bem-relaxed, and we share reproducibility packages for all results in https://github.com/barbagroup/inexact-gmres/. PubDate: 2022-05-10
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Abstract: Abstract This paper is devoted to an optimal control problem of a coupled spin drift-diffusion Landau–Lifshitz–Gilbert system describing the interplay of magnetization and spin accumulation in magnetic-nonmagnetic multilayer structures, where the control is given by the electric current density. A variational approach is used to prove the existence of an optimal control. The first-order necessary optimality system for the optimal solution is derived in one space-dimension via Lagrange multiplier method. Numerical examples are reported to validate the theoretical findings. PubDate: 2022-05-06
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Abstract: Abstract This paper is devoted to the study of a time-discrete scheme and its corresponding fully discretization approximating a d-dimensional chemotaxis model describing tumor invasion, d ≤ 3. This model describes the chemotactic attraction experienced by the tumor cells and induced by a so-called active extracellular matrix, which is a chemical signal produced by a biological reaction between the extracellular matrix and a matrix-degrading enzyme. In order to construct the numerical approximations and to control the chemo-attraction term in the tumor cells equation, we introduce an equivalent model with a new variable given by the gradient of the active extracellular matrix and use an inductive strategy. Then, we consider a first-order and non-linear time-discrete scheme which is mass-conservative and possesses the property of positivity for all the biological variables. After, we study the corresponding fully discrete finite element with “mass-lumping” approximation proving well-posedness, mass-conservation and the non-negativity of the extracellular matrix, the degrading enzyme, and the active extracellular matrix. In addition, we obtain uniform strong estimates required in the convergence analysis, and we prove optimal error estimates and convergence towards regular solutions. Finally, we provide some numerical results in agreement with our theoretical analysis with respect to the positivity and the error estimates. PubDate: 2022-05-03
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Abstract: Abstract In this paper, a novel three-point fourth-order compact operator is considered to construct new linearized conservative compact finite difference scheme for the symmetric regularized long wave (SRLW) equations based on the reduction order method with three-level linearized technique. The discrete conservative laws, boundedness and unique solvability are studied. The convergence order \(\mathcal {O}(\tau ^{2}+h^{4})\) in the \(L^{\infty }\) -norm and stability of the present compact scheme are proved by the discrete energy method. Numerical examples are given to support the theoretical analysis. PubDate: 2022-05-03
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Abstract: Abstract Recently, a Monte Carlo approach was proposed for processing highly redundant continuous frames. In this paper, we present and analyze applications of this new theory. The computational complexity of the Monte Carlo method relies on the continuous frame being so-called linear volume discretizable (LVD). The LVD property means that the number of samples in the coefficient space required by the Monte Carlo method is proportional to the resolution of the discrete signal. We show in this paper that the continuous wavelet transform (CWT) and the localizing time-frequency transform (LTFT) are LVD. The LTFT is a time-frequency representation based on a 3D time-frequency space with a richer class of time-frequency atoms than classical time-frequency transforms like the short time Fourier transform (STFT) and the CWT. Our analysis proves that performing signal processing with the LTFT has the same asymptotic complexity as signal processing with the STFT and CWT (based on FFT), even though the coefficient space of the LTFT is higher dimensional. PubDate: 2022-04-27
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Abstract: Abstract We present a method to stabilize bases with local supports by means of extension. It generalizes the known approach for tensor product B-splines to a much broader class of functions, which includes hierarchical and weighted variants of polynomial, trigonometric, and exponential splines, but also box splines, T-splines, and other function spaces of interest with a local basis. Extension removes elements that cause instabilities from a given basis by linking them with the remaining ones by means of a specific linear combination. The two guiding principles for this process are locality and persistence. Locality aims at coupling basis functions whose supports are close together, while persistence guarantees that a given set of globally supported functions, like certain monomials in the case of polynomial splines, remain in the span of the basis after extension. Furthermore, we study how extension influences the approximation power and the condition of Gramian matrices associated with the basis, and present a series of examples illustrating the potential of the method. PubDate: 2022-04-21
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Abstract: Abstract A priori error analysis of the finite element approximation of Stokes equations under slip boundary condition of friction type has been centered on the interpolation error on the slip zone. In this work, we propose a novel approach based on the approximation of the tangential component of traction force by a truncated (cutoff) function. More precisely, we carry out (i) a complete analysis of the truncated formulation from the continuous to discrete level in two and three dimensions. In particular, we show linear convergence rate of the finite element solution by assuming standard regularity of the weak solution. This improves all previous results. (ii) The description of our solution strategy, (iii) a verification of the convergence properties with analytic solution and benchmark tests. PubDate: 2022-04-21
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Abstract: Abstract In this paper, we are concerned about a time-domain carpet cloak model, which was originally derived in our previous work Li et al. (SIAM J. Appl. Math., 74(4), pp. 1136–1151, 2014). Some finite element schemes have been developed for this model and used to simulate the cloaking phenomenon in Li et al. (SIAM J. Appl. Math., 74(4), pp. 1136–1151, 2014) and Li et al. (Methods Appl. Math., 19(2), pp. 359–378, 2019). However, numerical stabilities for those proposed explicit schemes are only proved under the time step constrain τ = O(h2), which is impractical and too restricted. To overcome this disadvantage, we propose two new finite element schemes for solving this carpet cloak model: one is the implicit Crank-Nicolson (CN) scheme, and another one is the explicit leap-frog (LF) scheme. Inspired by a totally new energy developed for the continuous model, we prove the unconditional stability for the CN scheme and conditional stability for the LF scheme under the usual CFL constraint τ = O(h). Both numerical stabilities inherit the exact form as the continuous stability. Optimal error estimate is also established for the LF scheme. Finally, numerical results using the LF scheme are presented to support our analysis and demonstrate the cloaking phenomenon. PubDate: 2022-04-21
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Abstract: Abstract Simulating the flow of water in district heating networks requires numerical methods which are independent of the CFL condition. We develop a high order scheme for networks of advection equations allowing large time steps. With the MOOD technique, unphysical oscillations of nonsmooth solutions are avoided. In numerical tests, the applicability to real networks is shown. PubDate: 2022-04-18
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Abstract: Abstract Fast and high-order accurate algorithms for three-dimensional elastic scattering are of great importance when modeling physical phenomena in mechanics, seismic imaging, and many other fields of applied science. In this paper, we develop a novel boundary integral formulation for the three-dimensional elastic scattering based on the Helmholtz decomposition of elastic fields, which converts the Navier equation to a coupled system consisted of Helmholtz and Maxwell equations. An FFT-accelerated separation of variables solver is proposed to efficiently invert boundary integral formulations of the coupled system for elastic scattering from axisymmetric rigid bodies. In particular, by combining the regularization properties of the singular boundary integral operators and the FFT-based fast evaluation of modal Green’s functions, our numerical solver can rapidly solve the resulting integral equations with a high-order accuracy. Several numerical examples are provided to demonstrate the efficiency and accuracy of the proposed algorithm, including geometries with corners at different wavenumbers. PubDate: 2022-04-18
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Abstract: Abstract The purpose of this paper is twofold. The first is to give some new structural results for the invertibility of Bessel multipliers. Secondly, as applications of these results, we provide some conditions regarding the scaling sequence c = {cn}n which can be used in the role of the scalability of a given frame, a notion which has found more and more applications in the last decade. More precisely, we show that positive and strict scalability coincides for all frames Φ = {φn}n with \(\liminf _{n}\parallel \varphi _{n}\parallel >0\) which in particular provides some equivalent conditions for positive scalability of certain frames. Moreover, it is our objective to consider the effect of optimal frame bounds on the choice of scalings sequence. Along the way, the scalability of Riesz frames and Riesz bases are completely characterized and some necessary conditions for scalability of a near-Riesz basis are determined depending on its norm properties. Next, we turn our attention to the (c-)scalable bounded frame Φ and our results with the aid of the Feichtinger Conjecture give α and β, depending on the optimal frame bounds of Φ, such that the elements of the scaling sequence c should be chosen from the interval [α,β] for all but finitely many n. Finally, we introduce an explicit construction algorithm to produce desired invertible multiplier from the given one which is of interest in its own right. PubDate: 2022-04-14
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Abstract: Abstract We analyze rates of approximation by quantized, tensor-structured representations of functions with isolated point singularities in ℝ3. We consider functions in countably normed Sobolev spaces with radial weights and analytic- or Gevrey-type control of weighted semi-norms. Several classes of boundary value and eigenvalue problems from science and engineering are discussed whose solutions belong to the countably normed spaces. It is shown that quantized, tensor-structured approximations of functions in these classes exhibit tensor ranks bounded polylogarithmically with respect to the accuracy ε ∈ (0,1) in the Sobolev space H1. We prove exponential convergence rates of three specific types of quantized tensor decompositions: quantized tensor train (QTT), transposed QTT and Tucker QTT. In addition, the bounds for the patchwise decompositions are uniform with respect to the position of the point singularity. An auxiliary result of independent interest is the proof of exponential convergence of hp-finite element approximations for Gevrey-regular functions with point singularities in the unit cube Q = (0,1)3. Numerical examples of function approximations and of Schrödinger-type eigenvalue problems illustrate the theoretical results. PubDate: 2022-04-14
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Abstract: Abstract In this paper, we study a direct parallel-in-time (PinT) algorithm for first- and second-order time-dependent differential equations. We use a second-order boundary value method as the time integrator. Instead of solving the corresponding all-at-once system iteratively, we diagonalize the time discretization matrix B, which yields a direct parallel implementation across all time steps. A crucial issue of this methodology is how the condition number (denoted by Cond2(V )) of the eigenvector matrix V of B behaves as n grows, where n is the number of time steps. A large condition number leads to large roundoff error in the diagonalization procedure, which could seriously pollute the numerical accuracy. Based on a novel connection between the characteristic equation and the Chebyshev polynomials, we present explicit formulas for V and V− 1, by which we prove that Cond \(_{2}(V)=\mathcal {O}(n^{2})\) . This implies that the diagonalization process is well-conditioned and the roundoff error only increases moderately as n grows, and thus, compared to other direct PinT algorithms, a much larger n can be used to yield satisfactory parallelism. A fast structure-exploiting algorithm is also designed for computing the spectral diagonalization of B. Numerical results on parallel machine are given to support our findings, where over 60 times speedup is achieved with 256 cores. PubDate: 2022-04-06
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Abstract: Abstract In this work, an r-linearly converging adaptive solver is constructed for parabolic evolution equations in a simultaneous space-time variational formulation. Exploiting the product structure of the space-time cylinder, the family of trial spaces that we consider are given as the spans of wavelets-in-time and (locally refined) finite element spaces-in-space. Numerical results illustrate our theoretical findings. PubDate: 2022-04-06
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Abstract: Abstract We consider approximations formed by the sum of a linear combination of given functions enhanced by ridge functions—a Linear/Ridge expansion. For an explicitly or implicitly given objective function, we reformulate finding a best Linear/Ridge expansion in terms of an optimization problem. We introduce a particle grid algorithm for its solution. Several numerical results underline the flexibility, robustness and efficiency of the algorithm. One particular source of motivation is model reduction of parameterized transport or wave equations. We show that the particle grid algorithm is able to find a Linear/Ridge expansion as an efficient nonlinear model reduction. PubDate: 2022-04-04
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