arXiv subjects

Matteo Ferrari

Publications and source records attributed to Matteo Ferrari.

At least 19 recordsLinked to original sources

arXiv2026

Condition numbers of block Toeplitz matrices and stability of space-time IgA approximations for the wave and Schr\"odinger equations

In previous work by several authors, the behavior of the condition numbers of banded Toeplitz matrices was studied as the matrix size tends to infinity. In the present contribution, two main directions are pursued. As a first step, we extend this study to block Toeplitz matrices with blocks of fixed size $N$. As in the scalar case, we show that even when the symbol generates a Fredholm infinite Toeplitz operator, the condition numbers of the finite matrices may grow at least exponentially. Upper and lower bounds for the condition numbers are obtained, and examples showing that they may grow arbitrarily fast are presented. Then, as a second step, we apply the developed theory to the stability analysis of space-time Galerkin methods, where in time an Isogeometric approach is used with regularity $r$, $1\le r\le p-1$, $p$ being the employed polynomial degree. These stability issues are related exactly to the conditioning of block Toeplitz-like matrices with blocks of size $N=p-r$. Specific examples are treated in detail and related numerical experiments are presented and critically discussed. We finally present a short list of relevant open problems.

math.NA↗

arXiv2026

Convolution quadrature based on a truncated trapezoidal rule

We study the truncated trapezoidal rule, a family of A-stable second order multistep methods parametrized by an integer $J \ge 2$, a compromise between BDF2 and the trapezoidal rule. We obtain a closed-form expression for the coefficients that minimize the principal error constant under the A-stability constraint and we derive an explicit formula for the corresponding principal error constant. The latter decreases to the optimal Dahlquist value $1/12$ as $J$ increases, and is strictly smaller than the BDF2 constant for every $J \ge 2$. We apply the truncated trapezoidal rule within the convolution quadrature framework. Its analyticity in a neighbourhood of the closed unit disk yields milder regularity and perturbation requirements than those of the trapezoidal rule, while the error constant can be made arbitrarily close to the optimal one by increasing $J$. Numerical experiments show that convolution quadrature based on the truncated trapezoidal rule remains stable under symbol perturbations, where the trapezoidal rule fails, while achieving a smaller error constant than BDF2.

math.NA↗

arXiv2026

Financial Resilience Evaluation: From Conditional Expectations to Dynamic Convex Risk Measures

Financial resilience concerns the rate at which a position recovers, or further deteriorates, in response to adverse conditions. As a first step, Laeven, Ferrari, Rosazza Gianin, and Zullino (arXiv:2505.07502) introduced the resilience rate, defined as the expected instantaneous rate of (favorable) change of a price or risk-assessment process. Since this quantity captures only the conditional mean of future increments, it cannot distinguish between positions having the same expected recovery but different conditional risk profiles. We obtain a richer characterization by evaluating such increments through a genuine, possibly nonlinear, dynamic risk measure. More precisely, for an It\^o process $\pi$ and a normalized, cash-additive dynamic risk measure $\rho$, we define the resilience evaluation by \[\mathcal D_s^\rho\pi_t := L^1\text{-}\lim_{\varepsilon\to0^+} \frac{1}{\varepsilon}\rho_s(\pi_{t+\varepsilon}-\pi_t), \qquad 0\leq s\leq t<T,\] whenever the limit exists. When $\rho$ is a convex dynamic risk measure induced by a BSDE with a Lipschitz or quadratic driver, we prove that this limit is well-posed and admits an explicit dual representation. It is given by the worst-case conditional expectation, over a zero-penalty class of measure changes, of an effective drift combining the drift of $\pi$ with the risk adjustment assigned by $\rho$ to its volatility. We further establish attainment of the optimal scenario and illustrate the scope of the construction, as well as the role of the assumptions, through examples and counterexamples.

q-fin.MF↗

arXiv2026

A space-time sparse-grid method for the wave equation

We develop a fast space-time numerical scheme for approximating solutions to the linear wave equation. The approach is based on the sparse-grid combination technique applied to a coercive space-time discretization. Designed for tensor-product space-time discretizations, the method enables efficient parallelization of the resulting solver. We provide a rigorous theoretical analysis establishing convergence rates and computational complexity estimates. Numerical experiments validate the theoretical estimates and demonstrate the efficiency of the proposed method.

math.NA↗

arXiv2026

Stability, convergence, and geometric properties of second-order-in-time space-time discretizations for linear and semilinear wave equations

We revisit second-order-in-time space-time discretizations of the linear and semilinear wave equations by establishing precise equivalences with first-order-in-time formulations. Focusing on schemes using continuous piecewise-polynomial trial functions in time, we analyze their stability, convergence, and geometric properties. We consider first a weak space-time formulation with test functions projected onto discontinuous polynomials of one degree lower in time, showing that it is equivalent to the scheme proposed in [French, Peterson 1996] in the linear case, and extended in [Karakashian, Makridakis 2005] to the semilinear case. In particular, this equivalence shows that this method conserves energy at mesh nodes but is not symplectic. We then introduce two symplectic variants, obtained through Gauss-Legendre and Gauss-Lobatto quadratures in time, and show that they correspond to specific Runge-Kutta time integrators. These connections clarify the geometric structure of the space-time methods considered.

math.NA↗

arXiv2025

A matrix-based approach to the stability of a space-time isogeometric method for the linear Schr\"odinger equation

We propose a space-time isogeometric finite element method for the linear Schr\"odinger equation, and establish its unconditional stability through a matrix-based analysis. Although maximal-regularity splines in time provide higher accuracy per degree of freedom compared to piecewise continuous polynomials, the nonlocal support of the spline bases precludes the use of standard variational arguments in the stability proofs. To overcome this, we show that the resulting scheme is governed by a family of nearly Toeplitz system matrices and, by studying the condition number of these matrices, we prove that the family is weakly well-conditioned, which guarantees the unconditional stability of the method. Furthermore, the discrete scheme preserves mass and energy at the final time. Numerical experiments confirm our theoretical findings and illustrate the optimal convergence behavior of the scheme. Finally, we exploit an algebraic connection between our formulation and a recent first-order-in-time space-time isogeometric method for the wave equation to derive a complete matrix-based stability analysis for the latter.

math.NA↗

arXiv2025

Inf-sup stable space-time discretization of the wave equation based on a first-order-in-time variational formulation

In this paper, we present a conforming space-time discretization of the wave equation based on a first-order-in-time variational formulation with exponential weights in time. We analyze the method, showing its stability without imposing any restrictions on the mesh size or time step, and proving quasi-optimal convergence for any choice of space-time tensor product discrete spaces that satisfies standard approximation assumptions. Numerical examples are provided to support the theoretical findings.

math.NA↗

arXiv2025

Measuring Financial Resilience Using Backward Stochastic Differential Equations

We introduce the resilience rate as a measure of financial resilience. It captures the expected rate at which a dynamic risk measure recovers, i.e., bounces back, when the risk-acceptance set is breached. We develop the corresponding stochastic calculus by establishing representation theorems for expected time-derivatives of solutions to backward stochastic differential equations (BSDEs) with jumps, evaluated at stopping times. These results reveal that the resilience rate can be represented as a suitable expectation of the generator of a BSDE. We analyze the main properties of the resilience rate and the formal connection of these properties to the BSDE generator. We also introduce resilience-acceptance sets and study their properties in relation to both the resilience rate and the dynamic risk measure. We illustrate our results in several canonical financial examples and highlight their implications via the notion of resilience neutrality.

q-fin.MF↗

arXiv2025

Intrinsic unconditional stability in space-time isogeometric approximation of the acoustic wave equation in second-order formulation

We present a novel space-time isogeometric discretization of the acoustic wave equation in second-order formulation that is intrinsically unconditionally stable. The method relies on a variational framework inspired by [Walkington 2014], with an exponential weight introduced in the time integrals. Conformity requires at least $C^1$ regularity in time and $C^0$ in space. The approximation in time is carried out using spline functions. The unconditional stability of the space-time method for conforming discrete spaces arises naturally from the variational structure itself, rather than from any artificial stabilization mechanisms. The analysis of an associated ordinary differential equation problem in time yields error estimates with respect to the mesh size that are suboptimal by one order in standard Sobolev norms. However, for certain choices of approximation spaces, it achieves quasi-optimal estimates. In particular, we prove this for $C^1$-regular splines of even polynomial degree, and provide numerical evidence suggesting that the same behavior holds for splines with maximal regularity, irrespective of the degree. The error analysis is extended to the full space-time problem with tensor-product approximation spaces. Numerical results are provided to support the theoretical findings and demonstrate the sharpness of the estimates.

math.NA↗

arXiv2024

Unconditionally stable space-time isogeometric discretization for the wave equation in Hamiltonian formulation

We consider a family of conforming space-time discretizations for the wave equation based on a first-order-in-time formulation employing maximal regularity splines. In contrast with second-order-in-time formulations, which require a CFL condition to guarantee stability, the methods we consider here are unconditionally stable without the need for stabilization terms. Along the lines of the work by M. Ferrari and S. Fraschini (2024), we address the stability analysis by studying the properties of the condition number of a family of matrices associated with the time discretization. Numerical tests validate the performance of the method.

math.NA↗

arXiv2024

Inviscid Limit of the Stochastic Hyperviscous Navier-Stokes Equations and Invariant Measures for the Euler Equations in $\mathbb R^2$

We prove the existence and some moment estimates for an invariant measure $\mu$ for the two-dimensional ($2$D) deterministic Euler equations on the unbounded domain $\mathbb R^2$ and with highly regular initial data. The result is achieved by first showing the existence of Markov stationary processes which solve the hyperviscous $2$D Navier-Stokes equations with kinematic viscosity $\nu>0$ and an additive stochastic noise scaling as $\sqrt \nu$. We then study the inviscid limit and prove that, as $\nu$ tends to $0$, these processes converge, in an appropriate trajectory space, to a pathwise stationary solution to the Euler equations. Its law is the sought invariant measure $\mu$.

math.PR↗

arXiv2024

Some properties of a modified Hilbert transform

Recently, Steinbach et al. introduced a novel operator $\mathcal{H}_T: L^2(0,T) \to L^2(0,T)$, known as the modified Hilbert transform. This operator has shown its significance in space-time formulations related to the heat and wave equations. In this paper, we establish a direct connection between the modified Hilbert transform $\mathcal{H}_T$ and the canonical Hilbert transform $\mathcal{H}$. Specifically, we prove the relationship $\mathcal{H}_T \varphi = -\mathcal{H} \tilde{\varphi}$, where $\varphi \in L^2(0,T)$ and $\tilde{\varphi}$ is a suitable extension of $\varphi$ over the entire $\mathbb{R}$. By leveraging this crucial result, we derive some properties of $\mathcal{H}_T$, including a new inversion formula, that emerge as immediate consequences of well-established findings on $\mathcal{H}$.

math.CA↗

arXiv2024

Stability of conforming space-time isogeometric methods for the wave equation

We consider a family of conforming space-time finite element discretizations for the wave equation based on splines of maximal regularity in time. Traditional techniques may require a CFL condition to guarantee stability. Recent works by O. Steinbach and M. Zank (2018), and S. Fraschini, G. Loli, A. Moiola, and G. Sangalli (2023), have introduced unconditionally stable schemes by adding non-consistent penalty terms to the underlying bilinear form. Stability and error analysis have been carried out for lowest order discrete spaces. While higher order methods have shown promising properties through numerical testing, their rigorous analysis was still missing. In this paper, we address this stability analysis by studying the properties of the condition number of a family of matrices associated with the time discretization. For each spline order, we derive explicit estimates of both the CFL condition required in the unstabilized case and the penalty term that minimises the consistency error in the stabilized case. Numerical tests confirm the sharpness of our results.

math.NA↗

arXiv2024

New a priori estimate for stochastic 2D Navier-Stokes equation with applications to invariant measure

The paper deals with the stochastic two-dimensional Navier-Stokes equation for incompressible fluids, set in a bounded domain with Dirichlet boundary conditions. We consider additive noise in the form $G\, dW$, where $W$ is a cylindrical Wiener process and $G$ a bounded linear operator with range dense in the domain of $A^\gamma$, $A$ being the Stokes operator. While it is known that existence of invariant measure holds for $\gamma>1/4$, previous results show its uniqueness only for $\gamma > 3/8$. We fill this gap and prove uniqueness and strong mixing property in the range $\gamma \in (1/4, 3/8]$ by adapting the so-called Sobolevski\u{\i}-Kato-Fujita approach to the stochastic N-S equations. This method provides new \textit{a priori} estimates, which entail both better regularity in space for the solution and strong Feller and irreducibility properties for the associated Markov semigroup.

math.PR↗

arXiv2023

On the discriminator of Lucas sequences. II

The family of Shallit sequences consists of the Lucas sequences satisfying the recurrence $U_{n+2}(k)=(4k+2)U_{n+1}(k) -U_n(k),$ with initial values $U_0(k)=0$ and $U_1(k)=1$ and with $k\ge 1$ arbitrary. For every fixed $k$ the integers $\{U_n(k)\}_{n\ge 0}$ are distinct, and hence for every $n\ge 1$ there exists a smallest integer $D_k(n)$, called discriminator, such that $U_0(k),U_1(k),\ldots,U_{n-1}(k)$ are pairwise incongruent modulo $D_k(n).$ In part I it was proved that there exists a constant $n_k$ such that $D_{k}(n)$ has a simple characterization for every $n\ge n_k$. Here, we study the values not following this characterization and provide an upper bound for $n_k$ using Matveev's theorem and the Koksma-Erdos-Tur\'an inequality. We completely determine the discriminator $D_{k}(n)$ for every $n\ge 1$ and a set of integers $k$ of natural density $68/75$. We also correct an omission in the statement of Theorem 3 in part I.

math.NT↗

arXiv2023

Generalized convolution quadrature based on the trapezoidal rule

We present a novel generalized convolution quadrature method that accurately approximates convolution integrals. During the late 1980s, Lubich introduced convolution quadrature techniques, which have now emerged as a prevalent methodology in this field. However, these techniques were limited to constant time stepping, and only in the last decade generalized convolution quadrature based on the implicit Euler and Runge-Kutta methods have been developed, allowing for variable time stepping. In this paper, we introduce and analyze a new generalized convolution quadrature method based on the trapezoidal rule. Crucial for the analysis is the connection to a new modified divided difference formula that we establish. Numerical experiments demonstrate the effectiveness of our method in achieving highly accurate and reliable results.

math.NA↗

arXiv2023

A virtual element method for the solution of 2D time-harmonic elastic wave equations via scalar potentials

In this paper, we propose and analyse a numerical method to solve 2D Dirichlet time-harmonic elastic wave equations. The procedure is based on the decoupling of the elastic vector field into scalar Pressure ($P$-) and Shear ($S$-) waves via a suitable Helmholtz-Hodge decomposition. For the approximation of the two scalar potentials we apply a virtual element method associated with different mesh sizes and degrees of accuracy. We provide for the stability of the method and a convergence error estimate in the $L^2$-norm for the displacement field, in which the contributions to the error associated with the $P$- and $S$- waves are separated. In contrast to standard approaches that are directly applied to the vector formulation, this procedure allows for keeping track of the two different wave numbers, that depend on the $P$- and $S$- speeds of propagation and, therefore, for using a high-order method for the approximation of the wave associated with the higher wave number. Some numerical tests, validating the theoretical results and showing the good performance of the proposed approach, are presented.

math.NA↗

arXiv2022

Runge-Kutta convolution quadrature based on Gauss methods

An error analysis of Runge-Kutta convolution quadrature based on Gauss methods applied to hyperbolic operators is given. The order of convergence relies heavily on the parity of the number of stages, a more favourable situation arising for the odd cases than the even ones. Moreover, for particular situations the order of convergence is higher than for Radau IIA or Lobatto IIIC methods when using the same number of stages. We further investigate an application to transient acoustic scattering where, for certain scattering obstacles, the favourable situation occurs in the important case of the exterior Dirichlet-to-Neumann map. Numerical experiments and comparisons show the performance of the method.

math.NA↗