tdanalysis: Accelerated Time-Domain GW Inference

• tdanalysis is a time-domain Bayesian inference pipeline for gravitational-wave detection that evaluates the exact Gaussian likelihood without relying on frequency-domain approximations.
• It exploits structured Toeplitz covariance, Cholesky whitening, and the GSCE method to reduce computational complexity from O(N²) to roughly O(N log N).
• Benchmark tests demonstrate its flexibility in handling segmented data, sharp signal truncations, and data gaps, making it effective for multi-segment consistency tests.

Searching arXiv for the primary paper and closely related gravitational-wave time-domain inference context. tdanalysis is an accelerated implementation of a fully time-domain formulation of gravitational-wave Bayesian inference. It evaluates the Gaussian-noise likelihood directly in the time domain, rather than through the customary frequency-domain inner products, while exploiting Toeplitz noise structure, FFT-enabled linear algebra, and CPU/GPU acceleration to make stochastic sampling practical. Within the formulation described in "Accelerated Time-domain Analysis for Gravitational Wave Astronomy" (Prasad, 6 Mar 2026), tdanalysis is positioned as an end-to-end pipeline for exact time-domain likelihood evaluation under wide-sense stationary Gaussian noise, with particular advantages for sharply truncated signals, data gaps, and multiple disjoint analysis segments.

1. Definition and conceptual scope

tdanalysis is presented as a time-domain Bayesian inference pipeline for gravitational-wave data analysis, intended to make direct evaluation of the exact Gaussian likelihood computationally feasible for real parameter-estimation workflows. Its motivation is the long-standing dominance of frequency-domain methods, whose efficiency derives from FFT-based matched filtering and from the approximate diagonalization of stationary noise covariance in the Fourier basis, but which rely on the circulant and Whittle approximations and typically require tapering or windowing of analyzed segments (Prasad, 6 Mar 2026).

The framework is not a rejection of the Gaussian-noise paradigm. Rather, it retains the standard detector model

$d(t)=h(t;\theta)+n(t),$

but re-expresses inference in a domain where sharply bounded segments, data excision, and disjoint intervals are natural. The central claim is that a fully time-domain formulation is often more exact, more flexible for segmented data, and better suited to sharply truncated signals, data gaps, and nonstationary extensions. In this sense, tdanalysis occupies a methodological niche distinct from conventional frequency-domain implementations: it seeks exactness of the Gaussian likelihood in the native sample domain without giving up the computational benefits of structured linear algebra.

A recurrent misconception in this area is that time-domain inference is intrinsically prohibitive. The pipeline is designed precisely to challenge that view. Its contribution is not merely a formal restatement of the likelihood, but an implementation strategy that uses Toeplitz covariance structure, whitening operators, multiprocessing, and GPU support to make time-domain parameter estimation competitive enough for practical use in gravitational-wave astronomy.

2. Likelihood formulation and whitening

The statistical core of tdanalysis is the exact time-domain Gaussian likelihood. Under wide-sense stationarity, the discrete noise covariance depends only on lag,

$C_{ij}\equiv E[n_i n_j]=\rho(|i-j|),$

so the covariance matrix $C$ is symmetric Toeplitz rather than arbitrary dense. The likelihood is then written in quadratic form as

$$\ln \mathcal{L}(d\,|\,\theta,H)=\text{const}-\frac{1}{2}\,\langle d-h(\theta),d-h(\theta)\rangle,$$

with time-domain inner product

$\langle s,h\rangle=s^\top C^{-1}h.$

The paper explicitly contrasts this with the usual frequency-domain formulation and identifies it as the exact Gaussian likelihood, with no Whittle approximation required (Prasad, 6 Mar 2026).

For implementation, the covariance is factorized through a Cholesky decomposition,

$C=LL^\top,$

which defines the whitening operator

$W=L^{-1}, \qquad \bar v=Wv.$

The inner product becomes Euclidean in whitened coordinates,

$\langle s,h\rangle=\bar s^\top \bar h.$

For the residual $r=d-h$, the whitened residual is

$\bm{r}=W(d-h),$

and the likelihood simplifies to

$C_{ij}\equiv E[n_i n_j]=\rho(|i-j|),$0

This whitening-based expression is conceptually important because it separates statistical exactness from computational strategy. The probabilistic model remains classical Gaussian inference; the innovation lies in how the required action of $C_{ij}\equiv E[n_i n_j]=\rho(|i-j|),$1 or $C_{ij}\equiv E[n_i n_j]=\rho(|i-j|),$2 is computed efficiently. A plausible implication is that tdanalysis is best understood not as a new likelihood model, but as a new computational realization of a familiar one.

3. Structured linear algebra and acceleration strategy

The main computational obstacle in time-domain inference is the quadratic cost of dense covariance operations. A naive direct evaluation using $C_{ij}\equiv E[n_i n_j]=\rho(|i-j|),$3 scales as $C_{ij}\equiv E[n_i n_j]=\rho(|i-j|),$4. The paper describes a Cholesky-decomposed operator route in two variants: CDL, based on triangular solves, and CDO, based on caching $C_{ij}\equiv E[n_i n_j]=\rho(|i-j|),$5 or $C_{ij}\equiv E[n_i n_j]=\rho(|i-j|),$6 and applying dense matrix-vector multiplication. CDO is reported to be faster and more practical than repeated triangular solves, whereas CDL may require many iterations or be unstable in practice. Even so, dense likelihood evaluation remains memory-bandwidth-limited (Prasad, 6 Mar 2026).

The major algorithmic acceleration is the Gohberg–Semencul Circulant Embedding (GSCE) method. Using the Gohberg–Semencul theorem, the inverse Toeplitz covariance is represented as

$C_{ij}\equiv E[n_i n_j]=\rho(|i-j|),$7

and Toeplitz matrix-vector products are evaluated by circulant embedding,

$C_{ij}\equiv E[n_i n_j]=\rho(|i-j|),$8

This yields an $C_{ij}\equiv E[n_i n_j]=\rho(|i-j|),$9 route to the quadratic form without requiring the original covariance itself to be circulant. That distinction is central: the exact time-domain covariance may remain non-circulant, while the computation still exploits FFTs internally.

The paper also notes an SVD-based reduced-basis compression as possible future work, in which whitened data and signals would be represented in a basis of size $C$0, lowering repeated inner products to roughly $C$1. That component is not part of the reported implementation’s core benchmark path, but it clarifies the architectural direction of the framework.

From a systems perspective, the acceleration strategy is shaped as much by memory traffic as by flop count. On CPUs, AVX/SIMD, FMA, multithreading, and optimized BLAS/LAPACK are available, but dense likelihood evaluation saturates memory bandwidth quickly. The implementation therefore recommends process-based parallelism for sampling, often with OMP_NUM_THREADS=1, rather than aggressive shared-memory threading. On GPUs, the CDO path offloads the heavy whitening linear algebra: whitening operators and whitened data are cached in VRAM, the template waveform is transferred to the accelerator, and the main GEMV is executed there. Supported GPU backends include TensorFlow, PyTorch, CuPy on NVIDIA, and custom HIP kernels on AMD; the custom HIP kernel is reported to outperform the TensorFlow backend on AMD hardware. Because the autocorrelation has wide dynamic range, the paper recommends precomputing whitening operators in FP64 and then performing much of runtime evaluation in FP32 when safe (Prasad, 6 Mar 2026).

4. Segments, gaps, detectors, and software architecture

A defining feature of tdanalysis is its treatment of segmented data. For a finite analysis interval between sample indices $C$2, the code truncates the autocorrelation function and constructs a corresponding segment covariance $C$3. If data within an interval must be excluded, the implementation can set those samples to zero or equivalently remove the corresponding autocorrelation entries. The paper presents this as seamless in the time domain and explicitly identifies reduced memory traffic and computation as practical benefits. The same infrastructure extends naturally to multiple disjoint segments (Prasad, 6 Mar 2026).

This segment-aware design is scientifically consequential because it allows inference on inspiral only, merger only, ringdown only, the full inspiral–merger–ringdown signal, or several separated intervals at once. The paper emphasizes the relevance of this capability to the Multi-Segment Consistency Test (MSCT) and to waveform-systematics studies across different phases of the signal. In contrast to frequency-domain workflows, where tapering and boundary handling are persistent technical constraints, tdanalysis treats sharp truncation as a native operation rather than an approximation artifact.

The software stack is organized around a detector class, GWTransientTD, which stores data, whitening operators, waveform-generation interfaces, and the sampler interface. A multi-detector class aggregates several GWTransientTD instances and sums their log-likelihoods. Waveforms are supplied through PyCBC or lalsimulation. The pipeline includes waveform conditioning with zero-padding and geocentric peak-time alignment, data acquisition through gwdatafind, configuration-driven execution via ini files, a job factory for run generation and SLURM scripts, and diagnostic routines checking residual Gaussianity, overwhitening, and residual correlations.

The treatment of coalescence time is also notable. The paper states that the peak-time definition used in tdanalysis is more observer-based and model-independent than many waveform-model-specific coalescence-time conventions. This suggests a deliberate effort to keep the temporal parameterization aligned with measured detector-frame morphology rather than with internal conventions of particular waveform families.

5. Benchmarks and validation

The reported benchmarks compare time-domain likelihood evaluation against frequency-domain baselines on an AMD EPYC 7763 dual-socket CPU platform and on NVIDIA A100 and AMD MI100 GPUs. The results differentiate sharply between dense and structured time-domain methods. On CPUs, CDO is about 3–15× slower than frequency-domain likelihood evaluation for a single likelihood call, and about 10× slower than frequency-domain evaluation in multiprocessing throughput. On GPUs, CDO gives about 2–10× speedup over CPUs, but still remains about 1–5× slower than frequency-domain performance depending on configuration. By contrast, GSCE nearly matches frequency-domain performance in practice, is reported as only about a factor of ~2 slower than frequency-domain, and is roughly 10× faster than CDO on CPUs (Prasad, 6 Mar 2026).

These benchmark outcomes are central to the interpretation of tdanalysis. The paper does not claim universal superiority over frequency-domain pipelines. Instead, it argues that with the right structured solver, time-domain inference ceases to be prohibitively slow and becomes a realistic alternative when its modeling advantages matter. That balance between exactness and speed is one of the framework’s defining characteristics.

Validation is demonstrated through several inference studies. A GW230814-like zero-noise injection using NRSur7dq4, with $C$4, $C$5, $C$6, and $C$7 fixed, was run with 500 live points and stopping criterion $C$8; the reported runtime was about 78.825 hours on an NVIDIA A100 in 64-bit precision. A second GW250114-like injection was truncated at $C$9 before peak and analyzed in a full 15-dimensional parameter-estimation run using GSCE; the injection was recovered consistently, with injected SNR 43, recovered median SNR $$\ln \mathcal{L}(d\,|\,\theta,H)=\text{const}-\frac{1}{2}\,\langle d-h(\theta),d-h(\theta)\rangle,$$0, and total sampling time about 6.5 hours in 64-bit precision with 1000 live points. The paper also reports a Gaussian-noise injection produced by whitening-filter inversion, with results consistent with the injected model (Prasad, 6 Mar 2026).

A real-data analysis of GW250114 provides residual diagnostics. The paper reports a KS statistic of $$\ln \mathcal{L}(d\,|\,\theta,H)=\text{const}-\frac{1}{2}\,\langle d-h(\theta),d-h(\theta)\rangle,$$1, p-value $$\ln \mathcal{L}(d\,|\,\theta,H)=\text{const}-\frac{1}{2}\,\langle d-h(\theta),d-h(\theta)\rangle,$$2, Anderson–Darling statistic $$\ln \mathcal{L}(d\,|\,\theta,H)=\text{const}-\frac{1}{2}\,\langle d-h(\theta),d-h(\theta)\rangle,$$3, and whitened-residue standard deviation $$\ln \mathcal{L}(d\,|\,\theta,H)=\text{const}-\frac{1}{2}\,\langle d-h(\theta),d-h(\theta)\rangle,$$4, which it interprets as mild overwhitening at the $$\ln \mathcal{L}(d\,|\,\theta,H)=\text{const}-\frac{1}{2}\,\langle d-h(\theta),d-h(\theta)\rangle,$$5 level but overall consistency with standard-normal residual behavior. These diagnostics matter because the framework’s credibility depends not only on likelihood speed but on the adequacy of the whitening and covariance construction in realistic detector data.

6. Scientific role, advantages, and limitations

tdanalysis is presented as enabling several categories of gravitational-wave analysis: full inspiral–merger–ringdown parameter estimation in the time domain; sharply truncated segment inference; inspiral-, merger-, and ringdown-only analyses; multi-segment consistency tests; waveform-systematics studies across phases; ringdown and black-hole spectroscopy applications; and tests related to Hawking’s area theorem and the Kerr nature of black holes (Prasad, 6 Mar 2026). The paper also emphasizes improved access to time-localized signal morphology and precession information, reflecting the natural compatibility of the time-domain representation with localized waveform structure.

Its stated advantages are correspondingly specific. The framework evaluates the exact Gaussian likelihood in time domain; it requires no Whittle approximation; it avoids tapering and window functions; it handles sharp boundaries, gaps, and disjoint segments naturally; it supports sampling over peak time and other extrinsic parameters; and it can approach frequency-domain performance when GSCE is used. These properties make it especially suitable when temporal localization is itself part of the scientific question.

The limitations are equally explicit. Direct dense time-domain likelihood evaluation remains expensive at $$\ln \mathcal{L}(d\,|\,\theta,H)=\text{const}-\frac{1}{2}\,\langle d-h(\theta),d-h(\theta)\rangle,$$6. CDO is memory-bandwidth-limited. GPU performance depends on kernel overhead and transfer patterns. GSCE is specialized in that it requires exploiting Toeplitz and circulant structure. Some GPU workflows are still under testing. For moderate signal lengths, frequency-domain pipelines may still be faster or operationally simpler. The paper therefore does not portray tdanalysis as a universal replacement for frequency-domain inference.

The broader significance of the framework lies in the claim that time-domain inference need not remain a niche methodology. This suggests a shift in how exactness, segmentation flexibility, and computational tractability may be balanced in future gravitational-wave analysis. Within the scope established by the paper, tdanalysis serves as both a software implementation and a methodological statement: exact time-domain likelihoods, once considered too costly for routine use, can be made practical enough to support scientifically nontrivial Bayesian inference in contemporary gravitational-wave astronomy.